Dynamic Geometry

This work is a part of a larger study, which presents geometry through a daily life story using dynamic geometry software. It aimed in particular to enable students to feel the importance of geometry in daily life, to share in the process of formulating geometric statements and conjectures, to experience the geometric proof more than validating the correctness of geometric statements and to start with a real-life situation and go through seven steps to geometric proof. The content of the suggested approach was organized so that every activity was a prerequisite for entering the next one, either in the structure of geometric concepts or in the geometric story context. Some indications will be presented according to three Likert-type questionnaires, which were prepared by the researcher with the purpose of assessing students’ attitudes toward geometry and geometric proof, using computers in mathematics learning and the suggested approach. The analysis of single responses to questionnaire items showed significant changes in students’ beliefs about geometry, importance and functions of geometric proof and toward using the suggested approach.

Software design is increasingly being recognized as an important factor in student learning when interacting with interactive geometry software (IGS). A categorization of the operations possible within an IGS is used to identify and analyse design decisions made in a number of current IGS programs: Cabri II Plus, Cabri 3D, Cinderella, GeoGebra and Geometer’s Sketchpad. The analysis, in the context of exploring the area of a circle, is focused on construction, dragging, and alternative spatial and semantic views. A wide diversity of both design issues and individual design decisions was identified, illustrating both the scarcity of research in this area and a number of inevitable tensions, such as between functionality and complexity, and between static and dynamic geometry, related to which future research questions might be posed.

This paper reports the outcomes of an empirical study undertaken to investigate the effect of students’ cognitive styles on achievement in measurement tasks in a dynamic geometry learning environment, and to explore the ability of dynamic geometry learning in accommodating different cognitive styles and enhancing students’ learning. A total of 49 6th grade students were tested using the VICS and the extended CSA-WA tests (Peterson, Verbal imagery cognitive styles and extended cognitive style analysis-wholistic analytic test—Administration guide. New Zealand: Peterson, 2005) for cognitive styles. The same students were also administered a pre-test and a post-test involving 20 measurement tasks. All students were taught a unit in measurement (area of triangles and parallelograms) with the use of dynamic geometry, after a pre-test. As expected, the dynamic geometry software seems to accommodate different cognitive styles and enhances students’ learning. However, contrary to expectations, verbalisers and wholist/verbalisers gained more in their measurement achievement in the environment of dynamic geometry than students who had a tendency towards other cognitive styles. The results are discussed in terms of the nature of the measurement tasks administered to the students.

ABSTRACT. There are several possible approaches in which dynamic computerized environments play a significant, and possibly unique, role in supporting innovative learning trajectories in mathematics in general, and particularly in geometry. These approaches are influenced by the way ...

This paper presents new structural analysis tools based on limit state analysis for vaulted masonry buildings. Thrust lines are used to visualize the forces within the masonry and to predict possible collapse modes. The models are interactive and parametric to explore relationships between building geometry and possible equilibrium conditions in real time. Collapse analysis due to applied displacements is determined by combining kinematics and statics. The approach is largely two dimensional, though more complex three-dimensional problems are analyzed using the same methods. This paper presents a series of analytical tools that are fast and easy to use in real time, but at the same time rigorous and highly accurate. This work represents a significant improvement over traditional methods of thrust line analysis performed by hand, which are often tedious and time-consuming.

This is a snapshot about a student's discovery. Usually, dynamic geometry software is used in investigations of two-dimensional problems. In this snapshot we want to describe how Geometry Inventor (1994) was used by a high school student as a trigger to the discovery of a three-dimensional extension of the classical Ceva's theorem.

Our work introduces immersive collaborative learning to geometry education. More specifically, we present a system that uses collaborative augmented reality as a medium for teaching, and uses 3D dynamic geometry to facilitate mathematics and geometry education. Both these aspects are novel to geometry education. We describe improvements in the user interface and visual design of such an application. We also report on practical experiences with using our system for actual teaching of high school students, and present initial quantitative data on the educational value of such an approach.

