Neusa Branco

Exploratory Activity in the Mathematics Classroom

In this chapter, we show that mathematical explorations may be integrated into the core of the daily classroom mathematics activities instead of just being a peripheral activity that is carried out occasionally. Based on two episodes, one on the initial learning of the rational number at grade 5 and the other on the learning of algebraic language at grade 7, we show how teachers may invite students to get involved and interpret such tasks and how they may provide students with significant moments of autonomous work and lead widely participated collective discussions. Thus, we argue that these tasks provide a classroom setting with innovative features in relation to conventional education based on the exposition of concepts and procedures, presentation of examples, and practice of exercises and with much more positive results regarding learning.

Pensamento algébrico na formação inicial de professores

Este artigo descreve uma experiencia de formacao inicial orientada para o desenvolvimento do pensamento algebrico de educadores de infância e de futuros professores dos primeiros anos, realizada ao longo de um semestre numa disciplina do 3o ano da licenciatura em educacao basica. A formacao destes professores debate-se com dilemas muito especificos entre os quais avulta o modo separado ou integrado de tratar o conteudo e a didatica. Usando uma metodologia de design research, recorremos a dados de estudos de caso de tres formandos apresentando o modo como resolviam tarefas de algebra, nomeadamente envolvendo sequencias pictoricas antes, durante e depois da experiencia. Alem disso, tendo em vista fazer um balanco global da experiencia, descrevemos episodios relevantes do ponto de vista formativo ocorridos durante as aulas. Concluimos que o modo de trabalho exploratorio tendo por base situacoes matematicas e situacoes reais da sala de aula, com producoes escritas dos alunos e registros em video, e as oportunidades de reflexao e discussao proporcionadas, contribuiu para promover um efetivo desenvolvimento do conhecimento didatico das tres formandas e do conhecimento matematico da formanda com reduzida formacao matematica anterior.

Optimization Problems at School: Some Examples from the 1st to the 12th Grades

This paper presents a study involving three well-known optimization problems within the Operations Research community. These problems provide an interesting context for creating tasks for the mathematics classroom, from the 1st to the 12th grades, given the connections with real contexts that they establish. Studies, about approaches and experiences with students at the basic and high school levels, are presented and the processes standards are discussed. The three proposed problems (path optimization, packing optimization and linear programming), which can be solved at different grades, are discussed focusing on the resolution strategies and on the dynamics of the mathematics classroom, thus promoting reasoning, communication, representation, connections and problem solving processes. The results of this study show that these problems provide a rich context to involve students in their learning process and to promote mathematics discussions. They share different strategies to solve the problems, emerging and relate different representations (active, iconic and symbolic), depending on the problem and the grade. Also, they explore connections between mathematics and real life.