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We analyzed the written answers and explanations by 26 Brazilian high-school students who attempted to solve double and multiple proportionality problems using the cross-product algorithm. We focused on students’ awareness of the scalar and functional relations among the quantities described in verbal problems, an aspect that has been part of their school instruction. Our results show that, in their written work, the students included data tables or pairs of values, usually with their referents and connected by arrows indicating the scalar or the functional relations described in the problems. In most cases, during individual interviews, they explained their solutions in terms of these relations. Our findings suggest that students’ understanding of proportion relationships between quantities in verbal problems constitutes a basis for their understanding and correct use of algorithms.

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.

O artigo investiga as estratégias bem-sucedidas, mobilizadas pelos alunos do ciclo de alfabetização ao resolverem problemas de proporção simples. Solicitou-se a 483 alunos do 1º ao 3º ano que resolvessem seis problemas de proporção, sendo analisadas as respostas de 182 alunos que tiveram estratégias bem-sucedidas. Quando comparados os anos escolares, há aumento nos acertos de ano para ano, na classe de situações um para muitos, o que não se pode afirmar sobre a classe muito para muitos, que permaneceu estagnada. Pelos resultados obtidos, é possível que a estrutura multiplicativa esteja sendo pouco discutida, nos três primeiros anos do Ensino Fundamental., The article investigates the successful strategies mobilized by students in the literacy cycle when solving problems of simple proportion. 483 students from the 1st to the 3rd grade of Brazilian Elementary Education, were asked to solve six problems of proportion, and the responses of 182 students who had successful strategies were analyzed. When comparing school grades, there is an increase in hits from year to year, in the class of situations one-to-many, which cannot be said about the class for many, which remained stagnant. From the results obtained, it is possible that the multiplicative structure is being little discussed in the first three grades of elementary school.

The aim of this study is to contrast levels of student engagement in blended and non-blended learning scenarios. The theoretical framework addresses core elements for the implementation of blended learning scenarios and discusses their impacts on student engagement. A design based on the within-subject experimental model was utilized with a sample of 92 tenth grade students from four different groups at a state public secondary school in the Northeast of Brazil. Procedures comprised the application of a survey on familiarity with ICT tools followed by the implementation of two blocks of lessons separated by a two-month interval, each of them composed of computer-assisted blended and traditional lecture-based learning scenarios. Students' responses to the tasks in each lesson were captured through a screen recording application and levels of engagement determined from criteria based on observable quantifiable behaviors. Nonparametric tests show significantly higher levels of student engagement in blended learning scenarios not affected by significant differences in familiarity with ICT. Categories of engagement regarding the students’ use of resources available in the lessons are presented and discussion emphasizes behavioral engagement as a context-specific phenomenon, highlighting implications for teaching practices.

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.

The aim of this chapter is to contribute to the debate about the trends, objects of study, and theoretical and methodological assumptions that have marked and constituted the current identity of working group Cognitive and Linguistic Processes in Mathematics Education of the Brazilian Society of Mathematics Education. The considerations presented bring together a range of investigations that explore the cognitive and linguistic aspects involved in the teaching and learning of mathematics in different learning contexts and different levels of schooling. The work developed by the group has been characterised by investigations conducted on language and communication in the classroom and their sociocultural aspects, alongside studies into the cognitive processes involved in mathematical reasoning. The discussion about the theoretical-methodological questions underlying the reflections on the cognitive and linguistic processes in the Brazilian scenario has been divided into two parts. In the first, we present a historical review of the main trends considered during the first 10 years of working group’s history. In the second, explore the more recent objects of study. We describe how this development indicates a convergence of Brazilian researchers with different theoretical and methodological affiliations, as they search for theoretical models that can explain the role of language, cognition and cultural aspects in the teaching and learning of mathematics. The advances made in terms of knowledge of cognitive and linguistic processes in mathematics education within Brazil are also presented.

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.