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Rocha, H. Pre-service teachers’ knowledge: impact on the integration of mathematical applications on the teaching of mathematics. ATEE Winter Conference - Science and mathematics education in the 21st century. Brussels: ATEE and CIEd, 2019. 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.

Rocha, H. Teacher knowledge and the implementation of investigation tasks. Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education. Kiel, Germany: PME, 2013.
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.

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., E. Faggiano, and F. Mennuni. "Teachers as task designers in the digital age: Teaching using technology." Proceedings of the 10th ERME Topic Conference - MEDA 2020. Linz (Austria): ERME, 2020. Abstract2020_meda_rocha_faggiano_mennuni.pdf

The aim of the paper is to present and analyse the case of one teacher attempting to introduce his students to fractals using digital technology. His task design process has been made explicit through the writing of a storyboard. It has been analysed in order to focus on the stages of the process, identifying prominent elements in it by using the knowledge quartet framework. Results can be useful to inform teacher educators about his needs with respect to the development of his ability in task design. The importance of this aspect, particularly worth of note in the digital age in which teachers have many opportunities to access teaching resources online, has been amplified by the constraints to which educational systems have been subjected during the Covid-19 pandemic emergency.

Rocha, H. O formalismo matemático num contexto de utilização da tecnologia [Mathematical proof in a context of technology integration]. Atas do XXVI SIEM. Évora: APM, 2015. Abstract

The technology and how it tends to emphasize the intuitive and overshadow calculus and mathematical proof are the focus of this paper. The conclusions reached suggest that tasks where students might realize the usefulness of calculus as well as of more intuitive approaches are possible even when the technology is a reality in the classroom. They also suggest that proof may, among other things already identified in the literature, make an important contribution to the students’ understanding of fundamental aspects of mathematics.

A tecnologia e a forma como esta tende a enfatizar o intuitivo e a relegar para segundo plano o formal e a demonstração matemática são o foco deste artigo. As conclusões alcançadas sugerem que é possível colocar aos alunos situações onde estes se possam aperceber da vantagem de recorrer tanto a abordagens mais formais como a abordagens mais intuitivas e isto mesmo quando a tecnologia é uma realidade em sala de aula. Sugere ainda que a realização de demonstrações pode, entre outros aspectos já identificados na literatura, dar um contributo importante para a compreensão de aspectos basilares da Matemática.

Rocha, H. O formal da matemática e o intuitivo da tecnologia: que articulação?. Atas do ProfMat 2015. Évora, Portugal: APM, 2015. Abstractpaper.pdf

A tecnologia é cada vez mais indispensável no dia-a-dia, rodeando-nos constantemente. Para os nossos alunos é uma realidade que conhecem desde sempre e que tendem a encarar com uma naturalidade descontraída e intuitiva. A facilidade de acesso à tecnologia e o modo como esta tende a enfatizar o intuitivo e a relegar para segundo plano o formal e a demonstração matemática são o foco deste artigo. Partindo da análise de uma proposta de trabalho onde alunos de 10.º ano começam por uma abordagem intuitiva apoiada na calculadora gráfica e terminam a realizar uma demonstração da conjectura que formularam, procuro discutir a problemática. As conclusões alcançadas sugerem que é possível colocar aos alunos situações onde estes se podem aperceber da vantagem de recorrer tanto a abordagens mais formais como a abordagens mais intuitivas e isto mesmo quando a tecnologia é uma realidade em sala de aula. Sugere ainda que a realização de demonstrações pode, entre outros aspectos já identificados na literatura, dar um contributo importante para a compreensão de aspectos basilares da Matemática.

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. "Analyzing teachers’ knowledge based on their approach to the information provided by technology." European Journal of Science and Mathematics Education. 11.1 (2023): 132-145. AbstractWebsite

Teachers’ knowledge plays a central role in technology integration. In this study we analyze situations, where there is some divergence between the mathematical results and the information offered by the graphing calculator (lack of mathematical fidelity), putting the focus in the teachers and in their approaches. The goal of this study is to analyze, in the light of knowledge for teaching mathematics with technology (KTMT) model, the teachers’ professional knowledge, assuming the situations of lack of mathematical fidelity as having the potential to reveal some characteristics of their knowledge. Specifically, considering the teaching of functions at 10th grade (age 16), we intend to analyze: (1) What knowledge do the teachers have of technology and of its mathematical fidelity? (2) What can the teachers’ options related to situations of lack of mathematical fidelity tell us about their knowledge in other KTMT domains? The study adopts a qualitative and interpretative approach based on the case studies of two teachers. Data were collected by interviews and class observation, being the analysis guided by the KTMT model. The main result points to the relevance of the mathematics and technology knowledge. However, there is evidence of some difficulties to integrate the information provided by the technology with the mathematics, and also of some interference of the teaching and learning and technology knowledge, and specifically of the knowledge related to the students. This suggests that the analysis of the teachers’ actions in relation to situations of lack of mathematical fidelity, can be useful to characterize their KTMT.

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.

