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Journal Article
Martins, R., F. Viseu, and H. Rocha. "Functional Thinking: A Study with 10th-Grade Students." Education Sciences. 13.4 (2023): 1-22. AbstractWebsite

This study aims to understand the functional thinking of 10th-grade students while studying functions. Specifically, we intend to answer the following research questions: what are the functional thinking processes used by 10th-grade students when studying functions? What difficulties do students present while learning functions? In view of the nature of this research objective, we adopted a qualitative and interpretative approach. In order to answer these questions, data were collected from the written records produced by the students while solving the proposed tasks, from records of the oral interactions during discussions and from a questionnaire. The results show that functional thinking processes were implicit in the resolution of the tasks proposed to the students. The students expressed an understanding of how the variables were related, presenting evidence of their functional thinking while working on the new concepts represented by the functions addressed in the proposed tasks. Some students expressed difficulties in interpreting the different types of representations associated with the functions, in retaining the necessary information from a graphical representation that would help them to draw conclusions and establish correspondences, in explaining functional relationships, and in interpreting the information provided by algebraic expressions. These difficulties can reduce the recognition of the relationships between variables and their behavior in the different representations, becoming an obstacle to learning for some students.

Viseu, F., A. Silva, H. Rocha, and P. Martins. "The graphical representation in the learning of functions by 10th grade students." Educación Matemática. 34.1 (2022): 186-213. AbstractWebsite

A exploração de diferentes representações promove a compreensão dos tópicos de funções. Partindo deste pressuposto, com este estudo pretende-se analisar o contributo da representação gráfica na aprendizagem da noção de função inversa e da paridade de uma função por alunos do 10.º ano de escolaridade e identificar dificuldades na exploração dessa representação. Na procura de responder a este objetivo, adotou-se uma abordagem qualitativa e interpretativa para compreender as ações dos alunos na resolução das tarefas
propostas. A análise das resoluções mostra que a representação gráfica serviu de suporte para a instituição das definições dos tópicos em estudo. E isto apesar de alguns alunos revelarem dificuldades ao interpretar e ao construir gráficos; ao identificar imagens e imagens inversas em gráficos de funções; ao representar determinadas características gráficas associadas a alguns conceitos, como é o caso da relação entre a paridade de uma função e a simetria na sua representação gráfica (confundindo eixo de simetria e de reflexão). Globalmente, este estudo mostra como a abordagem de conceitos a partir da representação gráfica pode contribuir para a sua compreensão.

Viseu, F., and H. Rocha. "Perceptions of mathematics teachers on the teaching of functions and on the use of technological materials - Perceções de professores de matemática sobre o ensino de funções e sobre o uso de materiais tecnológicos." Educação Matemática Pesquisa. 20.2 (2018): 113-139. AbstractWebsite

This study intends to understand the perceptions of mathematics teachers from lower and upper secondary regarding the teaching of Functions and the use of technological materials. For that, a mixed methodology was adopted, and the perceptions of 129 teachers were collected through a questionnaire and four teachers through an interview. The main conclusions point to similarities in teachers' perceptions, but also to some differences related to the teaching level of lower or upper secondary. In the teaching of Functions, textbooks are widely used, but differently depending on the level being taught. The same happens with the representations and with the use that is made of the technologies. Involvement of students in work is another aspect considered important, but again there are differences. The assessment also has similarities, but differs in the valuation ascribed to group work.

Rocha, H., F. Viseu, and S. Matos. "Problem solving in a real-life context: an approach during the learning of inequalities." European Journal of Science and Mathematics Education. 12.1 (2024). AbstractWebsite

This study was conducted while 9th grade students learn to solve inequalities and seeks to understand their approach to solving problems with a real-life context. Specifically, the aim is to understand: (1) What are the main characteristics of the students’ approaches to the proposed problems? (2) What is the impact of the real context on the students’ resolutions? A qualitative and interpretative methodology is adopted, based on case studies, with data collected through documentary collection and audio recording of discussions between a pair of students while solving problems. The main conclusions suggest a trend to approach problems without establishing immediate connections with what was being done in the classroom, with students’ decisions being essentially guided by criteria of simplicity. The real context of the problems seems to have the potential to develop in students a more integrated mathematics, focused on understanding and not so much on the repetition of mechanical and meaning-independent procedures. The students’ familiarization with the context in question is one of the aspects highlighted by this study.

