Helena Rocha

Technology is recognized for its potential to implement exploration tasks. The ease and speed with which it becomes possible to observe many cases of a situation, allows the development of conjectures and brings conviction about their veracity. Mathematical proof, assumed as the essence of Mathematics, tends to appear to the students as something dispensable. Based on KTMT – Knowledge for Teaching Mathematics with Technology model, this study intends to understand the impact of the teachers’ knowledge on mathematical proof in a context of technology integration. The study adopts a qualitative and interpretative methodology, based on case study, analyzing the practice of one teacher. The conclusions emphasize the relevance of the teacher’s MTK – Mathematics and Technology Knowledge, and TLTK – Teaching and Learning and Technology Knowledge. The teacher's MTK guides her decisions, leading her to focus on helping students understand the meaning of conjecture and proof, valuing, at the same time, the relevance of algebraic manipulations. However, the teacher’s TLTK guides her practice, where the knowledge about the students is determinant. The study provides evidence about the difficulty of articulating proof and technology, but it also clarifies the relevance of this articulation and of how the teacher’s KTMT can impact the teacher’s decisions.

Technology is recognized by its potential to promote mathematical learning. However, achieving this potential
requires the teachers to have the knowledge to integrate it properly into their practices. Several authors have intended to characterize the teachers’ knowledge and developed several models, but this approach has often been criticized by its static approach, not attending neither valuing the teachers’ practice. In this study we adopt the KTMT – Knowledge for Teaching Mathematics with Technology model, assuming the teachers’ practice as the main scenario of analysis. We focus on the options guiding the teachers’ decisions when confronted with a situation of lack of mathematical concordance while teaching functions. The situations of lack of mathematical concordance (i.e., situations where the mathematics addressed by the students is different from the one intended by the teacher) are assumed as rich and encapsulating the potential to reveal significant aspects of the teachers’ KTMT. The main goal of the study is to understand what domains of the teachers’ KTMT are highlighted in these circumstances. A qualitative methodology is adopted and one episode of one 10th grade teacher’s practice is analyzed, based on the KTMT model. The conclusions reached show the relevance of different knowledge domains, but emphasize the Mathematics and Technology Knowledge (MTK). They also raise questions about the impact of the specific technology being used on the teachers’ KTMT.

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.

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.

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.

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.

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

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

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.

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.

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.

Perante as conhecidas dificuldades em alcançar uma adequada integração da tecnologia no processo de ensino e aprendizagem da Matemática, este estudo pretende, apoiando-se na formação ao nível da tecnologia ministrada em duas instituições europeias, identificar aspetos com potencial para promover a formação inicial, no âmbito da tecnologia, de professores de Matemática. Adota-se uma metodologia de índole qualitativa e interpretativa, sendo os dados recolhidos de natureza documental ou relativos a trabalhos de análise e reflexão crítica realizados por dois futuros professores (um de cada instituição). As principais conclusões alcançadas apontam para grandes diferenças entre os contextos de formação, com uma das instituições a valorizar de forma mais significativa a formação na área. Ainda assim, os futuros professores de ambas as instituições mostram alguma tendência para escolher tarefas onde a exploração que é feita da tecnologia fica aquém do seu potencial, onde o recurso ao papel e lápis está sempre presente, e onde a reflexão em torno das características das tarefas e da sua implementação parece ser algo superficial. Apesar da complexidade do processo de integração da tecnologia nas práticas, os aspetos referidos parecem-nos ser dignos de atenção em qualquer programa de formação inicial de professores de Matemática.

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.