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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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Giglione, S., E. Faggiano, and H. Rocha Investigating the impact of technology integration on Italian and Portuguese teachers’ perspectives concerning standardized assessment. MEDA – Mathematics Education at the Digital Age., In Press.
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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.

M
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.

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
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.

Morais, C., J. Terroso, and H. Rocha. "E de repente tudo mudou… - Editorial." Educação e Matemática. 155 (2020): 1.Website
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.

R
Rocha, H. "Using tasks to develop pre-service teachers’ knowledge for teaching mathematics with digital technology." ZDM Mathematics Education. 52.7 (2020): 1381-1396. AbstractWebsite

Teacher education is central to the development of the professional knowledge of pre-service teachers. The main goal of this paper is to refect on the development that the analysis (done by a group of pre-service secondary teachers) of a set of tasks, based on elements related to domains of KTMT—Knowledge for Teaching Mathematics with Technology—can bring to the knowledge of pre-service teachers of mathematics. Specifcally, the goal was to investigate the following questions: (1) What are the factors that guide the pre-service teachers’ task discussion? (2) Which KTMT domains are emphasized by pre-service teachers during task discussion? The elements taken into account are the characteristics of the tasks (focus on cognitive level, structuring level and technology role), the use of representations (focus on balance and articulation of representations), and the equilibrium between experimentation (focus on digital technology afordances) and justifcation (focus on argumentation and proof). The methodology of this case study involves a qualitative approach. The main conclusions suggest that infuences in the pre-service teachers’ discussion of tasks fell into the following categories: the potentialities of technology, the type of tasks, and the prospective teachers’ experience with a set of tasks, and analysis of some real students’ reports. With regard to KTMT, although it was possible to identify some global development, Teaching and Learning and Technology Knowledge was the domain in which stronger development took place.

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

MATHEMATICS TEACHING IN EDUCATION AND TRAINING COURSES
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. Utilização, uso ou integração da tecnologia: contributo para a clarificação de um conceito. Actas do XXV SIEM. Braga: APM, 2014. Abstract

Abstract. The recognition of the potential of technology for the teaching
and learning of mathematics has encouraged many studies around
technology. In all these studies, the integration, the utilization or the use of
technology is (or should be) necessarily an important element. In this paper
I consider the most common terminologies present in research and the
meaning assigned to them, based on a research review and on the analysis
of the studies presented in SIEM over the last five years. The conclusions
reached suggest a diversity of understandings and a lack of explicitness of
these understandings. However, different types of technology use seem to be
recognized, usually associated with continuity or change of practices. The
teacher's role and a more directive or more student-centered approach,
associated with a change in the proposed tasks, are also mentioned. In what
concerns to the terminology adopted, there is great diversity, with cases of
differentiation in terms of some of the elements listed and cases of adoption
of multiple terms with apparently identical meanings.

Resumo. O reconhecimento das potencialidades da tecnologia para o
ensino e aprendizagem da Matemática tem motivado diversos estudos em
torno da tecnologia. Em todos eles a integração, a utilização ou o uso que é
feito da tecnologia é (ou deveria ser) necessariamente um elemento
importante. Neste artigo procuro ponderar as terminologias mais comuns
na investigação e o significado que lhes é atribuído, partindo de uma
revisão de literatura e analisando os estudos apresentados no SIEM nos
últimos cinco anos. As conclusões alcançadas apontam para uma
diversidade de entendimentos e para uma ausência de explicitação desses
entendimentos. Ainda assim, parecem ser reconhecidos diferentes tipos de
utilização da tecnologia, geralmente associados à manutenção ou alteração
das anteriores práticas. O papel do professor e o assumir de uma postura
mais diretiva ou mais centrada no aluno, associada a uma alteração
relativamente às tarefas propostas, são igualmente referidos. Quanto à
terminologia adotada, a diversidade é grande, com casos de diferenciação
em função de alguns dos elementos referidos e com casos de adoção de
múltiplos termos aparentemente com significados idênticos.

Rocha, H. Processos de mudança associados às práticas de avaliação nos Cursos de Educação e Formação de Adultos. Atas do XII Congresso da SPCE. Vila Real, Portugal: UTAD e SPCE, 2014. Abstractpaper.pdf

Os cursos de Educação e Formação de Adultos prevêem uma avaliação que se afasta do tradicionalmente implementado nas escolas, propiciando o emergir de processos de mudança. Neste estudo analisa-se a forma como um formador concretiza a avaliação, ponderando continuidades e descontinuidades relativamente a práticas anteriores, com a intenção de caracterizar o inerente processo de mudança e os factores que o influenciam.
As conclusões obtidas sugerem um processo de mudança complexo, cuja necessidade não é verdadeiramente reconhecida, e onde parece ser determinante a reflexão do formador sobre os formandos, o contexto existente e algumas opções ao nível local da escola.

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. 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.

