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Resumo Diversos aspectos influenciam a resolução de problemas multiplicativos por crianças, como as características numéricas das tarefas e os suportes de representação disponibilizados durante os procedimentos de resolução. O presente estudo investiga o papel desempenhado por um outro aspecto que é constitutivo do raciocínio multiplicativo: a correspondência um para muitos. Esta correspondência pode estar implícita ou explicitamente mencionada no enunciado dos problemas. Neste estudo examinou-se a possibilidade de que o nível de explicitação da relação um para muitos seria fator importante na resolução de problemas multiplicativos, e que quanto mais explícitas fossem essas relações, melhor seria o desempenho das crianças. Para isso, problemas de multiplicação e de divisão foram solucionados por crianças do 3° e 5° ano do Ensino Fundamental. Os problemas apresentavam diferentes níveis de explicitação da correspondência uma para muitos. Observou-se que o desempenho era melhor nos problemas em que a correspondência uma para muitos era explicitamente mencionada no enunciado do que em problemas em que esta relação estava implícita. Isso foi observado em ambos os anos escolares. Os resultados evidenciam o papel facilitador que a explicitação da correspondência um para muitos desempenha na resolução de problemas de estrutura multiplicativa. Implicações teóricas e educacionaissão discutidas. Abstract The way in which children solve multiplicative problems is influenced by several aspects, such as the numerical characteristics of the tasks and the representation support that is available during the solution procedure. The present study investigates the role played by another aspect of multiplicative reasoning: the one-to-many correspondence. This correspondence may be implicitly or explicitly mentioned in the wording of the problem. This study examined whether the degree of explicitness of the one-to-many relationship is an important factor in the solution of multiplicative problems, and whether the more explicit this correspondence is, the better the child’s performance. With that in mind, 3rd and 5th grades elementary school students were asked to solve multiplication and division problems. In the wording of the problems, the one-to-many correspondence was presented in different degrees of explicitness. It was found that performance was better in problems in which the one-to-many correspondence was explicitly mentioned than in those in which this relationship was implicit, a factobserved in both school grades. Results show the facilitating role played by the explicitness of the one-to-many correspondence in the solution of problems of multiplicative structure. Theoretical and educational implications are discussed.

Background: In science, posing problems is considered as important as solving them, however, school has explored little this type of activity. Objective: To examine the features of mathematical problems posed by elementary school teachers, analysing aspects related to the statement of the problems and the types of problems formulated. Design: Descriptive, qualitative research. Setting and participants: Eighty-seven teachers (45 teaching 1 st and 2 nd grades, and 42 teaching 3 rd , 4 th , and 5th grades of elementary school) attending a teacher education course promoted by the Municipal Secretary of Education of Curitiba. Data collection and analysis: The teachers were asked to formulate four problems involving addition, subtraction, multiplication, and division. The types of the quantities involved, the necessary information, the number of steps required for solving the problems, and the types of problems from the theory of conceptual fields were analysed. Results: The problems presented a clear language, sufficient information, required a single operation for their solution, involved discrete quantities, and presented few challenges. The problems of addition and subtraction involved situations of composition and transformation, those of multiplication were of simple proportion, and those of division were of partitive problems. Conclusions: The results suggest that the teachers have a limited conception about the formulation of problems, emphasising the need to promote teacher training courses that develop a greater understanding of the properties of the mathematical concept involved in the problems to be formulated and about resolution procedures to be adopted

Resumo O presente estudo investiga como professores do Ensino Fundamental concebem e formulam situações-problema inseridas no campo conceitual das estruturas multiplicativas. Trinta e nove professores do 1° ao 9° ano de escolas públicas foram solicitados a formular problemas matemáticos que pudessem ser resolvidos por meio de multiplicação e/ou de divisão. Os resultados mostram que os professores investigados compreendem o que uma situação multiplicativa significa e formulam problemas apropriadamente, sendo poucos os enunciados em que se omitem informações ou que apresentam imprecisões linguísticas. Verificou-se que a maioria dos problemas era de um mesmo tipo e envolviam apenas um passo para sua resolução. A pouca variabilidade foi observada em relação a todos os professores, independentemente do ano em que lecionavam. Concluiu-se que os professores têm dificuldade em formular problemas que envolvam diferentes relações no âmbito das estruturas multiplicativas, sendo necessário desenvolver no professor do Ensino Fundamental a habilidade de formular problemas.

