Proof in classroom social practice

How does the construction of proof relate to the social practice developed in the mathematics classroom? This report addresses the role of diagrams in order to focus the complementarity of participation and reification in the process of constructing a proof and negotiating its meaning. The discussion is based on the analysis of the mathematical practice developed by a group of four 9th grade students and is inspired by the social theory of learning

Reasoning and proof in classroom (9th grade)

This paper is research oriented and pretends to contribute toward giving empirical evidence about how students develop their reasoning and how they achieved to a proof construction in school context. Its main theme is epistemology. It describes the way in which four students in 9th Grade explored a task related with the discovery of symmetry axes in various geometric figures. The proof constructed by students had essentially an explaining function and it was related with the symmetry axes of regular polygons. The teacher’s role in meaning negotiation of the proof and its need is described through illustrative episodes. The paper presents part of a study which purpose is to analyse the nature of mathematical proof in classroom, its role and the nature of the relationship between the construction of a proof and the social interactions. Assuming a social perspective, attention is focussed on the social construction of knowledge and on the structuring resources that shape mathematical experience. The study’s methodology has an interpretative nature. One outcome of the study discussed here is that students develop first a practical understanding with no awareness of the reasons founding mathematical statements and after a theoretical one leading them to a proof elaboration.

Producing an explanatory proof in a 9th grade classroom

Este artigo apresenta parte de um estudo fundamentado na problemática da demonstração na matemática escolar. Descreve o modo como quatro alunos do 9.º ano exploraram uma tarefa relacionada com a descoberta de eixos de simetria em várias figuras geométricas. A demonstração, que os mesmos construíram, teve essencialmente uma função explicativa. O papel da professora na negociação do significado de demonstração e da sua necessidade é igualmente analisado. Os alunos desenvolvem primeiro uma compreensão prática sem consciência das razões que fundamentam as afirmações matemáticas e só depois uma compreensão teórica que os conduz à construção de uma demonstração.

e-learning in the classroom?

As teachers, we are challenged everyday to solve pedagogical problems and we have to fight for our students’ attention in a media rich world. I will talk about how we use ICT in Initial Teacher Training and give you some insight on what we are doing. The most important benefit of using ICT in education is that it makes us reflect on our practice. There is no doubt that our classrooms need to be updated, but we need to be critical about every peace of hardware, software or service that we bring into them. It is not only because our budgets are short, but also because e‐learning is primarily about learning, not technology. Therefore, we need to have the knowledge and skills required to act in different situations, and choose the best tool for the job. Not all subjects are suitable for e‐learning, nor do all students have the skills to organize themselves their own study times. Also not all teachers want to spend time programming or learning about instructional design and metadata. The promised land of easy use of authoring tools (e.g. eXe and Reload) that will lead to all teachers become Learning Objects authors and share these LO in Repositories, all this failed, like previously HyperCard, Toolbook and others. We need to know a little bit of many different technologies so we can mobilize this knowledge when a situation requires it: integrate e‐learning technologies in the classroom, not a flipped classroom, just simple tools. Lecture capture, mobile phones and smartphones, pocket size camcorders, VoIP, VLE, live video broadcast, screen sharing, free services for collaborative work, save, share and sync your files. Do not feel stressed to use everything, every time. Just because we have a whiteboard does not mean we have to make it the centre of the classroom. Start from where you are, with your preferred subject and the tools you master. Them go slowly and try some new tool in a non‐formal situation and with just one or two students. And you don’t need to be alone: subscribe a mailing list and share your thoughts with other teachers in a dedicated forum, even better if both are part of a community of practice, and share resources. We did that for music teachers and it was a success, in two years arriving at 1.000 members. Just do it.

