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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.

Rocha, H. "Graphical representation of functions using technology: a window to teacher knowledge." Teaching Mathematics and its Applications. 39.2 (2020): 105-126.Website
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).

Rocha, H. The impact of technologies on the teacher's use of different representations. Proceedings of 12th International Conference onTechnology in Mathematics Teaching. Faro: Universidade do Algarve, 2015. Abstract

This study intends to characterize how the teacher uses and integrates the different representations provided by the graphing calculator on the process of teaching and learning functions at the secondary level. Specifically, it intends to understand the balance established between the use of the different representations, and the way these representations are articulated. The conclusions reached point to an active use of the graphic and algebraic representations and to a scarce use of the tabular representation. The conclusions also point to a flexible articulation between the two representations usual used, assuming different forms and frequently an interactive approach, repeatedly switching between representations.

Rocha, H. The impact of technology on the teachers’ use of different representations. CERME. Utrecht, Holanda: ERME, 2019. Abstract

The potential of using different representations is widely recognized, but not much is known about how teachers use them nor about the impact of the technology on such use. The goal of this study is to characterize the teachers’ representational fluency when teaching functions at high school level, discussing, at the same time, the impact in the use of representations resulting from the use of technology. Adopting a qualitative approach, I analyze one teacher’s practice. The results suggest that algebraic and graphical representations are seen as more important, that tabular representation is assumed as irrelevant and that the access to technology impacts the learning, the representations used and how they are used.

Rocha, H. The impact of the cultural context on the professional practice of the teacher. Proceedings of 8th Annual International Conference of Education, Research and Innovation. Seville, Spain: ICERI, 2015. Abstract


H. Rocha

Universidade NOVA de Lisboa, Faculdade de Ciências e Tecnologia (PORTUGAL)

The professional knowledge is a key element of the teacher’s practice. This knowledge is naturally influenced by the teacher’s beliefs and conceptions and by his training, but the context where he develops his practice is perhaps the most decisive influence. At this level, the school where the teacher works and his colleagues are a powerful influence, but the characteristics of his students are even a stronger influence. The cultural diversity of the students and specifically the linguistic diversity are highly relevant elements. A classroom where different languages converge is always a complex context which requires a deeper professional knowledge with inevitable repercussions over the teacher’s practice.

This study focuses on a teacher working with a mathematics’ class of foreign students with heavy linguistic limitations on the language of instruction and it intends to analyze the impact of this context on the teacher’s practice. In particular, it intends to analyze how this context interferes with the characteristics of the tasks proposed by the teacher and with the way how mathematical concepts are presented to the students.

The study adopts a qualitative and interpretative methodological approach, undertaking one teacher case study. Data were collected during one school year by semi-structured interviews, class observation, and documental data gathering. All interviews and classes observed were audio taped and transcribed. Data analysis was conducted in an interpretative way.

The conclusions reached point to an increase on the appreciation of mechanization, to a large reduction in the use of problematic situations and to a presentation of Mathematics as calculation, disconnected from any application, and where reasoning appears as a marginal element or is even missing. The use of several examples becomes a key element of the practice of this teacher. The main finding of this study suggests that language limitations caused a strong impact on the practice of a teacher who considers the understanding and the development of reasoning from the discussion around mathematical ideas as central to the teaching of this subject. It was also possible to identify that the need to find a way to communicate reinforced the formalism of the mathematical language, placing it in the center of the learning process.

Keywords: cultural context, teacher’s practice, mathematics.

Rocha, H. The influence of teacher’s knowledge for teaching mathematics with technology on the implementation of investigation tasks. Proceedings of 8th International Technology, Education and Development Conference. Valencia, Spain: INTED, 2014.
Coelho, T., and H. Rocha A interdisciplinaridade em contexto de integração da tecnologia: o conhecimento profissional de professores de Matemática e de Físico-Química. Atas do Encontro de Investigação em Educação Matemática., In Press.
Rocha, H. Interdisciplinary tasks: pre-service teachers’ choice and approach. ATEE Winter Conference - Science and mathematics education in the 21st century. Brussels: ATEE and CIEd, 2019. Abstract

This study focusses on the criteria used by pre-service teachers of Mathematics to choose interdisciplinary tasks. The pre-service teachers’ knowledge is assumed as the basis of the actions taken and used as the origin of the choices and approaches observed. The study adopts a qualitative and interpretative methodology and the data were collected using class observation and interviews. The analysis is guided by the Application and Pedagogical Content Knowledge, a model inspired on TPACK (from Mishra and Koehler) and MKT (from Ball and colleagues). The conclusions point to an appreciation of the mathematical part of the tasks and to a devaluation of the remaining components. This suggests difficulty in articulating and integrating different domains of knowledge and points to a fragmented view of the potential of using mathematical applications.