Resumo O artigo trata da interpretação que professores e futuros professores fazem dos erros de alunos do ensino fundamental na solução de problemas de estrutura multiplicativa. Os participantes foram 12 futuros professores e 12 professores de matemática do ensino fundamental. Em entrevista semiaberta, foram-lhes apresentadas seis cartelas, cada uma contendo o enunciado de um problema (três de produto de medidas, três de isomorfismo de medidas) cuja solução incorreta deveria ser interpretada. Os entrevistados identificaram erros de natureza procedimental, linguística e conceitual. Futuros professores e professores viram os erros nos problemas de produto de medidas, sobretudo como conceituais, e os erros nos problemas de isomorfismo de medidas, sobretudo como linguísticos. Como o mesmo padrão de resultados quantitativos foi encontrado para os dois grupos, conclui-se que no ensino de matemática, o tipo de problema tem papel relevante na forma de interpretar erros, mais que a formação e a experiência dos professores. Abstract This study investigates how in-service and pre-service teachers interpret the errors made by students when solving problems of multiplicative structure. Participants were 12 pre-service and 12 Mathematics Elementary School teachers. In a semi-structured interview, each participant was presented six cards, each one containing a mathematical problem (three product of measures and three isomorphism of measures problems) which incorrect solutions should be interpreted. According to the participants, the errors were procedural, linguistic and conceptual in nature. Pre-service and in-service teachers considered the errors in the product of measures problems mostly as conceptual in nature, while errors in isomorphism of measures problems were mostly linguistic in nature. As the two groups showed the same pattern of quantitative results, was concluded that in Mathematics teaching, more than professional training and experience, the nature of the problem has a relevant role in the way that errors are interpreted.

This study examined the possibility that children can learn how to summarize texts. Forty-four children (10 years old) were divided into an Experimental and a Control Group. All participants were given a pre-test and a post-text in which they had to write a summary of a story. While children in the the CG went ahead with the usual teaching practice that did not include any type of activity related to summaries, children in the EG were provided with a classroom intervention that emphasized the principles of summary composition (brevity, clarity, faithfulness to the theme, and the presence of the main ideas of the textbase). No significant differences were found between groups in the pre-test. However, in the post-test children in the EG produced more elaborated summaries than those in the CG. Also only children in the EG did significantly better in the post-test than in the pre-test. The conclusion was that children can be taught how to write summaries and that the intervention helped them to overcome their difficulties regarding summary writing.

This study involves a teaching experiment aimed to improve children´s text comprehension. Children with difficulties in text comprehension were divided into an Experimental and a Control Group. Children in the EG were given inferential training that required readers to relate textual information to their previous knowledge and to integrate information mentioned in the text. No significant differences were found between the two groups in the pre-test. However, in the post-test children in the EG were significantly more successful than those in the CG. Only the children in the EG showed improvement of their comprehension skills when compared in the pre- and in the post-test. The nature of the intervention and its implications for education are discussed in terms of finding useful ways of teaching comprehension in the school setting.

This study explored the strategies 7- and 8-year-old children use when estimating probability. An experiment was carried out in which children had to order three sets of blue and pink marbles according to their degree of chance of drawing a target colour. Neither age group had received any previous instruction in probability. The task in this experiment was opened to solutions in terms of part–part or part–whole comparisons. We tested the idea that children would solve the tasks by using part–part relations to estimate probability rather than part–whole comparisons. The analysis of children’s strategies and their verbal justifications revealed that children ignored the role played by the possible cases by making correct ordering on the basis of part–part relations in which they made comparisons between favourable and unfavourable cases in each set of marbles. The results support the findings reported by Spinillo & Bryant, 1991 , Spinillo & Bryant, 1999 on proportional reasoning, and the idea claimed by Resnick and Singer (1993) and Singer, Kohn, and Resnick (1997) about the early intuitive origins of relational concepts termed protoquantitative schemes. The informal knowledge represented by the use of part–part comparisons plays an important role in early probabilistic thinking.

3 experiments are reported, all of which show the crucial importance of the “half” boundary in children's proportional judgments. In the first experiment 4-, 5-, 6-, and 7-year-old children had to judge which of 2 boxes of blue and white bricks was represented in a small picture. The proportion of blue to white bricks was different in the 2 boxes, and the question was whether the children could use these proportions to make their choices. The 6- and 7-year-old children solved the problem much more successfully when the proportions crossed the “half” boundary (e.g., 3/8 blue vs. 5/8 blue). The second experiment showed that discriminations involving half (e.g., 1/2 blue vs. 1/4 blue) are also easier than those that do not cross the “half” boundary for the 6- and 7-year-olds. The third experiment confirmed the results of the first 2 with pictures of different absolute sizes from each other. We conclude that “half” plays a crucial role in children's early proportional reasoning, and that the “half” boundary is to some extent similar to, though not as powerful as, the category boundaries discovered in speech perception.