Rocha, H. "Pre-service teachers’ knowledge and the use of different technologies to teach Mathematics." Perspectives and Trends in Education and Technology. Springer, 2022. 505-515. Abstract

Teachers are central to the choice of tasks proposed to the students. And the teachers’ knowledge is one of the important elements guiding these choices. Despite the different models that conceptualize the teachers’ knowledge to integrate technology in their practices, research has focused essentially on the integration of a single technology. Little is known about how the work with different technologies can contribute to promote the development of the professional knowledge of pre-service teachers (PTs) or how the use of different technologies mobilizes different domains of the PTs’ knowledge. The main goal of this study is to deepen the understanding about the relation between the PTs’ Knowledge for Teaching Mathematics with Technology (KTMT) and their choice of tasks. The study adopts a qualitative and in-terpretative methodology based on one case study. The main conclusions suggest a strong impact of the PTs’ Learning and Teaching Technology Knowledge (a knowledge related to the impact of technology on the teaching and learning process) and a not so strong impact of their Mathematical and Technological Knowledge (a knowledge related to the impact of technology on the mathematical knowledge). The conclusions also point to the potential of the work with different technologies to deepen the PTs reflections and analysis of tasks.

Rocha, H. O recurso a diferentes representações no ensino das funções com o apoio da tecnologia. Actas do XXIII SIEM – Seminário de Investigação em Educação Matemática. Coimbra, Portugal: APM, 2012.
Rocha, H. Práticas de avaliação sumativa na disciplina de Matemática Aplicada de um Curso de Educação e Formação. Atas do VI Seminário Luso-Brasileiro Educação, Trabalho e Movimentos Sociais. Lisboa, Portugal: IE-UL, 2014. Abstractpaper.pdf

Os Cursos de Educação e Formação (CEF) foram concebidos tendo presente o elevado número de jovens em situação de abandono escolar, alunos usualmente marcados por experiências de insucesso, em particular a Matemática. O programa de Matemática Aplicada tem em conta esta realidade, tanto ao nível das aprendizagens como das metodologias e das características da avaliação a implementar. Relativamente à avaliação sumativa, é valorizado o trabalho desenvolvido pelo aluno, a sua apresentação, discussão e melhoria. As indicações dadas ao professor afastam-se da opção tradicional do teste de avaliação. O papel do professor na gestão curricular não é contudo negligenciado, sendo valorizada a adequação das propostas às características dos alunos. Este estudo pretende analisar as concepções de alunos e professores relativamente à avaliação sumativa, procurando compreender a forma como se influenciam mutuamente e como afectam a prática de avaliação do professor. Foram realizados três estudos de caso, incidindo sobre alunos e respectivo professor. Os dados foram recolhidos através de entrevistas, observação de aulas e recolha documental. Os resultados alcançados sugerem uma forte valorização dos testes por parte dos alunos, sendo notória a influência sobre as opções assumidas pelo professor. Determinantes parecem ser as concepções dos alunos relativamente ao papel de alunos e professores no que à avaliação respeita.

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. "Pre-service teachers’ knowledge: impact on the integration of mathematical applications on the teaching of mathematics." Science and mathematics education in the 21st century. Eds. L. Leite, and et al. Brussels: ATEE and CIEd, 2019. 26-37. Abstract

Mathematics is present everywhere. However, uncovering the relevance of Mathematics requires, from the teachers, a special kind of knowledge. This study tries to characterize the knowledge used by pre-service teachers when developing a mathematical task intending to promote the students’ exploration of barcodes. 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 some difficulties to see the potential of the situation to promote mathematical learning. The knowledge on the mathematical content seems to be dominant on the options assumed and operated in a rigid way that prevent the pre-service teachers from exploring the richness of the situation on the tasks they developed.

Rocha, H., and M. Botelho Teachers’ knowledge for teaching Mathematics with technology: an analysis of different frameworks. INTED - 15th annual International Technology, Education and Development Conference. IATED, 2021. Abstract

Teacher education is central to promote the development of the professional knowledge of teachers, and
to help them achieve an appropriate integration of digital technologies, an issue that has proved to be a
difficult one. Several authors refer difficulties in the integration of the technology, emphasizing the central
role played by the teachers’ knowledge in classroom use. In this paper we discuss three models (TPACK
– Technological Pedagogical and Content Knowledge, KTMT – Knowledge for Teaching Mathematics with
Technology, PTK / MPTK - Mathematical Pedagogical Technology Knowledge), intending to identify the
main contributions of each model to a deeper understanding of how to promote the teachers’ integration
of technology in the teaching of Mathematics. The study is based on a literature review and on an analysis
of the similarities and differences among the models and its use. On this analysis we identify common
influences among the models as well as influences from other research areas. The main conclusions
achieved point to a common base to all the models considered, but also to several differences among
them, being that some of the models emphasize the role of technology and its impact on Mathematics
learning, but others go further, intending to integrate in the model elements based on the research on
technology or even other theories such as the one on instrumental genesis.