Making Associativity Operational

The purpose of this paper is to propose an operational idea for developing algebraic thinking in the absence of alphanumeric symbols. The paper reports on a design experiment encouraging preschool children to use the associative property algebraically. We describe the theoretical basis of the design, the tasks used, and examples of algebraic thinking in 5–6-year-old children. Theoretically, the paper makes a critical distinction between operational and structural meanings of the notion of equality. We argue that mathematical thinking involving equality among young learners can comprise both an operational and a structural conception and that the operational conception has a side that is productively linked to the structural conception. Using carefully designed hands-on tasks, the crux of the paper is the realization of algebraic thinking (in verbal mathematics) as operationally experienced in the ability to transform one number structure, with a quantity that is subject to change, into another through equality-preserving transformations

Reconhecimento antecipado de problemas ortográficos em escreventes novatos: quando e como acontecem

A aprendizagem da ortográfica constitui um processo complexo, envolvendo questões lexicais e gramaticais. Muitos estudos sobre essa aprendizagem tratam os problemas ortográficos de modo independente e separado da produção textual. Neste estudo defendemos a importância de se analisar a aprendizagem da ortografia a partir da perspectiva proposta pela Genética Textual, colocando em destaque a gênese do processo de escritura e criação textual. Apresentamos o Sistema Ramos, metodologia de investigação que registra o processo de escritura em tempo e espaço real da sala de aula. Esse Sistema oferece informações multimodais (fala, escrita, gestualidade) sobre o que alunos, em duplas, reconhecem como problemas ortográficos (PO) e os comentários espontâneos feitos quando estão escrevendo o texto. Este estudo analisa justamente o momento em que aconteceram esses reconhecimentos e os comentários de duas alunas no 2º ano de escolaridade, durante a produção de seis histórias inventadas. Mais do que uma análise quantitativa dos tipos de PO identificados no produto, apresentamos uma análise enunciativa e microgenética de reconhecimentos de PO e seus comentários, particularmente aqueles PO antecipados pelas escreventes. Os resultados indicam: i. Reconhecimentos ensejam comentários nem sempre relacionados ao PO identificado; ii. Reconhecimentos e comentários estão relacionados aos conteúdos ortográficos ensinados em sala de aula; iii. Alguns PO reconhecidos envolvem a articulação de diferentes níveis linguísticos. Esses aspectos podem contribuir para a compreensão da aprendizagem da ortografia em situações didáticas propiciadas pela escrita colaborativa a dois.The acquisition of spelling competence is a complex process, involving lexical and grammatical questions. Research, however, almost always places the spelling from an autonomous point of view and disconnected from the other components of writing. In this text, we present the relevance of the Ramos System that captures students in an ecological situation of text production in pairs, allowing access to the processes for solving orthographic problems. Collaborative writing also grants access to comments made by students during the process of textual linearization. Our study focuses on the recognition of spelling problems (SP) and the comments made regarding such problems by two 2nd grade students during the production of six invented stories. More than a quantitative analysis of the types of SP identified in the product, we were interested in making a qualitative and fine analysis of oral recognitions of SP, particularly those SP anticipated by the writers. Our results indicate that: i. Recognition motivates comments that are not always related to the identified SP; ii. Recognition and comments are related to the orthographic contents taught in the classroom; iii. Some of the recognized SP involve the articulation between different linguistic levels. These aspects can contribute for the comprehension of orthographic learning in didactic situations provided by collaborative writing.publishe

The contribution of logical reasoning to the learning of mathematics in primary school

It has often been claimed that children's mathematical understanding is based on their ability to reason logically, but there is no good evidence for this causal link We tested the causal hypothesis about logic and mathematical development in two related studies. In a longitudinal study, we showed that (a) 6-year-old children's logical abilities and their working memory predict mathematical achievement 16 months later; and (b) logical scores continued to predict mathematical levels after controls for working memory, whereas working memory scores failed to predict die same measure after controls for differences in logical ability. In our second study, we trained a group of children in logical reasoning and found that they made more progress in mathematics than a control group who were not given this training. These studies establish a causal link between logical reasoning and mathematical learning. Much of children's mathematical knowledge is based on their understanding of its underlying logic. © 2007 The British Psychological Society