This paper describes a study of the cognitive complexity of young students, in the pre-formal stage, experiencing the dragging tool. Our goal was to study how various conditions of geometric knowledge and various mental models of dragging interact and influence the learning of central concepts of quadrilaterals. We present three situations that reflect this interaction. Each situation is characterized by a specific interaction between the students' knowledge of quadrilaterals and their understanding of the dragging tool. The analyses of these cases offer a prism for viewing the challenge involved in changing concept images of quadrilaterals while lacking understanding of the geometrical logic that underlies dragging. Understanding dragging as a manipulation that preserves the critical attributes of the shape is necessary for constructing the concept images of the shapes.

The notion of mediation, widely used in the current mathematics education literature, has been elaborated into a pedagogical model describing the contribution of integrating tools to the human activity, and to teaching and learning mathematics in particular. Following the seminal idea of Vygotsky, and elaborating on it, we postulate that an artifact can be exploited by the teacher as a tool of semiotic mediation to develop genuine mathematical signs, that are detached from the use of the artifact, but that nevertheless maintain with it a deep semiotic link. The teaching organization proposed in this paper is modeled by what we have called the didactical cycle. Starting from assuming the centrality of semiotic activities, collective mathematical discussion plays a crucial role: during a mathematical discussion the intentional action of the teacher is focused on guiding the process of semiotic mediation leading to the expected evolution of signs. The focus of the paper is on the role of the teacher in the teachinglearning process centered on the use of artifacts and in particular a dynamic geometry environment. Some examples will be discussed, drawn from a long-term teaching experiment, carried out over the past years as part of a National project. The analysis is accomplished through a Vygotskian perspective, and it mainly focuses on the process of semiotic mediation centered on the use of artifacts and on the role of the teacher in this process.

The aim of this study was to investigate the effects of mathematics instruction supported by dynamics geometry activities on students’ achievement in area of quadrilaterals and students’ achievements according to their van Hiele geometric thinking levels. The study was conducted in a public elementary school in KırĢehir in 2012 – 2013 spring semester and lasted two weeks. The participants in the study were 76 seventh grade students. The study was examined through nonrandomized control group pretest-posttest research design. In order to gather data, Readiness Test for Area and Perimeter Concepts (RTAP), Area of Quadrilaterals Achievement Test (AQAT) and van Hiele Geometric Thinking Level Test (VHLT) were used. A twoway analysis of variance (ANOVA) procedure was employed to answer research questions. The result of the study indicated that there was a significant interaction between the effects of method of teaching and van Hiele geometric thinking level on scores of AQAT. In addition, mathematics instruction supported by dynamic geometry activities had significant effects on seventh grade students’ achievement on area of quadrilaterals topic. The results also revealed that students in experimental group were significantly more successful in AQAT than students in comparison group when the students were in second level of van Hiele geometric thinking.

ÖZ: Bilgisayarın e itim dünyasında hayat bulması, olu turmacı bilgi kuramı ile birlikte kullanılmaya ba lamasıyla gerçekle mi tir. Bilgisayarın matematik ö retiminde etkili bir ekilde kullanılabilmesi için geli tirilen bilgisayar destekli ö retim program ve tasarımlarının güçlü ö renme kuramlarının üzerine in a edilmesi gerekir. Bu çalı mada, bir dinamik geometri yazılımı olan Cabri Geometri kullanılarak Piaget'in adaptasyon kuramına uygun, ö renci merkezli ortamların nasıl kurulabilece inin örneklenmesi amaçlanmı tır. Bu amaç do rultusunda geli tirilen bilgisayar destekli etkinlikler, Trabzon ili içerisinde 2 farklı ilkö retim okulunda 8. sınıf ö rencilerine uygulanmı , ortaya çıkan ö renme ürünleri ve bilgi kurma süreçleri de erlendirilmi tir. Çalı manın son kısmında, geometri ö retiminde, geli tirilen tasarımın etkili bir biçimde kullanılabilmesi için bazı önerilerde bulunulmu tur.

In his work on human cognition, Bruner (The culture of education, Harvard University Press, Cambridge, 1996) distinguishes between narrative and paradigmatic modes of thinking. While the latter is closely associated with mathematics, Bruner’s writings suggest that the former contributes non-trivially to the learning of mathematics. In this paper, we argue that the very nature of dynamic mathematical representations—being intrinsically temporal, occurring over time—offer very different opportunities for narrative thinking than do the static diagrams and pictures traditionally available to learners. Using examples from our research, we analyse these opportunities both in terms of their potential for enhancing understanding and for their relation to the kind of paradigmatic thinking that usually constitutes mathematical knowledge.