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.

Rocha, H. Knowledge for Teaching Mathematics with Technology - a new framework of teacher knowledge. Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education. Kiel, Germany: PME, 2013. Abstract

Knowledge for Teaching Mathematics with Technology (KTMT) is a theoretical model that seeks to articulate previously existing models on professional knowledge and the conclusions that the investigation around the integration of technology has achieved. KTMT is a dynamic knowledge, informed by the practice, that develops from the knowledge on the base domains (Mathematics, Teaching and Learning, Technology and Curriculum), evolving as knowledge in the base domains interacts and as this promotes the development of inter-domain knowledge, which continue to interact, strengthening relations and leading to the development of an integrated knowledge, where knowledge on the base domains and on the two sets of inter-domains appears deeply integrated into a global knowledge.

Rocha, H. Knowledge for teaching mathematics with technology and the search for a suitable viewing window to represent functions. Proceedings of Cerme 9. Prague, Czech Republic: ERME, 2015. Abstract

The usual difficulties of students regarding the choice of an appropriate window when using the graphing calculator in the study of functions and the importance of the teachers’ knowledge to overcoming them, led to this study. The main goal was to characterize the way teachers address the viewing window in the classroom, trying to infer aspects of the Knowledge for Teaching Mathematics with Technology that can justify that practice. The conclusions reached point to the importance of a set of specific knowledge where I highlight the knowledge of the students’ difficulties, the knowledge of mathematical content necessary to understand the impact of the viewing window on the graphic, and the knowledge of teaching strategies that address both the students’ difficulties and the relevant mathematical knowledge.

Rocha, H. "Mathematical knowledge for teaching with technology: episodes of one teacher’s practice - Conhecimento matemático para ensinar com tecnologia: episódios da prática de uma professora." Educação Matemática Debate. 5.11 (2021): 1-22. AbstractWebsite

Research has highlighted the potential of technology to transform the teaching of Mathematics, but also the relevance of teachers and their professional knowledge. In this article, a qualitative methodology is adopted and two episodes of the practice of one teacher are analyzed in the scope of the study of functions in the 10th grade, based on the model of Knowledge for Teaching Mathematics with Technology (KTMT). The goal is to characterize the teacher's knowledge from her practice, simultaneously understanding how this contributes to promoting the development of the teacher's knowledge. The conclusions reached show the importance of including in the KTMT conception aspects highlighted by the research on technology integration. These aspects are determinant to characterize the teacher's knowledge. They also show the relevance of the practice for the development of the teacher's knowledge and the dynamic character of the vision of knowledge offered by KTMT.

Rocha, H. "Mathematical proof: from mathematics to school mathematics." Philosophical Transactions of the Royal Society A. 377.2140 (2019). AbstractWebsite

Proof plays a central role in developing, establishing, and communicating mathematical knowledge. Nevertheless it is not such a central element in school mathematics. This article discusses some issues involving mathematical proof in school, intending to characterize the understanding of mathematical proof in school, its function and the meaning and relevance attributed to the notion of simple proof. The main conclusions suggest that the idea of addressing mathematical proof at all levels of school is a recent idea that is not yet fully implemented in schools. It requires an adaptation of the understanding of proof to the age of the students, reducing the level of formality, and allowing the students to experience the different functions of proof and not only the function of verification. Among the different functions of proof, the function of explanation deserves special attention due to the illumination and empowerment that it can bring to the students and their learning. The way this function of proof relates to the notion of simple proof (and the related aesthetic issues) seems relevant enough to make it, in the future, a focus of attention for the teachers who address mathematical proof in the classroom.

