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Babo, A., and H. Rocha. "Problem solving in linear programming: a study in a vocational course - Resolução de problemas de Programação Linear: um estudo no ensino profissional." Revista de Estudios e Investigación en Psicología y Educación. E.1 (2017): 41-46. Abstract

The development of meaningful learning becomes possible when students are actively involved in solving real problems. Thus, this study intends to investigate how students of the 11th grade of a vocational course solve problems of Linear Programming, using the graphing calculator. The conclusions reached indicate that: the interpretation of the conditions of the problems is the most delicate point; the graphical approach using technology is dominant; and the difficulties raised by the problem as well as the need to discuss the results achieved are the basis for the interactions both among the students and between them and the teacher.

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. Setúbal: APM, 2022.
Botelho, M. C., T. Coelho, and H. Rocha How the use of different technologies mobilises different domains of professional knowledge. Cerme 13. Budapest, Hungary, 2023.
Botelho, M., and H. Rocha Aspectos da comunicação matemática na resolução de problemas. Atas do XXVI SIEM. Évora, Portugal: APM, 2015. Abstract

The influence of mathematical communication over the students’ learning led to this research, whose main goal is to understand the impact on problem solving of the students’ communication difficulties. The study adopts a qualitative and interpretative methodology, undertaking two case studies of 10th grade students. The reached conclusions point to the students’ difficulties at the interpretation of the problem, namely at the interpretation of figures, and at the interpretation of the available data, especially when part of them is irrelevant to the problem. Some difficulties were also identified at the communication level, in relation to the arguments used by the students to support their ideas, where a clear preference to restrict them to mathematic calculations was identified.

Resumo
A importância da comunicação matemática sobre a aprendizagem dos alunos, levou à realização desta investigação que pretendeu compreender o impacto sobre a resolução de problemas das dificuldades de comunicação evidenciadas pelos alunos. Optou-se por uma metodologia de natureza qualitativa e interpretativa e pela realização de estudos de caso envolvendo dois alunos do 10.º ano. As conclusões alcançadas apontam para dificuldades na interpretação do enunciado, nomeadamente relativamente às figuras e a dados em quantidade superior ao necessário. Também ao nível da comunicação da resolução foram identificadas dificuldades em fundamentar ideias, evidenciando uma preferência pelo recurso ao cálculo.

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., 2022.
Botelho, M. C., and H. Rocha Conceptualizações do conhecimento do professor para a integração da tecnologia no ensino da matemática. XVI Congresso Internacional Galego-Portugués de Psicopedagogia. Braga, Portugal, 2021. Abstract

The potential of technology for teaching and learning mathematics has been widely recognized. However, research has shown that the integration of technology into teaching practice has proved to be difficult, with teachers' knowledge and professional development being identified as a determinant factor. Although there tends to be a focus on the teacher's knowledge of technology, the literature suggests that attention needs to be given to how to integrate technology into teaching practice, thereby involving other types of knowledge. This awareness has led several authors, inspired by Shulman's work, to develop models or conceptualizations of the knowledge required for effective integration of technology.
This study is based on a literature review of the research conducted on the models TPACK - Technological Pedagogical and Content Knowledge, KTMT - Knowledge for Teaching Mathematics with Technology, and PTK/MPTK - Mathematical Pedagogical Technology Knowledge and aims to carry out an analysis of the similarities and differences between the models considered. The main conclusions reached suggest the existence of a common basis for the models considered, but also point to several differences between them, with some highlighting the role of technology and its impact on the teaching and learning process, while others aim to integrate into the model elements based on research on technology or theories related to technology integration, such as instrumental genesis.

Botelho, M. C., T. Coelho, and H. Rocha Fluência representacional: a Matemática na resolução de problemas de Física. Atas do EIEM 2023 – Encontro em Investigação em Educação Matemática. Aveiro: SPIEM, 2023.
Botelho, M., and H. Rocha A comunicação matemática na avaliação da resolução de problemas. Atas do ProfMat 2015. Évora, Portugal: APM, 2015. Abstractpaper.pdf

A aprendizagem dos nossos alunos é fortemente influenciada pelas caraterísticas das tarefas que lhes propomos e a resolução de problemas é frequentemente apontada como uma das tarefas com mais potencial para promover aprendizagens ricas. Mas aprender implica ser capaz de desenvolver raciocínios, de comunicar as nossas ideias e de compreender as dos outros num processo argumentativo e reflexivo. A avaliação das aprendizagens num contexto de resolução de problemas envolve assim, necessariamente como parte importante do processo, uma análise da comunicação que se estabelece entre todos os envolvidos.
Nesta comunicação iremos focar-nos precisamente na comunicação que se estabelece durante a resolução de problemas, abordando as dificuldades dos alunos e dando atenção à interpretação que fazem do enunciado, à compreensão que manifestam das figuras apresentadas, à relação que conseguem estabelecer entre a situação em causa e a informação disponibilizada através de um gráfico, à forma como conseguem explicitar o seu raciocínio e à linguagem matemática que utilizam no decurso do processo de argumentação. Para tal vamos basear-nos num conjunto de problemas propostos a alunos do 10.º ano de escolaridade no decorrer do estudo de funções.