The overall aim of our research project is to explore the impact of dynamic geometry environments (DGEs) on children's geometrical thinking. The point of departure for the study presented in this paper is the analytically and empirically grounded assumption that as the geometric discourse develops, the direct visual identification of geometric shapes gives way to discursively mediated identification, that is to a process in which one needs to perform a discursive procedure, prescribed by a formal definition of the shape, in order to ascertain the name of the shape. Previous research, conducted in static geometry environments, has already shown that many children, even in the middle school grades, rely on static, visual prototypes when identifying geometric shapes and that formal definitions, even if known, play no role in this process. Our study aimed at testing the conjecture that DGEs, in which the shapes can be continuously transformed, may flex the routine of identification, allowing for greater diversity in the shapes recognized as deserving a given name (e.g. triangle). This, we believed, would be an important step toward the discursively mediated routine of identification. The study, conducted among 4-5 year-old children working with Sketchpad, furnished some supporting evidence. In this paper, the focus is on one 30-min lesson during which the children observed, described, created and transformed triangles of different sizes, proportions, and orientations. During this one meeting the children's thinking evolved, in that the diversity of three-sided polygons they were prepared to call 'triangle' grew substantially. Not surprisingly, however, this rapidly-induced change was local and object-level rather than meta-level: it changed the children's use of a specific word rather than causing a transition to a discourse-mediated routine of identification.

In this study, we considered dynamic geometry software (DGS) as the tool that mediates students' strategies in solving and posing problems. The purpose of the present study was twofold. First, to understand the way in which students can solve problems in the setting of a dynamic geometry environment, and second, to investigate how DGS provides opportunities for posing new problems. Two mathematical problems were presented to six pre-service teachers with prior experience in dynamic geometry. Each student participated in two interview sessions which were video recorded. The results of the study showed that DGS, as a mediation tool, encouraged students to use in problem solving and posing the processes of modeling, conjecturing, experimenting and generalizing. Furthermore, we found that DGS can play a significant role in engendering problem solving and posing by bringing about surprise and cognitive conflict as students use the dragging and measuring facilities of the software.

In the first part we summarize our development of a system that uses collaborative augmented reality as a medium for teaching, and uses 3D dynamic geometry to facilitate mathematics and geometry education. Geometry education has proven as one powerful means of improving spatial abilities, an important component of human intelligence. A number of training studies have shown the usefulness of virtual reality (VR) in training spatial ability. Our immersive collaborative educational application, specifically developed for geometry education, serves as the basis of a comprehensive evaluation study regarding its efficacy in training spatial abilities.

Dynamic geometry software provides tools for students to construct and experiment with geometrical objects and relationships. On the basis of their experimentation, students make conjectures that can be tested with the tools available. In this paper, we explore the role of software tools in geometry problem solving and how these tools, in interaction with activities that embed the goals of teachers and students, mediate the problem solving process. Through analysis of successful student responses, we show how dynamic software tools can not only scaffold the solution process but also help students move from argumentation to logical deduction. However, by reference to the work of less successful students, we illustrate how software tools that cannot be programmed to fit the goals of the students may prevent them from expressing their (correct) mathematical ideas and thus impede their problem solution.

This paper sits within the research on the affordances of new technologies in the mathematics classroom and focuses on a specific feature that is available in dynamic geometry environments, i.e. measuring tools, within the context of conjecturing and proving in open geometry problems. We develop a classification of different modalities of measuring, based on our previous work on dragging. The modalities are illustrated through the analysis of 15–16 year-old students’ proving processes, which focuses on how these modalities relate to the moves between the spatio-graphical field and the theoretical field and may either support or hinder the proving process. The classification of the modalities of measuring enables researchers to access students’ cognitive processes and teachers to be aware of the different possible uses and interpretation of measuring, giving them tools to support students when difficulties arise.

In this paper we propose a taxonomy of the pedagogical opportunities supported by mathematics analysis software: the wide class of software capable of performing the algorithmic processes necessary to carry out routine mathematical procedures from any branch of mathematics. This taxonomy has been structured at three levels from the perspective of the teacher who must think about the subject, their classroom and student learning tasks. It is presented in the form of a 'pedagogical map', a chart to give a visual indication of potential or actual teaching practices. The map draws attention to possibilities of teaching (e.g. in lesson planning or in teacher education) and gives a graphic display of practice (e.g. for research, or for mapping professional growth). It has diverse uses, including reminding teachers and researchers of available options, identifying where teachers need professional development, and being a catalyst for professional discussion.