Rocha, H. Mathematics teaching in Education and Training Courses. Proceedings of the International Conference on Education and New Learning Technologies. Barcelona, Spain: EduLearn, 2014. Abstract

H. Rocha

Universidade Nova de Lisboa, Faculdade de Ciências e Tecnologia (PORTUGAL)

Education and Training Courses have been specifically designed to the high number of young people in a situation of school dropout and in transition to working life, particularly those who enter the labor market early with insufficient levels of schooling and professional training. Mathematics is one of the curriculum components of these courses, for its contribution to the exercise of citizenship in a democratic society. Being an important part of the cultural legacy of our society is too often seen by students as a source of exclusion. It is known that young people who enter these courses often had an experience of underachievement in the discipline, what justifies that motivating students is at once the great challenge faced by the teacher. The program suggests taking a more concrete and linked to reality approach, allowing students to learn to recognize the mathematics in the world around them and using technology to promote that learning. However, it is the teacher who is responsible for managing its implementation, shaping the learning situations and integrating them in a coherent and articulated way in the specific course that students attend. In what concerns to assessment, the program also takes into account the usual characteristics of the students. Thus, the assessment includes a strong appreciation of students’ work, its presentation and discussion and further improvement of that work. The directions given to the teacher diverge from the traditional option of the evaluation test, providing guidelines to the form that each evaluation can take depending on the contents in study. However, once again, the teacher's role in curriculum management is not neglected, being valued the adequacy of proposals to the characteristics of the students.
The study presented here had as its main goal to analyze and understand the choices made by the teacher during the different stages of his practice, giving attention to the dilemmas he faced and to the reasons he took into account when making decisions.

The study adopts a qualitative and interpretative methodological approach, undertaking one teacher case study. Data collection included semi-structured interviews, classroom observation and document collection. Data analysis was based on the evidence gathered in the light of the problem under study.

The conclusions of the study point to the important role of technology and suggest that the reduction of prerequisites, the intention of taking into account the students’ interests and the desire of improving students culture is central in what concerns to task selection; while the active involvement of students characterized the implementation of the classes. The dilemmas faced by the teacher focus mainly on the relative importance and on the demanding level that he should give to each content, as well as the articulation that he should promote between formal and intuitive knowledge. In what concerns to assessment, the results achieved highlight the impact that students ideas can have on teacher’s practice, conducting to the inclusion of tests as an assessment element, against the teacher’s intentions.

keywords: education and training courses, mathematics, innovation, technology.

Rocha, H. "Moving from one representation to another: different ways of doing it, different mathematical learning." The Scottish Mathematical Council Journal. 47 (2017): 40-48.Website
Rocha, H., and I. Oitavem. "A mudança que abala o mundo – Editorial." Educação e Matemática.116 (2012): 1.
Rocha, H. Múltiplas abordagens, múltiplas representações: um contributo para incrementar a relevância da representação algébrica [Multiple approaches, multiple representations: a contribute to increase the relevance of algebraic representation]. Atas do Encontro de Investigação em Educação Matemática. Bragança, Portugal: SPIEM, 2015. Abstract

A tecnologia e o impacto que esta pode ter sobre as diferentes representações utilizadas e, em particular, sobre a representação algébrica são o foco deste artigo. Procura-se assim compreender como é que o professor enquadra a representação algébrica no trabalho em sala de aula e como a procura tornar relevante para os alunos num contexto de utilização da tecnologia. As conclusões alcançadas apontam para a opção por uma estreita articulação entre as representações algébrica e gráfica e para uma criteriosa escolha de tarefas, envolvendo múltiplas abordagens, onde a representação algébrica vem disponibilizar informação fundamental e tendencialmente inacessível a partir de outras representações.

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.

Botelho, M. C., and H. Rocha O conhecimento do professor de matemática e a integração das tecnologias na sua prática. Atas do Encontro de Investigação em Educação Matemática., In Press.
Botelho, M. C., and H. Rocha O conhecimento profissional do professor de matemática na integração de diferentes tecnologias. Atas do XXXII Seminário de Investigação em Educação Matemática., In Press.
Coelho, T., and H. Rocha O conhecimento profissional do professor e a interdisciplinaridade em contexto de integração com a tecnologia. XXXII Seminário de Investigação em Educação Matemática., In Press.
Rocha, H., M. C. Costa, and H. Jacinto O desenvolvimento curricular e a formação de professores. Atas do Encontro de Investigação em Educação Matemática., In Press.