Contributos de um ambiente de geometria dinâmica (geogebra) e do geoplano na compreensão das propriedades e relações entre quadriláteros: um estudo com alunos do 4.º ano

O estudo insere-se no âmbito da educação matemática, mais especificamente na área da geometria. Com esta investigação pretende-se compreender qual o contributo do Ambiente de Geometria Dinâmica (AGD), GeoGebra, e do material manipulável (geoplano) na identificação das propriedades e relações entre quadriláteros: trapézio, paralelogramo, retângulo, losango e quadrado. De acordo com o objetivo do estudo formularam-se três questões: Qual o papel das representações na identificação das propriedades dos quadriláteros? Que influência tem a visualização na identificação das propriedades dos quadriláteros? Quais os contributos da utilização do AGD, GeoGebra, e do material manipulável (geoplano) na compreensão das propriedades e relações entre os quadriláteros? Atendendo à problemática em estudo, optou-se por uma metodologia de investigação predominantemente qualitativa, baseada em três estudos de caso. A recolha de dados empíricos foi realizada numa turma de 4.º ano do 1.º ciclo do ensino básico onde se implementou uma experiência de ensino, envolvendo todos os alunos e recorrendo ao geoplano e ao GeoGebra. Essa recolha incidiu na observação participante, em sala de aula, no registo de notas da investigadora, nos documentos produzidos pelos alunos e nos registos áudio e vídeo das discussões nos grupos e no coletivo da turma. Em termos de resultados, salienta-se que a utilização de representações evidenciou a compreensão que os alunos têm dos conceitos, facilitou a identificação das propriedades dos quadriláteros e a compreensão das relações entre eles. Os alunos identificaram as propriedades com base nas representações, no entanto focaram-se em casos particulares, de acordo com a imagem mental que têm da figura, especificamente o protótipo, indiciando a influência da visualização. Dos resultados sobressai também a dificuldade que os alunos sentiram em considerar uma figura como representante de uma classe e em distinguir atributos essenciais e não essenciais. Da análise dos dados ressalta também que tanto o geoplano como o GeoGebra foram uma mais-valia na concretização da experiência de ensino deste estudo. Os dois possuem vantagens e limitações e podem ser usados de maneira que um complemente o outro, podendo dessa forma contribuir, favoravelmente, para a aprendizagem da geometria. - ABSTRACT The study falls within the scope of mathematics education, specifically in the area of geometry. This research aims to understand the contribution of the Dynamic Geometry Environment (GeoGebra) and manipulative material (Geoboard) in the identification of properties and relations of quadrilaterals: trapezium, parallelogram, rectangle, diamond and square. According to the purpose of the study three questions were formulated: What is the role of the representations in the identification of the properties of quadrilaterals? What is the influence of visualization on the identification of the properties of quadrilaterals? What are the contributions on the use of Dynamic Geometry Environment (GeoGebra) and manipulative material (Geoboard) to understand the properties and relationships among quadrilaterals? Given the problem under study, it was chosen a predominantly qualitative research methodology based on three case studies. The empirical data collection was carried out on a fourth primary grade class, where it was implemented a teaching experience, involving all students and the use of Geoboard and GeoGebra. This collection was focused on participant observation in the classroom, in the record of the researcher’s notes, the documents produced by the students and the audio and video recordings of group discussions and collective class. In terms of results, it is noted that the use of representations showed the students understanding of the concepts and that it has facilitated the identification of the properties of quadrilaterals and understanding of the relationships between them. The students identified properties based on representations, however they focused in particular cases, according to the mental image that they have of the figure, specifically the prototype, indicating the influence of visualization. Results also highlights the difficulty that students felt in considering a figure as a representation of a group and distinguishing essential and nonessential attributes. Data analysis also highlights that both Geoboard and GeoGebra have been an asset in the delivery of teaching experience in this study. Both have advantages and limitations and they may be used to complement each other and contribute favorably to the learning of geometry.