Rocha, H., I. Oitavem, F. Viseu, and S. Palha. "Reinvenção do ensino a distância: a inovação ao ritmo de cada professor." Educação e Matemática. 155 (2020): 16-20. AbstractWebsite

A suspensão das aulas presenciais na sequência da pandemia que estamos a atravessar trouxe para primeiro plano o ensino a distância. Neste artigo partilhamos algumas ideias e conceptualizações relativas a este tipo de ensino, abordamos aquilo que alguns autores que se têm dedicado à temática apontam como importantes desafios e oportunidades que se lhe encontram associados e, por fim, partilhamos algumas possíveis opções e recursos que pensamos poderem ser úteis para todos os professores que estão a viver a sua primeira experiência de ensino a distância.

Viseu, F., H. Rocha, and J. Monteiro. "Rethinking digital technology versus paper and pencil in 3D Geometry." Journal of Learning for Development. 9.2 (2022): 267-278. AbstractWebsite

Recognising the relevance of learning Geometry, and in particular 3D Geometry, this study aims to discuss the contributions that digital technology and paper and pencil approaches can bring to students’ learning. We seek, therefore, to identify the differences between the two approaches, and specifically: What factors are relevant in one and the other approach? What does one approach facilitate over the other? A quantitative and a qualitative and interpretive methodology was adopted, and based on a didactic intervention, the students' resolutions of the proposed tasks were analysed. The results obtained show that the experience and prior knowledge of the students with each of the solids involved seems to be decisive in the approach with paper and pencil. However, technology emerges as an enhancing resource when prior knowledge is more fragile. The study also shows differences between the representations supported by the two resources, suggesting the mobilisation of different knowledge by the students in relation to each of the resources.

Viseu, F., S. Campos, J. Fernandes, and H. Rocha. "The use of graphing calculator in the exploration of nonlinear continuous models." Revemat. 11.2 (2016): 79-98. AbstractWebsite

The integration of the graphing calculator in mathematical activity encourages students to express many of their processes and ways of thinking. Since some of the activities at the high school level are carried out with the graphing calculator, we intend to investigate the contribution of this resource to promote the learning of nonlinear continuous models in the 11th grade. By adopting a qualitative methodology, we collected and analysed the students‟ writing productions. At first, students used to present the information given by the calculator with no justification. As they acquire skills in the use of this resource, they usually set up the viewing window in order to visualize the graphical representations of functions that model the problem situation they are working on and also relate the different existing menus in the study of those functions characteristics. Such procedures make students to present the data collected in the calculator with a justification of their arguments and a validation of their conjectures.

Conference Proceedings
Campos, S., F. Viseu, H. Rocha, and J. A. Fernandes The graphing calculator in the promotion of mathematical writing. Proceedings of 12th International Conference onTechnology in Mathematics Teaching. Faro, Portugal: Universidade do Algarve, 2015. Abstract

Through writing, students express many of their processes and ways of thinking. Since at high school level some of the activities are carried out with the graphing calculator, we intend to investigate the contribution of this resource to promote the mathematical writing in the learning of continuous nonlinear models at 11th grade. Adopting a qualitative methodology, we collected and analyzed the students’ writing productions. What they write when using the calculator gives evidence about the information valued (when they sketch graphics without any justification); about the strategies used (when they define the viewing window and relate different menus on the graphing calculator); and about the reasoning developed (when they justify the information given by the calculator and the formulation of generalizations and conjectures validation).

Viseu, F., P. Mendes, and H. Rocha The notion of function by basic education preservice teachers. ATEE Winter Conference ‘Science and mathematics education in the 21st century. Brussels: ATEE and CIEd, 2019. Abstract

The current curricular guidelines for mathematics education in Portugal emphasize the relevance of working with different representations of functions to promote understanding. Given this relevance, we seek understanding about the notion of function held by 37 basic education pre-service teachers in their first year of a master’s course. Data were collected through a task focusing on identifying functions in situations based on different representations. The content analysis technique was then adopted in the search for an understanding of the justifications given by the participants. The results achieved suggest it is easier for the pre-service teachers to identify examples that are not functions than examples that are functions. There is also a tendency for greater accuracy in the identification of examples expressed by tables than by algebraic expressions. The justifications presented show a notion of function as a relation between values of two non-empty sets, but without guaranteeing that this relation is single-valued.