Geometry education has proven as one powerful means of improving spatial abilities, an important component of human intelligence. In the first part of this paper we summarize our development of a system that uses collaborative augmented reality as a medium for teaching, and uses 3D dynamic geometry to facilitate mathematics and geometry education. Our immersive collaborative educational application, specifically developed for geometry education, serves as the basis of a comprehensive evaluation study regarding its efficacy in training spatial abilities. The main contribution is the description of evaluation design including the test instruments, learning tasks and practical experiences with using our system for actual training of high school students. Results of a pre-study with spatial ability tests in high schools are presented. They point to interesting genderspecific differences of strategies when solving spatial ability tests, which have not been reported in literature before.

In this paper we report on two complementary research results. In the first we describe a tool that allows different modes of synchronous distance teaching of mathematics. In the second we report the preliminary results of a pilot study conducted using this tool to teach geometry, ...

The idea of 'interpretative flexibility' underpins new approaches to studying technological artefacts and curricular resources in use. This paper opens by reviewing -in this light -the evolving design of dynamic geometry, its pioneering use within classroom projects, and early sketches of its mainstream use in ordinary classrooms. After examining curricular context and its instrumental dimension, the paper then reports a study of teacher constructions of dynamic geometry in classroom practice, conducted in professionally well-regarded mathematics departments in English secondary schools. From departmental focus-group interviews, four teacher-nominated examples of successful practice were selected for study in depth through lesson observation and post-lesson interview. Iterative thematic analysis was employed, first to establish a narrative outline of each case, and then the ideas and issues salient across cases. The study illustrates the interpretative flexibility surrounding the emergent use of dynamic geometry. It found important differences in practical elaboration of the widespread idea of employing dynamic geometry to support guided discovery. The process of evaluating the costs and benefits of student software use was influenced by the extent to which such use was seen as providing experience of a mathematical reference model, and more fundamentally as promoting mathematically disciplined interaction. Approaches to handling apparent mathematical anomalies of software operation depended on whether these were seen as providing opportunities to develop students' mathematical understanding, in line with a more fundamental pedagogical orientation towards supporting learning through analysis of mathematical discrepancies. Such variation was associated with differences in positioning dynamic geometry in relation to curricular norms and in privileging a mathematical register for framing figural properties. Across all cases, however, incorporating dynamic manipulation into mathematical discourse moved implicitly beyond established norms when dragging was used to focus attention on continuous dynamic variation, rather than being treated as an efficient means of generating multiple static figures.

In vivo determination of 3-D and dynamic geometries of alveolar structures with adequate resolution is essential for developing numerical models of the lung. A thorax window is prepared in anesthetized rabbits by removal of muscle tissue between the third and fourth rib without harming the parietal pleura. The transparent parietal pleura allows contact-free imaging by intravital microscopy ͑IVM͒ and 3-D optical coherence tomography ͑3-D OCT͒. We demonstrate that dislocation of the lung surface is small enough to observe identical regions in the expiratory and inspiratory plateau phase, and that OCT in this animal model is suitable for generating 3-D geometry of in vivo lung parenchyma. To our knowledge, we present a novel thorax window preparation technique for 3-D imaging of alveolar dynamics for the first time. The 3-D datasets of the fine structure of the lung beneath the pleura could provide a basis for the development of 3-D numerical models of the lung.

The aim of this study was twofold: First to examine the impact of the intuitive rule "more A-more B" on Cypriot students while dealing with geometric tasks and secondly to investigate whether teaching with the use of dynamic geometry may help children overcome the effects of the rule. The study was consisted of two parts. The first one was conducted with seventy-seven 6 th graders who were presented with seven tasks concerning the sum of angles in triangles and quadrilaterals. The second was conducted with two 5 th grade classes which were taught with two different teaching approaches, the traditional teaching approach and the approach with the use of a Dynamic Geometry software (DGs) called Euclidraw Jr. The results indicated that the intuitive rule has great impact on students' reasoning and that the use of DGs was more effective than traditional teaching.

