Helena Rocha

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.

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.

This study was conducted while 9th grade students learn to solve inequalities and seeks to understand their approach to solving problems with a real-life context. Specifically, the aim is to understand: (1) What are the main characteristics of the students’ approaches to the proposed problems? (2) What is the impact of the real context on the students’ resolutions? A qualitative and interpretative methodology is adopted, based on case studies, with data collected through documentary collection and audio recording of discussions between a pair of students while solving problems. The main conclusions suggest a trend to approach problems without establishing immediate connections with what was being done in the classroom, with students’ decisions being essentially guided by criteria of simplicity. The real context of the problems seems to have the potential to develop in students a more integrated mathematics, focused on understanding and not so much on the repetition of mechanical and meaning-independent procedures. The students’ familiarization with the context in question is one of the aspects highlighted by this study.

The development of meaningful learning becomes possible when students are actively involved in solving real problems. Thus, this study intends to investigate how students of the 11th grade of a vocational course solve problems of Linear Programming, using the graphing calculator. The conclusions reached indicate that: the interpretation of the conditions of the problems is the most delicate point; the graphical approach using technology is dominant; and the difficulties raised by the problem as well as the need to discuss the results achieved are the basis for the interactions both among the students and between them and the teacher.

This study seeks reflection on the approaches of 11th grade students to Linear Programming problems, discussing the approaches taken at different moments of the teaching process. It aims to analyze:
How is the students’ mathematical competence characterized in relation to problemsolving;
What differences can be identified in the resolutions at different moments of the teaching and learning process.
We adopt a qualitative and interpretative methodology, analyzing the approaches of two pairs of students with different mathematical backgrounds. The analysis is guided by P´olya’s stages of solving a problem and aspects of the understanding of mathematical competence. The results show different approaches to the problems depending on the teaching moment and different competences. The mathematical background impacts the students’ success when they implement routine procedures, however it does not seem to determine the students’ competence to reason about a problem.

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 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.

A suspensão das aulas presenciais na sequência da pandemia que estamos a atravessar trouxe para primeiro plano o ensino a distância. Neste artigo partilhamos algumas ideias e conceptualizações relativas a este tipo de ensino, abordamos aquilo que alguns autores que se têm dedicado à temática apontam como importantes desafios e oportunidades que se lhe encontram associados e, por fim, partilhamos algumas possíveis opções e recursos que pensamos poderem ser úteis para todos os professores que estão a viver a sua primeira experiência de ensino a distância.

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 assessment and the role it should be assumed by the summative and formative component are often a reason for discussion. It is therefore important to understand how the teacher assessment practices are characterized and what influences them. That is, identify aspects taken into account when planning assessment; the (dis)continuities between assessment and learning; the divergences/consonances between assessment planned and implemented. The conclusions reached point to a strong influence of peers, to the assessment criteria of the school and to the students’ characteristics, in a scenario where the test is the dominant element in assessment.

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.

STUDENTS’ CONCEPTIONS ABOUT THE USE OF GRAPHING CALCULATORS ON TESTS

H. Rocha

Faculdade de Ciências e Tecnologia, Universidade NOVA de Lisboa (PORTUGAL)

The assessment is considered a key element of the teaching and learning process and is often divided into two types: formative and summative. The distinction between these two types of assessment is usually made based on the moments in which it occurs and the objectives it has. Nevertheless, there are some continuities between these two types of assessment, and this leads some authors to question whether these two types of assessment should be seen as fully disjoint. Despite this, the prevailing understanding of summative assessment is that it takes place at the end of the learning process and that it is intended to classify the students.

The technology and, in particular, the graphing calculator is recognized for the impact it may have on the students’ approaches to solve mathematical questions. When technology is available, several studies point to an higher relevance of the understanding of the mathematical concepts, to an increase in graphical approaches to mathematical questions and to an increment in the use of exploratory approaches to solve the problems that are posed. Of course, all these changes will have its impact also on summative assessment moments, and specifically in testing.

Students’ conceptions about the use of technology have a deep impact on how they actually use the technology. The relevance usually attributed to tests, makes it important to understand what determines the performance of students in these moments.

This study focuses on the use of the graphing calculator at assessment moments such as tests, intending to understand the students’ conceptions related to that use. Namely it intends to analyze the impact of the students’ conceptions about Mathematics, about the use of technology to learn, and about teachers’ perspectives.