Rocha, H., and F. Viseu O ensino de Funções no 3.º ciclo e no ensino secundário: que diferenças? [Teaching Functions at lower and upper secondary: what is different?]. EIEM. Coimbra: SPIEM, 2018. Abstract

Neste estudo analisamos as perceções que professores do 3.º ciclo e do ensino secundário têm da sua prática no âmbito do ensino de Funções, com o objetivo de as caracterizar e de identificar as diferenças existentes entre estes dois grupos de professores. Um aspeto particularmente relevante se tivermos em conta que se tratam de dois grupos de professores com formações iniciais idênticas. Adotamos uma metodologia mista, com uma vertente quantitativa apoiada na aplicação de questionários e uma vertente qualitativa baseada na realização de entrevistas. As principais conclusões alcançadas apontam para semelhanças nas perceções dos professores, mas também para algumas diferenças em função do ciclo de ensino. Na planificação das aulas os manuais são amplamente utilizados, mas de forma diferente consoante o ciclo de ensino do professor. Os professores de ambos os ciclos de ensino estabelecem conexões entre diferentes representações, mas valorizam de diferentes formas as representações disponíveis. O envolvimento dos alunos nas atividades da aula é outro aspeto destacado pelos professores, mas uma vez mais existem diferenças. Na avaliação o recurso ao teste é enfatizado pelos dois grupos de professores, mas já existem diferenças quanto à importância atribuída ao trabalho de grupo.

Rocha, H., and F. Viseu Teachers’ perspectives on the use of technology to teach Functions at lower and upper secondary. Proceedings of the 5th ERME Topic Conference - MEDA 2018. Copenhagen, Denmark: ERME, 2018. Abstractmeda_rocha_2018.pdf

This study aims to understand the perceptions of lower and upper secondary age teachers of mathematics regarding the use of technology to teach functions. For that, a mixed methodology was adopted, and the perceptions of 129 teachers were collected through a questionnaire (quantitative section) and four teachers through an interview (qualitative section). The main conclusions point to similarities in teachers' perceptions, but also to some differences related to the level that they taught. Teachers show conviction about their knowledge on technology and about the potential of technology in what concerns their teaching and the students’ learning. However, they are not so clear about the best way to articulate technology and paper-and-pencil methods, nor about the use of technology in assessment.

Book Chapter
Viseu, F., and H. Rocha. "Interdisciplinary technological approaches from a mathematics education point of view." Science and mathematics education for 21st century citizens: challenges and ways forward. Eds. L. Leite, E. Oldham, A. Afonso, F. Viseu, L. Dourado, and H. Martinho. Nova Science Publishers, 2020. Abstract

Mathematics has a strong presence in the school curriculum, often justified by its usefulness in social life, in the world of work and by its connections with other sciences. This interdisciplinary connection, in particular when it requires constructing and refining mathematical models and discussing their applications to solve problems of other sciences, can assist students to understand why mathematics is so important in school. In the development of interdisciplinary activities, the characteristics of the tasks emerge as an important aspect. The emphasis is on the use of technological materials and the way they can support the development of concepts, provide different representations and support deeper understandings, and offer a multifaceted support to collect data and simulate experiences. Based on these assumptions, the aim of this chapter is to present, analyse and discuss tasks that promote interdisciplinary technological approaches from a mathematical point of view. In this chapter we assume interdisciplinarity as a complex construct, and in order to clarify its meaning we will discuss several types of conceptions, from multidisciplinary, to interdisciplinary, and to transdisciplinary. We will then address related concepts, such as modelling and STEM, highlighting similarities and differences between them, to reach an understanding of interdisciplinarity. In the process of the interdiciplinary approach, digital technologies arise as a central element. Based on a set of tasks on mathematics and on different sciences, we discuss what can change on an interdisciplinary approach to the teaching and learning of mathematical content and on the articulation between subjects.