In this paper I report a lengthy episode from a teaching experiment in which 15 Year 12 Greek students negotiated their definitions of tangent line to a function graph. The experiment was designed for the purpose of introducing students to the notion of derivative and to the general case of tangent to a function graph. Its design was based on previous research results on students' perspectives on tangency, especially in their transition from Geometry to Analysis. In this experiment an instructional example space of functions was used in an electronic environment utilising Dynamic Geometry software with Function Grapher tools. Following the Vygotskian approach according to which students' knowledge develops in specific social and cultural contexts, students' construction of the meaning of tangent line was observed in the classroom throughout the experiment. The analysis of the classroom data collected during the experiment focused on the evolution of students' personal meanings about tangent line of function graph in relation to: the electronic environment; the pre-prepared as well as spontaneous examples; students' engagement in classroom discussion; and, the role of researcher as a teacher. The analysis indicated that the evolution of students' meanings towards a more sophisticated understanding of tangency was not linear. Also it was interrelated with the evolution of the meaning they had about the inscriptions in the electronic environment; the instructional example space; the classroom discussion; and, the role of the teacher.

A computational model incorporating physiological motion and uniform transient wall deformation of a branchless right coronary artery (RCA) was developed to assess the influence of artery compliance on wall shear stress (WSS). Arterial geometry and deformation were derived from modern medical imaging techniques, whereas the blood flow was solved numerically employing a moving-grid approach using a well-validated inhouse finite element code. The simulation results indicate that artery compliance affects the WSS in the RCA heterogeneously, with the distal region mostly experiencing these effects. Under physiological inflow conditions, coronary compliance contributed to phase changes in the WSS time history, without affecting the temporal gradient of the local WSS nor the bounds of the WSS magnitude. Compliance does not cause considerable changes to the topology of WSS vector patterns nor to the localization of WSS minima along the RCA. We conclude that compliance is not an important factor affecting local hemodynamics in the proximal region of the RCA while the influence of compliance in the distal region needs to be evaluated in conjunction with the outflow to the myocardium through the major branches of the RCA.

Research in math education emphasises -since TIMSS and PISA -on extended problem solving skills and mathematical processes. But, to solve complex and real life problems students also need to be proficient in the "basic" skills like solving systems of equations or constructing simple geometrical figures. These more or less technical "routine skills" have to be trained individually and then evaluated by the teacher. Limited resources, however, often prohibit an adequate evaluation of such skills. A possible solution to this problem is a wider application of Intelligent Assessment tools.

Pellionisz and Llinás proposed, years ago, a geometric interpretation towards understanding brain function. This interpretation assumes that the relation between the brain and the external world is determined by the ability of the central nervous system (CNS) to construct an internal model of the external world using an interactive geometrical relationship between sensory and motor expression. This approach opened new vistas not only in brain research but also in understanding the foundations of geometry itself. The approach named tensor network theory is sufficiently rich to allow specific computational modeling and addressed the issue of prediction, based on Taylor series expansion properties of the system, at the neuronal level, as a basic property of brain function. It was actually proposed that the evolutionary realm is the backbone for the development of an internal functional space that, while being purely representational, can interact successfully with the totally different world of the so-called "external reality". Now if the internal space or functional space is endowed with stochastic metric tensor properties, then there will be a dynamic correspondence between events in the external world and their specification in the internal space. We shall call this dynamic geometry since the minimal time resolution of the brain (10-15 ms), associated with 40 Hz oscillations of neurons and their network dynamics, is considered to be responsible for recognizing external events and generating the concept of simultaneity. The stochastic metric tensor in dynamic geometry can be written as five-dimensional space-time where the fifth dimension is a probability space as well as a metric space. This extra dimension is considered an imbedded degree of freedom. It is worth noticing that the above-mentioned 40 Hz oscillation is present both in awake and dream states where the central difference is the inability of phase resetting in the latter. This framework of dynamic geometry makes it possible to distinguish one individual from another. In this paper we shall investigate the role of dynamic geometry in brain function modeling and the neuronal basis of consciousness.