The study adopts a qualitative and interpretative methodological approach, undertaking two students’ case studies. Data were collected during one school year by semi-structured interviews, students’ observation at testing moments, and documental data gathering. All interviews were audio recorded and transcribed and the students’ observation was video recorded. Data analysis was conducted in an interpretative way.

The conclusions reached suggest that students welcome the possibility of using the graphing calculator during testing. The way this technology allows them to avoid errors, both in the calculations and in the formulas to be used, is the main reason advanced by the students. The speed of resolution, which they consider very important during testing, is another of the valued aspects. The idea of Mathematics as something that you need to understand and where knowing the right formula is not enough to achieve the right answer is pointed as the main justification for the use of this technology in tests. Nevertheless, the idea that technology should not be used seems to be always present. The impact of family ideas and, in particular, the idea that one can become dependent of the graphing calculator, seems to have some influence over the students conceptions about the use of this technology. However, the one that is undoubtedly the decisive reason for this conception is what they consider to be the opinion of a teacher. For the students, a teacher cannot agree with the use of graphing calculators in tests. And the reason given for this is related to the idea that a teacher will not be able to actually understand the students’ mathematical knowledge if he uses the graphing calculator.

Keywords: summative assessment, students’ conceptions, technology, mathematics.

Teaching statistics is often based on an approach focused on teaching theoretical aspects, disconnected from
practical relevance and from interpretation of results, and where the use of technology lies behind its potential. In
this context, it is important to analyze how the teachers’ knowledge is characterized and to identify aspects of this
knowledge that mark the professional practice. The conclusions reached emphasize the impact of content
knowledge and its influence on knowledge of content and teaching. Knowledge of curriculum is also relevant, as
well as the way how it seems to prevent the development of other types of knowledge.

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.

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.

This study aims to understand the perceptions of lower and upper secondary age teachers of mathematics regarding the use of technology to teach functions. For that, a mixed methodology was adopted, and the perceptions of 129 teachers were collected through a questionnaire (quantitative section) and four teachers through an interview (qualitative section). The main conclusions point to similarities in teachers' perceptions, but also to some differences related to the level that they taught. Teachers show conviction about their knowledge on technology and about the potential of technology in what concerns their teaching and the students’ learning. However, they are not so clear about the best way to articulate technology and paper-and-pencil methods, nor about the use of technology in assessment.

This study focus on the different representations provided by graphing calculators, intending to characterize how the teacher uses and integrates them on the process of teaching and learning functions at the secondary level. The methodology adopted is qualitative and interpretative, undertaking two case studies. The main conclusions point to different levels of flexibility in the use of the different representations depending on the teacher, but suggest a strong preference for the graphical and the algebraic representations, a use of the numerical representation based on the graph of the function and a total lack of use of the tabular representation.

This study focuses on teacher’s Knowledge for Teaching Mathematics with Technology (KTMT), paying a special attention to teacher’s representational fluency. It intends to characterize how the teacher uses and integrates the different representations provided by the graphing calculator on the process of teaching and learning functions at the high school level. Specifically, it intends to understand the balance established between the use of the different representations, and the way these representations are articulated. The study adopts a qualitative approach undertaking one teacher case study. Data were collected for two school years, at 10th and 11th grades, and included class observation, semi-structured interviews and documents gathering. Data analysis was mainly descriptive and interpretive in nature, considering the problem under study. The conclusions reached reveal an active use of the graphical and algebraic representations and a scarce use of the tabular representation. The lack of balance on the use of representations also includes the work within a representation. In this case the graphical representation is the only one that was explored. The conclusions also indicate a flexible articulation between the two representations usually used. It was possible to identify different patterns on the use of the representations and a frequent use of an interactive approach, marked by repeated alternations between representations. Globally, this study emphasizes teacher’s KTMT and raises questions about the impact of technology on teacher´s representational fluency and about the difference between a numerical and a tabular representation.

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.

Reconhecendo o potencial do jogo para a aprendizagem matemática, este estudo pretende analisar o envolvimento e as aprendizagens dos alunos, com o objectivo de caracterizar os jogos com maior potencial para os promover.
Adoptando uma metodologia de índole qualitativa e envolvendo a realização de estudos de caso sobre alunos do 7.ºano, as conclusões alcançadas sugerem que os jogos de computador são particularmente apelativos para os alunos. Contudo, as características determinantes para o envolvimento dos alunos e consequente promoção da aprendizagem prendem-se com a possibilidade de jogar com diferentes níveis de conhecimento e com a obtenção de bons resultados no jogo.