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A
Rocha, H. Analyzing the teacher’s knowledge for teaching mathematics with technology. ICTMT. Lyon, France, 2017. Abstract

The teacher’s knowledge has long been viewed as a strong influence on the students’ learning. Several authors have sought to develop procedures to assess this knowledge, but this has proved to be a complex task. In this paper I present an outline of a conceptualization to analyze the teacher's knowledge, based on the model of the Knowledge for Teaching Mathematics with Technology (KTMT) and a set of tasks. These tasks are chosen by the teacher taking into account the potential of the tasks to take advantage of the technology’s potential. The analysis of the teacher’s KTMT is based on the characteristics of the tasks chosen by the teacher; the balance established between the representations provided by the technology that the tasks advocate; the way how the tasks pay attention to the new issue of seeking for a suitable viewing window; and also the way how the tasks take into account the expectable difficulties of the students in the process of looking for the window.

Rocha, H. As diferentes representações de funções e a compreensão de alunos do ensino secundário num contexto de integração da tecnologia. XV Congresso Internacional Galego-Portugués de Psicopedagogia. Corunha, Espanha: Asociación Científica Internacional de Psicopedagogía, 2019. Abstract

The different representations of functions are assumed as central on the development of the concept of function. Being widely recognized the complexity of this concept, the different representations allow the student to understand in a representation what could not be understood in another representation. And the integration of technology into the teaching and learning process provides an easy and quick way to access different representations. This study intends to analyse the understanding of upper secondary students about the information transmitted by each of the representations of functions usually available on technology. Specifically, it intends to understand which transitions between representations are more easily understood by the students and which ones are more difficult to perform. It also intended to identify some aspects that may contribute to this. This study adopts a quantitative methodology in which the answers given by a class to a test focused on the transition from one representation to another are analysed; and a qualitative methodology based on interviews to three of the students in the class, as a way of seeking comprehension about their answers. The results achieved suggest a greater ease of understanding associated to the graphical representation and a greater difficulty associated to the tabular representation. The reasons for this seem to be related to the specific characteristics of each representation, but fundamentally with aspects related to the experiences lived by the students on the mathematics classes, being the integration of technology an influence not to neglect.

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.

Rocha, H. Aspectos da utilização da tecnologia no ensino da Matemática. Actas do XII Congresso Internacional Galego-Português de Psicopedagogia. Braga, Portugal: CIEd - IEUM, 2013. Abstract

O potencial da tecnologia para o ensino e a aprendizagem é há muito reconhecido. Contudo, cada vez mais surgem estudos que indiciam que a sua utilização fica aquém das expectativas. O acesso à tecnologia, o papel que lhe é atribuído na aprendizagem e as características das tarefas em que é utilizada, encontram-se entre as principais influências identificadas sobre a utilização que é feita da tecnologia e potencialmente responsáveis pelas características dessa utilização. O estudo que aqui se apresenta teve como principal objectivo analisar e compreender a utilização que os professores fazem da tecnologia à luz dos aspectos referidos, procurando identificar de que forma estes definem diferentes tipos de utilização. A abordagem metodológica adoptada foi de natureza qualitativa e interpretativa, com a realização de estudos de caso de duas professoras de Matemática que utilizavam a calculadora gráfica. A recolha de dados foi concretizada através de entrevistas semi-estruturadas, observação de aulas e recolha documental, sendo a análise de dados orientada pelo quadro teórico, conciliado com a interpretação destes. As conclusões do estudo sugerem alguma diversidade tanto nas características das tarefas em que as professoras recorrem à tecnologia (exercícios, explorações, problemas, modelação de situações reais), como no papel que lhe é atribuído (obter informação, efectuar cálculos, experimentar), mas apontam também para uma diversidade com características que se traduzem em perfis de utilização da tecnologia distintos: um mais exploratório e outro mais prescritivo.

Roque, C., and H. Rocha Avaliação formativa com recurso à tecnologia [Formative assessment using technology]. SIEM. Almada, Portugal: APM, 2018. Abstract

The present study aims to understand the potentialities and implications,
to the teacher and her practice, of the use of formative assessment with the support
of educational technology.
Regarding the research methodology, this study is part of the research on own
practice. The participants were the teacher, who was simultaneously a researcher,
and the students of a 9th grade class.
In the course of this experience it was found that the use of formative assessment
allows, on the one hand, the student to realize what he manages to understand, and
what he has to do to overcome what are less consolidated parts of the content in
study; and, on the other, the teacher to detect in a timely manner the difficulties of
the student and to change strategies to allow the student to overcome his difficulties.
The lack of time, the difficulties in managing the curriculum and the existence of
national exams are three of the main obstacles mentioned by the teachers for the
non-realization of formative assessment. In this experience it was found that the use
of new technologies turns possible to overcome these limitations.
This type of assessment had a very positive impact on teacher’s practice and in the
learning of the students.
Keywords: assessment; formative assessment; new technologies.

