The role of information and communication technologies (ICT) in education is nowadays well recognised. The Web Geometry Laboratory, is an e-learning, collaborative and adaptive, Web environment for geometry, integrating a well known dynamic geometry system. In a collaborative session, teachers and students, engaged in solving collaboratively a given set of problems, can exchange geometrical and textual information between them. In a normal work session (stand-alone mode), all the geometric steps done by the students are recorded, allowing, in a latter stage, their teachers to “play back” the students sessions. This information, alongside the navigation and chat information, can be used, later on, to assert the students level of geometric knowledge, adjusting the teaching strategies to each individual student. Teachers can register and begin using the public servers, defining students, preparing materials to be released to the students, open collaborative sessions, etc. Students can work in WGL, defining his/her own working space, sharing geometric constructions between themselves. From the case studies already conducted it was possible to conclude that, using WGL, the students improved their achievement in mathematics, in the classroom and doing homework. In this paper an in-depth, full description of the WGL system in its current version, is made, covering all the features and functioning modes, from the perspective of teachers and students.

"WGL," a Web Laboratory for Geometry.

Unlike some other formal systems, the proof system Metamath has no built-in concept of "decimal number" in the sense that arbitrary digit strings are not recognized by the system without prior definition. We present a system of theorems and definitions and an algorithm to apply these as basic operations to perform arithmetic calculations with a number of steps proportional to an arbitrary-precision arithmetic calculation. We consider as case study the formal proof of Bertrand's postulate, which required the calculation of many small primes. Using a Mathematica implementation, we were able to complete the first formal proof in Metamath using numbers larger than 10. Applications to the mechanization of Metamath proofs are discussed, and a heuristic argument for the feasability of large proofs such as Tom Hales' proof of the Kepler conjecture is presented.

Arithmetic in Metamath, Case Study: Bertrand's Postulate

The Web Geometry Laboratory, (WGL), is a blended-learning, collaborative and adaptive, Web environment for geometry. It integrates a well known dynamic geometry system. In a collaborative session, exchange of geometrical and textual information between the user engaged in the session is possible. In a normal work session (stand-alone mode), all the geometric steps done by the students are recorded, alongside the navigation information, allowing, in a latter stage, their teachers to "play back" the students sessions, using that info to assert the students level and adjust the teaching strategies to each individual student. Teachers can register and begin using one of the public servers, defining students, preparing materials to be released to the students, open collaborative sessions, etc. Using an action research methodology the WGL system is being developed, validated through case-studies, and further improved, in a cycle where the implementation steps are intertwined with case studies.

A Web Environment for Geometry

A necessidade de desenvolver ferramentas educacionais de ensino para apoiar a aprendizagem da matemaatica e bem reconhecida As possibilidades abertas pelas utilizcao de ferramentas das tecnologias da informacao e comunicacao sao muitas, ambientes adaptativos, colaborativos, sincronos e assincronos, contribuem para reforcar a aprendizagem de materias complexas como a Matematica A plataforma Laboratorio de Geometria na Rede (em Ingles, Web Geometry Laboratory, WGL) carateriza-se por ser um ambiente de ensino presencial/nao presencial, colaborativo, adaptativo e integrando um sistema de geometria dinâmica A plataforma visa contribuir para o melhorar do nivel de raciocinio geometrico do aluno Neste artigo descreve-se de forma breve a plataforma e, atraves da descricao de um estudo de caso, a sua utilizacao para a realizacao, de forma colaborativa, de trabalhos para casa

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Laboratório de geometria na rede (WGL) em contexto de trabalho de casa

The area of geometry with its very strong and appealing visual contents and its also strong and appealing connection between the visual content and its formal specification, is an area where computational tools can enhance, in a significant way, the learning environments. 
The dynamic geometry software systems (DGSs) can be used to explore the visual contents of geometry. This already mature tools allows an easy construction of geometric figures build from free objects and elementary constructions. The geometric automated theorem provers (GATPs) allows formal deductive reasoning about geometric constructions, extending the reasoning via concrete instances in a given model to formal deductive reasoning in a geometric theory. 
An adaptative and collaborative blended-learning environment where the DGS and GATP features could be fully explored would be, in our opinion a very rich and challenging learning environment for teachers and students. 
In this text we will describe the Web Geometry Laboratory a Web environment incorporating a DGS and a repository of geometric problems, that can be used in a synchronous and asynchronous fashion and with some adaptative and collaborative features. 
As future work we want to enhance the adaptative and collaborative aspects of the environment and also to incorporate a GATP, constructing a dynamic and individualised learning environment for geometry.

Integrating DGSs and GATPs in an Adaptative and Collaborative Blended-Learning Web-Environment

The Web Geometry Laboratory is a collaborative blended-learning Web-environment integrating a dynamic geometry system and where each user has a personalised scrapbook of geometric constructions. All the base features needed to implement an adaptive module and to allow the integration of a geometry automated theorem prover are also already implemented. The actual testing of the Web Geometry Laboratory by high-school teachers is underway and a first field-test with high-school students is due to the spring term of 2013.

Vanda Santos Cisuc

Papers

Collaborative environment for geometry