We present an extension of the original progressive mesh algorithm for large dynamic meshes that contain a mix of triangle and quadrilateral elements. The demand for this extension comes from the visualisation of dynamic finite element simulations, such as car crashes or metal sheet punch operations. These methods use meshes, which consist mainly of quadrilaterals, due to their increased numerical stability during the simulation. Furthermore, these meshes have a dynamic geometry with about 25 to 100 animation steps. Accordingly, we extend the original progressive mesh algorithm in two aspects: First, the edge collapse operation is extended for meshes with a mixture of triangle and quadrilateral elements. Second, we present an algorithm on how to extend quadric error metrics for the simplification of large dynamic meshes with many animation steps. The results are dynamic progressive triangle-quadrilateral meshes -a progressive multi-resolution mesh structure that has two interactive degrees of freedom: simulation time and mesh resolution. We show that our method works on meshes with up to one million vertices and 25 animation steps. We measure the approximation error and compare the results to other algorithms.

Research has shown that the tools provided by dynamic geometry systems (DGSs) impact students’ approach to investigating open problems in Euclidean geometry. We particularly focus on cognitive processes that might be induced by certain ways of dragging in Cabri. Building on the work of Arzarello, Olivero and other researchers, we have conceived a model describing some cognitive processes that can occur during the production of conjectures in dynamic geometry and that seem to be related to the use of specific dragging modalities. While describing such cognitive processes, our model introduces key elements and describes how these are developed during the exploratory phase and how they evolve into the basic components of the statement of the conjecture (premise, conclusion, and conditional link between them). In this paper we present our model and use it to analyze students’ explorations of open problems. The description of the model and the data presented are part of a more general qualitative study aimed at investigating cognitive processes during conjecture-generation in a DGS, in relation to specific dragging modalities. During the study the participants were introduced to certain ways of dragging and then interviewed while working on open problem activities.

, an open web-based tool to determine equations of geometric loci specified using Cabri Geometry and The Geometer's Sketchpad, is described. A geometric construction of a locus is uploaded to a Java Servlet server, where two computer algebra systems, CoCoA and Mathematica, following the Groebner basis method, compute the locus equation and its graph. Moreover, an OpenMath description of the geometric construction is given. GLI can be efficiently used in mathematics education, as a supplement of the locus functions of the standard dynamic geometry systems. The system is located at

We present a new, three-dimensional (3D) plotting library with advanced features, and support for standard and enhanced display devices. The library -S2PLOT -is written in C and can be used by C, C++, and FORTRAN programs on GNU/Linux and Apple/OSX systems. S2PLOT draws objects in a 3D (x,y,z) Cartesian space and the user interactively controls how this space is rendered at run time. With a PGPLOTinspired interface, S2PLOT provides astronomers with elegant techniques for displaying and exploring 3D data sets directly from their program code, and the potential to use stereoscopic and dome display devices. The S2PLOT architecture supports dynamic geometry and can be used to plot time-evolving data sets, such as might be produced by simulation codes. In this paper, we introduce S2PLOT to the astronomical community, describe its potential applications, and present some example uses of the library.

We present a new, general, and real-time technique for soft global illumination in low-frequency environmental lighting. It accumulates over relatively few spherical proxies which approximate the light blocking and re-radiating effect of dynamic geometry. Soft shadows are computed by accumulating log visibility vectors for each sphere proxy as seen by each receiver point. Inter-reflections are computed by accumulating vectors representing the proxy's unshadowed radiance when illuminated by the environment. Both vectors capture low-frequency directional dependence using the spherical harmonic basis. We also present a new proxy accumulation strategy that splats each proxy to receiver pixels in image space to collect its shadowing and indirect lighting contribution. Our soft GI rendering pipeline unifies direct and indirect soft effects with a simple accumulation strategy that maps entirely to the GPU and outperforms previous vertex-based methods.

This paper draws on results of a research project InterActive Education: Teaching and Learning in the Information Age. The overall aim of the project is to examine the ways in which new technologies can be used in educational settings to enhance learning. To this end the project centres around the design and evaluation of teaching and learning initiatives for pupils from the age of eight to eighteen. Within this paper we report on our work with teachers and pupils, who have developed learning initiatives for English, mathematics and music.