Word learning and text organisation: an intervention case

In the present study we focus on the interaction between the acquisition of new words and text organisation. In the acquisition of new words we emphasise the acquisition of paradigmatic relations such as hyponymy, meronymy and semantic sets. We work with a group of girls attending a private school for adolescents in serious difficulties. The subjects are from disadvantaged families. Their writing skills were very poor. When asked to describe a garden, they write a short text of a single paragraph, the lexical items were generic, there were no adjectives, and all of them use mainly existential verbs. The intervention plan assumed that subjects must to be exposed to new words, working out its meaning. In presence of referents subjects were taught new words making explicit the intended relation of the new term to a term already known. In the classroom subjects were asked to write all the words they knew drawing the relationships among them. They talk about the words specifying the relation making explicit pragmatic directions like is a kind of, is a part of or are all x. After that subjects were exposed to the task of choosing perspective. The work presented in this paper accounts for significant differences in the text of the subjects before and after the intervention. While working new words subjects were organising their lexicon and learning to present a whole entity in perspective.

A transição de aluno a professor: dificuldades e desafios

Dissertação apresentada à Escola Superior de Educação de Lisboa para a obtenção do Grau de Mestre em Ciências da Educação - especialidade Supervisão em Educação

Facilitadores e obstáculos à colaboração entre os professores titulares de turma e os professores de educação especial no 1º ciclo

Dissertação apresentada à Escola Superior de Educação de Lisboa para obtenção de grau de mestre em Ciências da Educação - Especialidade Educação Especial

Solution enthalpies of 1,4-dioxane: Study of solvent effects through quantitative structure-property relationships

Solution enthalpies of 1,4-dioxane have been obtained in 15 protic and aprotic solvents at 298.15 K. Breaking the overall process through the use of Solomonov's methodology the cavity term was calculated and interaction enthalpies (Delta H-int) were determined. Main factors involved in the interaction enthalpy have been identified and quantified using a QSPR approach based on the TAKA model equation. The relevant descriptors were found to be pi* and beta, which showed, respectively, exothermic and endothermic contributions. The magnitude of pi* coefficient points toward non-specific solute-solvent interactions playing a major role in the solution process. The positive value of the beta coefficient reflects the endothermic character of the solvents' hydrogen bond acceptor (HBA) basicity contribution, indicating that solvent molecules engaged in hydrogen bonding preferentially interact with each other rather than with 1,4-dioxane. (C) 2013 Elsevier B.V. All rights reserved.

Gestão de comportamentos sociais adequados à sala de aula no 1.º ciclo do ensino básico

Relatório de Estágio apresentado à Escola Superior de Educação de Lisboa para obtenção de grau de mestre em Ensino do 1.º e 2.º Ciclo do Ensino Básico

Indisciplina na sala de aula: a perspetiva de professores do 3º Ciclo do Ensino Básico e do Ensino Secundário

Dissertação apresentada à Escola Superior de Educação de Lisboa para obtenção de grau de mestre em Ciências da Educação

How short and light can a simple pendulum be for classroom use?

We compare the period of oscillation of an ideal simple pendulum with the period of a more 'real' pendulum constituted of a rigid sphere and a rigid slender rod. We determine the relative error in the calculation of the local acceleration of gravity if the period of the ideal pendulum is use?d instead of the period of this real pendulum. © 2014 IOP Publishing Ltd..

Use of quantitative structure-property relationships to study the solvation process of 18-crown-6

Solution enthalpies of 18-crown-6 have been obtained for a set of 14 protic and aprotic solvents at 298.15 K. The complementary use of Solomonov's methodology and a QSPR-based approach allowed the identification of the most significant solvent descriptors that model the interaction enthalpy contribution of the solution process (Delta H-int(A/S)). Results were compared with data previously obtained for 1,4-dioxane. Although the interaction enthalpies of 18-crown-6 correlate well with those of 1,4-dioxane, the magnitude of the most relevant parameters, pi* and beta, is almost three times higher for 18-crown-6. This is rationalized in terms of the impact of the solute's volume in the solution processes of both compounds. (C) 2015 Elsevier B.V. All rights reserved.