Compact periods of Eisenstein series of orthogonal groups of rank one

Let G=O(n+3) be an orthogonal group of rank one and H=O(n+2) an anisotropic subgroup. We unwind the period along H of a spherical Eisenstein series of G against a cuspform of H into an Euler product and evaluate the local factors at odd primes.Comment: 19 pages; submitted. Changes from version 1: old sections 3 and 7 removed for conciseness. All else: minor mistakes fixed; presentation substantially revised (results unchanged); discussion at bad odd primes adde

Figurative elements in mosaics and roman painting at Algarve (Portugal)

The figurative mosaics with marine fauna at Algarve region, in the south of Portugal, are various and well known, particularly from Roman villa of Milreu. But connected to the sea there are also human representations in the mosaics of the region which, such as the marine themes, are characterized by a strong stylistic link to the Roman art of the North Africa. Recently both figurative themes, animal or human, also arise in parietal painting of the Algarve, specifically in a maritime villa specialized in fish-salting with strong North African links well attested by pottery import profile. This paper aims to explore affinities with both types of artistic representations in the region south of the former province of Lusitania, as well as to see North African links

Investigating mathematics and learning to teach mathematics

This paper deals with an idea that plays an increasing role in teaching and in teacher education—investigating as a powerful paradigm of knowledge construction. Investigations may be carried out both in learning mathematics and in learning how to teach mathematics at preservice and inservice levels. I look into investigations in mathematics and in the mathematics curriculum, pointing out some issues that teachers face proposing them in the classroom. Then, I discuss teacher education and professional development, stressing the value of investigations about practice as a means of developing knowledge. I conclude with examples of work done by preservice and inservice teachers and by teams of teachers and researchers focusing on pupils’ investigative work in mathematics classes that illustrate the educational value of this activity and discuss the roles of the teacher.Este artigo baseia-se numa ideia que desempenha um papel crescente no ensino e na formação de professores – investigar constitui um paradigma poderoso de construção do conhecimento. Tanto podem ser realizadas investigações no ensino da Matemática como na formação inicial e contínua do professor de Matemática. Assim, analiso o papel das investigações em Matemática e no currículo de Matemática, apontando algumas questões que os professores enfrentam quando as propõem na sala de aula. De seguida, discuto a formação de professores e o desenvolvimento profissional, dando ênfase ao valor das investigações sobre a prática como meio de desenvolver novo conhecimento. Concluo com exemplos de trabalho realizado por professores em formação inicial e contínua e por equipas de professores e investigadores que se centram no trabalho investigativo dos alunos realizado nas aulas de Matemática, exemplos esses que ilustram o valor educacional desta actividade e permitem discutir os papéis do professor

Investigations and explorations in the mathematics classroom

In Portugal, since the beginning of the 1990s, problem solving became increasingly identified with mathematical explorations and investigations. A number of research studies have been conducted, focusing on students’ learning, teachers’ classroom practices and teacher education. Currently, this line of work involves studies from primary school to university mathematics. This perspective impacted the mathematics curriculum documents that explicitly recommend teachers to propose mathematics investigations in their classrooms. On national meetings, many teachers report experiences involving students’ doing investigations and indicate to use regularly such tasks in their practice. However, this still appears to be a marginal activity in most mathematics classes, especially when there is pressure for preparation for external examinations (at grades 9 and 12). International assessments such as PISA and national assessments (at grades 4 and 6) emphasize tasks with realistic contexts. They reinforce the view that mathematics tasks must be varied beyond simple computational exercises or intricate abstract problems but they do not support the notion of extended explorations. Future developments will show what paths will emerge from these contradictions between promising research and classroom reports, curriculum orientations, professional experience, and assessment frameworks and instruments