Rocha, H., and I. Oitavem. "Barcodes: The Mathematics of everyday life." The Scottish Mathematical Council Journal. 49 (2019).Website
Rocha, H., E. Faggiano, A. Sacristan, and M. Santacruz-Rodríguez Professional development for the integration of technology in the teaching of Mathematics: in search of pragmatic theories - Desenvolvimento profissional para a integração da tecnologia no ensino da Matemática: em busca de teorias pragmáticas. XXXI Seminário de Investigação em Educação Matemática. Santarém: APM, 2021. Abstractsiem_2021.pdf

This paper presents part of a study that aimed to make more explicit the pragmatic theories that inform the design of professional development programs with an emphasis on the integration of digital technologies in the practices of mathematics teachers. The analysis carried out was based on a set of projects considered representative and implemented in four countries – Colombia, Italy, Mexico and Portugal. Based on this analysis, we identify relevant elements (e.g., similarities and differences, barriers and opportunities) and develop recommendations to be taken into account in the design of future professional development programs. In this process, we identified a set of aspects and sub-aspects, as well as several interconnections between them, which emerged in relation to five main themes and allowed us to reveal our pragmatic theories. Thus, this work provides a framework to support the design of future projects for the professional development of mathematics teachers regarding the use of digital technology.

Rocha, H., M. C. Costa, and H. Jacinto O desenvolvimento curricular e a formação de professores. Atas do Encontro de Investigação em Educação Matemática. SPIEM, 2022.
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.
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.

Rocha, H. A perspectiva de futuros professores sobre a demonstração matemática. XV Congresso Internacional Galego-Portugués de Psicopedagogia. Corunha, Espanha: Asociación Científica Internacional de Psicopedagogía, 2019. Abstract

The mathematical proof is assumed as a central element in the development of Mathematics. However, proof is conceived in different ways and assumed as having different functions in Mathematics. And when we move from mathematics to its teaching, the multiplicity of perspectives becomes even more significant. This diversity can have an impact on the students and on the relationship they establish with Mathematics. In these circumstances, this study seeks knowledge over the perspectives of future teachers regarding the mathematical demonstration. Specifically, it intends to achieve a deeper knowledge over the future teachers’ perspectives about what is a mathematical proof and about its functions. The study adopts a qualitative approach and uses interviews to collect data. The conclusions reached point to a traditional perspective of mathematical proof, closely tied to mathematical formalism and the validation function, where the teaching context introduces some changes, adjusting the formalism to the level of the students and highlighting the understanding function of proof, but maintaining the dominant character of the algebraic language.

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.

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Sacristán, A., E. Faggiano, M. Santacruz-Rodríguez, and H. Rocha. "Policies and implementations for technology use in mathematics education: perspectives from around the world." Handbook of digital resources in mathematics education. Springer, 2024. 1-35. Abstract

In this chapter, we examine the role of policies and other factors affecting digital technology (DT) integration in mathematics education. In particular, we develop a cross-national analysis of the impact on DT implementation in four countries: two countries in Europe (Italy and Portugal) and two countries in Latin America (Colombia and Mexico). We analyze the role that policies, political changes, reforms, curricula, educational organization and systems, sociocultural aspects, and teachers’ training, knowledge, and beliefs play toward possible DT implementations. We observe that there is a discourse in policies to promote digital technologies’ use, but in practice the availability and integration of such resources in mathematics classrooms is still scarce. We also note that the efforts done during the pandemic did not change this, promoting general ICT use, rather than DT resources that might enhance mathematics teaching and learning.

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Teixeira, P., C. Martins, and H. Rocha STE(A)M approach: Distinguishing and discussing meanings. EduLearn. Spain: IATED, 2022. Abstract

The STE(A)M approach has been recognized by several authors for its potential in assisting teaching and learning, and several curriculum standards already value its application in the classroom. This approach is based on the articulation between different areas, the clarification, and the deepening of the concepts being studied. Although there are different approaches, according to the fields involved, STEM and STEAM are two among the most often mentioned in the literature. STEM is based on learning that integrates the following areas of knowledge: Science, Technology, Engineering, and Mathematics. The conceptualization of the STE(A)M approach is not consensual and uniform. There are different models focusing on problem-solving based learning, project-based-learning, design-based learning, and engineering models. Still, different authors present different conceptualizations of this approach. In this paper, we relied on the existing literature to discuss the different understandings of the STE(A)M approach. We will also pay attention to mathematics and how different authors see the disciplines’ role within a STE(A)M approach and discuss the evolution of the mentioned authors’ positions throughout time. Thus, methodologically, we undertook the following steps: (i) literature search based on the selected keywords; (ii) selection of the texts, considering the authors and time gap, in order to analyze the evolution of the research and (iii) collection and organization of the relevant topics for the study. This study aims to present the meanings, conceptualizations, and possible influences present in different models and for understand the evolution of the STEM and STEAM approaches over time. The main findings suggest a focus on the interdisciplinary or transdisciplinary approach as opposed to the primeval years of investigations in STEM and STEAM when many authors advocated a multidisciplinary approach. This change in thinking is due to the need to train students in an integral and holistic manner, developing citizens with transversal knowledge and skills prepared for the current societal challenges.

Teixeira, P., C. Martins, and H. Rocha Abordagem STEAM: articulação disciplinar e práticas letivas de professores. Atas do Encontro de Investigação em Educação Matemática., 2022.
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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.

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.

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., 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.