Helena Rocha

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.

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.

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 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.

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.

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.

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.

This study aims to discuss similarities and differences between knowledge models focusing on technology, developing the Global Model. This is not a new model, but rather a contribution to the articulation between different conceptual frameworks.

Este estudo visa discutir semelhanças e diferenças entre modelos de conhecimento com foco na tecnologia, desenvolvendo a partir destes o Modelo Global. Este não é um novo modelo, mas antes um contributo para a articulação entre diferentes quadros conceptuais.

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.

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.

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.

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.

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.

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.

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.

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.

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.