Un micromundo para el estudio de paralelismo con triángulos y cuadriláteros en la escuela secundaria Educación Matemática, vol. 17, núm. 3, diciembre, 2005, pp. 77-104 Grupo Santillana México Distrito Federal, México Disponible en: http://www.redalyc.org/articulo.oa?id=40517305 Cómo citar el artículo Número completo Más información del artículo Página de la revista en redalyc.org Sistema de Información Científica Red de Revistas Científicas de América Latina, el Caribe, España y Portugal Proyecto académico sin fines de lucro, desarrollado bajo la iniciativa de acceso abierto EDUCACIÓN MATEMÁTICA, vol. 17, núm. 3, diciembre de 2005, pp. 77-104 © Santillana 77

The notion of mediation, widely used in the current mathematics education literature, has been elaborated into a pedagogical model describing the contribution of integrating tools to the human activity, and to teaching and learning mathematics in particular. Following the seminal idea of Vygotsky, and elaborating on it, we postulate that an artifact can be exploited by the teacher as a tool of semiotic mediation to develop genuine mathematical signs, that are detached from the use of the artifact, but that nevertheless maintain with it a deep semiotic link. The teaching organization proposed in this paper is modeled by what we have called the didactical cycle. Starting from assuming the centrality of semiotic activities, collective mathematical discussion plays a crucial role: during a mathematical discussion the intentional action of the teacher is focused on guiding the process of semiotic mediation leading to the expected evolution of signs. The focus of the paper is on the role ...

Graphical user interfaces (GUIs) of dynamic or interactive geometry software (DGS) allow users to interact with the DGS by using a computer mouse. Clicking on a GUI icon performs an action like choosing a construction tool or manipulating an object. For novices, it may be difficult to recognize and recall the icons needed for a task. Learning mathematics and learning the use of a dynamic geometry system at the same time could lead to cognitive overload. Several DGS systems try to solve this problem by offering different GUIs: expert users can choose between a wide range of icons, while for novice users only the most basic icons are presented. By preselecting a specific set of icons, a teacher can adapt a DGS to create a tool, which meets specific pedagogical demands. Two experiments were conducted to investigate the effects of reducing GUIs of a DGS. In experiment 1, which was carried out with full and reduced interfaces of the DGS Cinderella, the eye movements and gaze points of the users were recorded by an eye tracker. The time taken by users to find given icons in different types of interfaces was measured. In experiment 2, students measured the angle sums of polygons using the DGS Cinderella with a full or a reduced interface. No significant effects of GUI reduction were found in both experiments. The results are discussed and ideas for future research are presented.

In this contribution, we present the European project Intergeo whose aims are first to develop a common language for a description of geometric figures that will ensure interoperability of the main existing dynamic geometry systems, and second, to gather and to make available pedagogical resources of a good quality. This text focuses on the quality process for dynamic geometry resources aiming at their perpetual improvement.

The number of high resolution space imageries, in the civilian market is growing fast.This images have great interest in the photogrammetric and remote sensing communities. The problem with this images for many users, at the present time is the lack of sensor calibration information and precise ephemeris data. Consequently it is not possible to apply the precise mathematical models such as orbital parameter model to determine the dynamic geometry of them at the time of imaging. Therefore for, there is a need for an alternative approach to extract 2D and 3D spatial information from this images. This paper presents a mathematical solution to extract planimetric information as accurate as possible from IKONOS. This method uses Genetic Algorithm technique to find the best terms of global polynomial that provides the best fitness model of the imagery to the ground space. Using the output of global polynomial in multiquadric transformation gives better results in term of accuracy. The result of this method is compared with other 2D mathematical models such as global polynomial, 2D direct linear transformation and multi-quadric models. Also the model is applied to the IRS-1C imagery.

Dynamic geometry systems are tools for geometric visualization. They allow the user to define geometric elements, establish relationships between them and explore the dynamic behavior of the remaining geometric elements when one of them is moved. The main problem in dynamic geometry systems is the ambiguity that arises from operations which lead to more than one possible solution. While the user is defining the geometric construction, he is responsible to resolve these ambiguities. However, when the user is dragging a geometric element, the system is responsible to choose the intended solution, that is, the same solution that the user would select if we could ask him again.

We describe a framework for computationally embedded physical modeling kits to support interfacing with 3D geometry for science, engineering, and design applications and introduce an example project called FlexM that supports dynamic geometry construction and feedback with a hub and strut physical model.