Helena Rocha

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.

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.

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.

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.

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.

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.

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.

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.

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.

A exploração de diferentes representações promove a compreensão dos tópicos de funções. Partindo deste pressuposto, com este estudo pretende-se analisar o contributo da representação gráfica na aprendizagem da noção de função inversa e da paridade de uma função por alunos do 10.º ano de escolaridade e identificar dificuldades na exploração dessa representação. Na procura de responder a este objetivo, adotou-se uma abordagem qualitativa e interpretativa para compreender as ações dos alunos na resolução das tarefas
propostas. A análise das resoluções mostra que a representação gráfica serviu de suporte para a instituição das definições dos tópicos em estudo. E isto apesar de alguns alunos revelarem dificuldades ao interpretar e ao construir gráficos; ao identificar imagens e imagens inversas em gráficos de funções; ao representar determinadas características gráficas associadas a alguns conceitos, como é o caso da relação entre a paridade de uma função e a simetria na sua representação gráfica (confundindo eixo de simetria e de reflexão). Globalmente, este estudo mostra como a abordagem de conceitos a partir da representação gráfica pode contribuir para a sua compreensão.

This study intends to understand the perceptions of mathematics teachers from lower and upper secondary regarding the teaching of Functions and the use of technological materials. For that, a mixed methodology was adopted, and the perceptions of 129 teachers were collected through a questionnaire and four teachers through an interview. The main conclusions point to similarities in teachers' perceptions, but also to some differences related to the teaching level of lower or upper secondary. In the teaching of Functions, textbooks are widely used, but differently depending on the level being taught. The same happens with the representations and with the use that is made of the technologies. Involvement of students in work is another aspect considered important, but again there are differences. The assessment also has similarities, but differs in the valuation ascribed to group work.

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.