B
Rocha, H., and I. Oitavem. "Barcodes: The Mathematics of everyday life." The Scottish Mathematical Council Journal. 49 (2019).Website
C
Rocha, H. "A calculadora gráfica e a utilização que delas fazemos." Educação e Matemática.112 (2011): 41-42.
Rocha, H. A calculadora gráfica no ensino das funções: implicações sobre aspectos da prática de uma professora. Actas do EIEM – Encontro de Investigação em Educação Matemática. Póvoa do Varzim, Portugal: SPIEM, 2011.
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.

Rocha, H. Consensos e dilemas no ensino de matemática a alunos de um curso de Decoração e Pintura Cerâmica. Atas do VI Seminário Luso-Brasileiro Educação, Trabalho e Movimentos Sociais. Lisboa, Portugal: IE-UL, 2014. Abstractpaper.pdf

A Matemática é uma das áreas que integra o plano curricular dos Cursos de Educação e Formação (CEF), pelo contributo para o exercício da cidadania em sociedades democráticas e tecnologicamente avançadas, mas esta é, também, frequentemente fonte de exclusão. O programa reconhece-o e enfatiza uma aprendizagem mais ligada ao concreto e à realidade. Mas reconhece também que é ao professor que compete gerir a sua implementação, dando forma às situações de aprendizagem e integrando-as de forma coerente e articulada no curso específico que os alunos frequentam. O estudo que aqui se apresenta teve como principal objectivo analisar e compreender as opções efectuadas pelo professor no decorrer das diferentes etapas da sua prática, dando atenção aos dilemas que enfrentou e às razões que valorizou na tomada de decisões. A abordagem metodológica adoptada é de natureza qualitativa e interpretativa, com a realização dum estudo de caso do professor de Matemática Aplicada dum CEF de Decoração e Pintura Cerâmica. A recolha de dados foi concretizada através de entrevistas, observação de aulas e recolha documental, sendo a análise de dados orientada pelo quadro teórico, conciliado com a interpretação destes. Nas conclusões do estudo a redução dos pré-requisitos, a preocupação em partir dos interesses dos alunos e a intenção de alargar a cultura dos alunos surgem como centrais na selecção das tarefas; enquanto o envolvimento activo dos alunos caracteriza a implementação das aulas. Os dilemas centram-se fundamentalmente na valorização relativa e aprofundamento a atribuir a cada conteúdo e na articulação entre formal e intuitivo.

D
Rocha, H. Demonstração matemática versus demonstração no ensino da Matemática – a perspetiva de professores [Mathematical proof versus proof on mathematics teaching – the teachers’ point of view]. SIEM. Almada, Portugal: APM, 2018. Abstract

This study intends to analyze the perspectives of teachers of different levels regarding proof and its functions in Mathematics and Mathematics teaching. Adopting a methodology of a qualitative nature, and based on interviews, the perspectives of teachers of upper secondary, higher education and training teachers of Mathematics were collected. The conclusions reached suggest that teachers seem to share a formal conception of mathematical proof, recognizing the need to introduce some simplification when considering proof in Mathematics teaching as well as the importance of their functions of validation, contribution to learning and even a cultural function.

Rocha, H. Desenvolver o conhecimento de futuros professores sobre as características das tarefas e o papel que a tecnologia pode assumir nestas. SIEM. Castelo Branco, Portugal: APM, 2019. Abstract

The main goal of this work is to characterize how the knowledge of pre-service teachers about the characteristics of the tasks and the role of technology evolves. Based on a case study carried out around a pair of pre-service teachers, the main conclusions point to the contribution of the reflection around a set of six tasks on Functions selected by the pre-service teachers. Central to this reflection was an analyze of the role technology can play in tasks, the comments made by the colleagues to their tasks and some experiences on modeling and open-ended tasks. These elements provided the development of a greater awareness regarding aspects such as the level of structuring of the task and its degree of challenge. And this was determinant for an appropriation of the different characteristics of the tasks and to the development of the pre-service teachers’ knowledge.

Rocha, H. Different representations in mathematics teaching with technology. Proceedings of the 38th Conference of the International Group for the Psychology of Mathematics Education. Vancouver, Canada: PME, 2014. Abstract

The main focus of this paper is the teacher’s representational fluency in a context of graphing calculator use. The conclusions reached point to a more intensive use of some representations over the others, suggesting that technology turns numerical or tabular representation into two different representations.

Moreira, C., S. Lopes, and H. Rocha Dos jogos à aprendizagem. Atas do ProfMat 2015. Évora, Portugal: APM, 2015. Abstractpaper.pdf

Neste texto apresentamos os jogos no ensino da matemática como uma forma de aprendizagem de conteúdos e não apenas como um recurso que cada professor pode usar nas suas aulas para tornar a aula diferente. Analisamos dois jogos desenvolvidos por nós e que utilizámos com alunos dos 7.º e 10.º anos de escolaridade, procurando não só apresentar os jogos, mas também aspetos da sua implementação em sala de aula, ponderando o contributo que trouxeram à aprendizagem dos alunos.
Aprender matemática depende de um grande número de variáveis, o que torna o ensino um processo complexo, pois é necessário que se desenvolva o raciocínio lógico, além de estimular o desenvolvimento das mais variadas capacidades transversais, tais como o pensamento autónomo, a criatividade, o sentido de estratégia e a capacidade de resolver problemas.
Duas das dificuldades frequentemente encontradas pelos professores passam pela falta de motivação para a aprendizagem e pelo desinteresse pela Matemática. A solução para estes problemas pode passar pela utilização de jogos para complementar o estudo, mas também para a aquisição de novos conteúdos. No entanto, apenas a implementação dos jogos não basta. O papel do professor é de extrema importância e a planificação e orientação da aula são fundamentais para que se alcancem os objetivos pretendidos.