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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).

Caneco, R., and H. Rocha O uso de exemplos na demonstração: um estudo com alunos do 11.º ano. SIEM. Castelo Branco, Portugal: APM, 2019. Abstract

This article focuses the choice and use of examples by two students of the 11th grade to prove or refute a set of statements. The use of representations of sequences and functions is also considered. The study adopts a qualitative approach and data were collected by interviews and documental gathering. The conclusions suggest most of the examples used were well-known sequences or functions. However, the students sought different purposes for the use of examples, such as understanding the conjecture, demonstrate the falsity or truthfulness of the statement and conveying a general argument. The students made a satisfactory articulation between the various types of representations but relied mostly in the cartesian graph.

Coelho, T., and H. Rocha As características do jogo e o conhecimento profissional do professor: que pontos de contacto?. XVI Congresso Internacional Galego-Portugués de Psicopedagogia. Braga, Portugal, 2021. Abstract

For most teachers, the use of games is not a common practice. This limited professional experience forces the teacher to reflect on the use of games, on how to integrate them into class work, and on how to enhance the development of learning based on them. Our objective is to analyse a set of game scenarios and study aspects that characterise each of them, relating these characteristics to aspects of the professional knowledge to teach with technology. The study adopts an interpretative and qualitative methodology that seeks to understand the relationship between teachers' choices and their professional knowledge. The main findings suggest that the different characteristics of the game scenarios imply the creation of different working environments in the classroom; enhance the emergence of different student difficulties, requiring different ways to support their work; allowing/promoting different conceptual approaches; and the work with different representations. These differences between the game scenarios point to different emphases on the teaching and learning knowledge and the chemistry knowledge, suggesting different mobilizations of the teacher's professional knowledge.

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. Setúbal: APM, 2022.
Coelho, E., and H. Rocha O raciocínio dedutivo de alunos do 10.º ano de escolaridade [The deductive reasoning of students in the 10th grade]. XXVII SIEM. Porto, Portugal: APM, 2016. Abstract

Deductive reasoning, being central in mathematics, is also usually a source of difficulties for students, more used to the empirical approaches. In this study we focus on mathematical proof and we try to give attention to how this kind of reasoning is envisaged by the students, to the options they assume when asked to develop a deductive reasoning and to the factors affecting the implementation of this kind of reasoning. The study follows a qualitative and interpretative methodological approach, including the completion of two case studies of students of the 10th grade. Data were collected in work sessions and through interviews. The main findings point to a devaluation of mathematical proof and a strong preference for empirical approaches. Yet students show ability to develop different approaches. The preference for the mathematical subject and the attention given in class to the deduction work, appears to be relevant factors when considering the students' ability to develop a deductive reasoning when involved on a mathematical proof.

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Faggiano, E., H. Rocha, A. Sacristan, and M. Santacruz-Rodríguez. "Towards pragmatic theories to underpin the design of teacher professional development concerning technology use in school mathematics." Mathematics Education in the Digital Age: Learning, Practice and Theory . Eds. A. Donevska-Todorova, E. Faggiano, J. Trgalova, H. - G. Weigand, and A. Clark-Wilson. Routledge, 2021. 42-68. Abstract

This chapter aims to make more explicit the grounded or ‘pragmatic theories’ that inform the design of mathematics teachers’ professional development (PD) to exploit technological affordances. It uses aspects of some representative projects that took place in four countries (Colombia, Italy, Mexico, and Portugal) to illustrate lessons learned (e.g., similarities and differences, barriers and opportunities) and provide important insights to inform future PD implementations. To do this, we have identified a set of aspects (and sub-aspects) that emerged in relation to five major themes and reveal our ‘pragmatic theories’ alongside a consideration of the interconnections between these aspects. Our contribution offers a methodological frame to support future PD designs for teachers of mathematics concerning digital technology uses.

Faggiano, E., A. Sacristán, H. Rocha, and M. Santacruz-Rodríguez Addressing the congruence and similarity of figures with technology: A cross-national comparison. ICTMT., 2023.
Ferreira, G., H. Rocha, and A. Rodrigues As conexões matemáticas na resolução de problemas. Atas do EIEM 2023 - Encontro em Investigação em Educação Matemática. Aveiro: SPIEM, 2023.
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Kahle, R., I. Oitavem, and H. Rocha. "Os problemas de Hilbert." Educação e Matemática. 130 (2014): 23-27.Website
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Lopes, S., and H. Rocha O jogo como promotor da comunicação e aprendizagem matemática [Games to promote communication and mathematical learning]. XXVII SIEM. Porto, Portugal: APM, 2016. Abstract

Games are commonly appointed as a methodological tool capable of promoting students’ effective learning. In this context, this study intends to analyze the impact of mathematical discussions developed while
playing a polynomial game. Namely it intends to analyze the impact on the consolidation of mathematical concepts previously worked in the classroom and on the communications skills. Two case studies where developed involving 10th grade students. Data gathering was based on direct observation and an inquiry. The main conclusions suggest that the game encouraged the discussion about the mathematical contents and therefore promoted the development of the mathematical discourse. Besides that, it allowed a deeper apprehension of mathematical concepts, and the overcome of some difficulties.