E
Morais, C., J. Terroso, and H. Rocha. "E de repente tudo mudou… - Editorial." Educação e Matemática. 155 (2020): 1.Website
Martinho, H., and H. Rocha. "A escrita matemática e a intuição em Geometria [Mathematical writting and intuition in geometry]." Educação e Matemática. 149-150 (2018): 34-38.Website
Martinho, H., and H. Rocha A escrita matemática na resolução de um problema de geometria por alunos de licenciatura em Educação Básica [Mathematical writing in solving a geometry problem by undergraduate students in Basic Education]. EIEM. Lisboa, Portugal: SPIEM, 2017. Abstract

Apesar da escrita ter, habitualmente, uma maior expressão no ensino da Matemática que a própria oralidade, os alunos não estão habituados a explicitar raciocínios e a utilizar linguagem matemática apropriada. A comunicação matemática escrita tem algumas particularidades que podem ser diretamente trabalhadas com os alunos. Por exemplo, a escrita ajuda os alunos a dar sentido à Matemática e a melhorar o próprio discurso. As produções dos alunos transportam informações para o professor contribuindo para a planificação e concretização da sua prática profissional. Assim, e apesar de frequentemente ser descurada, a escrita matemática pode ser trabalhada na sala de aula, em particular, com futuros professores. Este artigo reporta parte de uma experiência realizada com uma turma da Licenciatura em Educação Básica, tendo por base a resolução em grupo de um problema de Geometria e o registo escrito do processo de resolução elaborado pelos alunos. Pretendeu-se desta forma caraterizar a comunicação escrita dos alunos e identificar contributos desta para a compreensão por parte do professor dos conhecimentos dos alunos. A análise da escrita matemática dos alunos, tendo por base um conjunto de critérios previamente definidos, permitiu identificar a preferência destes pelo recurso à representação verbal, dificuldades em fundamentar adequadamente as respostas apresentadas e uma forte tendência para desvalorizar as abordagens prévias que não conduziram à resposta ao problema. Permitiu ainda identificar uma tendência para não explicitar o entendimento das questões que lhes eram colocadas. A forma como os conceitos matemáticos surgem nas repostas escritas permite identificar aspetos relevantes do conhecimento dos alunos.

F
G
Rocha, H. Games and the learning of mathematics outside the classroom. Proceedings of the International Conference on Education and New Learning Technologies. Barcelona, Spain: EduLearn, 2014. Abstract

GAMES AND THE LEARNING OF MATHEMATICS OUTSIDE THE CLASSROOM
H. Rocha

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

Playing games is a recreational activity that is also highly recognized as a potentially rich activity for the teaching and learning. It is an activity that involves the recognition and observance of rules, as well as the development of strategies to achieve victory. It is thus an activity that encourages compliance with rules but also the development of learning and therefore has a socializing character while stimulating critical thinking and analysis of situations. This is why many authors think about playing games as a problem-solving activity with great potential for the learning of mathematics. However, a review of the literature suggests that mathematical learning does not always occur, pointing to the relevance of the specific features of the game and the circumstances in which it is used. Looking to contribute to a better understanding of these issues, the project that was the basis of this study focuses on the use of games by middle school students, intending to promote their mathematical learning in a voluntary and informal context, outside the classroom. The games were available in MatLab, a room of the school supervised by mathematics teachers, which students could visit in their leisure time. In this communication I intend to analyze how the visits to MatLab contributed to the mathematical learning of students, considering the influence of specific characteristics of the games and the atmosphere created in MatLab, given the students’ previous mathematical knowledge.

The study adopts a qualitative and interpretative methodological approach, undertaking two student case studies. Data collection was completed over three months and included observation of twenty visits of these students to MatLab. Data collection was made through the development of a logbook, audio record of the students’ visits and two interviews to the students and to their teacher. Data analysis was based on the evidence gathered in the light of the problem under study.

The conclusions reached stress the importance of certain features of the games to promote student engagement, leading to a desire for self-improvement, very important for the development of sustained learning. Computer games have proven to have a stronger potential to engage students than board games. Nevertheless, the most important characteristics of a game seem to be related to the possibility of playing at different mathematical levels (without getting blocked by lack of knowledge) and to the possibility of keep getting better marks (without the existence of a maximum level from which evolution is not possible). In what concerns to achievement in mathematics’ classes, the students’ teacher reports an improvement in mathematics knowledge (more evident in the average achiever student) as well as an increase in students’ involvement in class work (more evident in the low achiever student).

keywords: game-based learning, mathematics, informal learning.

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).

I
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

THE IMPACT OF THE CULTURAL CONTEXT ON THE PROFESSIONAL PRACTICE OF THE TEACHER

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.