Helena Rocha

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.

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.

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.

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.

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.

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.

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.

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.

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.

The teacher’s knowledge has long been viewed as a strong influence on the students’ learning. Several authors have sought to develop procedures to assess this knowledge, but this has proved to be a complex task. In this paper I present an outline of a conceptualization to analyze the teacher's knowledge, based on the model of the Knowledge for Teaching Mathematics with Technology (KTMT) and a set of tasks. These tasks are chosen by the teacher among the ones he prepared for his students taking into account the potential of the tasks to take advantage of the technology’s potential. The analyze of the teacher’s KTMT is based on the characteristics of the tasks chosen by the teacher; the balance established between the representations provided by the technology that the tasks advocate; the way how the tasks pay attention to the new issue of seeking for a suitable viewing window; and also the way how the tasks take into account the expectable difficulties of the students in the process of looking for the window.

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.

This study focusses on the criteria used by pre-service teachers of Mathematics to choose interdisciplinary tasks. The pre-service teachers’ knowledge is assumed as the basis of the actions taken and used as the origin of the choices and approaches observed. 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 an appreciation of the mathematical part of the tasks and to a devaluation of the remaining components. This suggests difficulty in articulating and integrating different domains of knowledge and points to a fragmented view of the potential of using mathematical applications.

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.

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.

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.

O potencial da tecnologia para o ensino e a aprendizagem é há muito reconhecido. Contudo, cada vez mais surgem estudos que indiciam que a sua utilização fica aquém das expectativas. O acesso à tecnologia, o papel que lhe é atribuído na aprendizagem e as características das tarefas em que é utilizada, encontram-se entre as principais influências identificadas sobre a utilização que é feita da tecnologia e potencialmente responsáveis pelas características dessa utilização. O estudo que aqui se apresenta teve como principal objectivo analisar e compreender a utilização que os professores fazem da tecnologia à luz dos aspectos referidos, procurando identificar de que forma estes definem diferentes tipos de utilização. A abordagem metodológica adoptada foi de natureza qualitativa e interpretativa, com a realização de estudos de caso de duas professoras de Matemática que utilizavam a calculadora gráfica. A recolha de dados foi concretizada através de entrevistas semi-estruturadas, observação de aulas e recolha documental, sendo a análise de dados orientada pelo quadro teórico, conciliado com a interpretação destes. As conclusões do estudo sugerem alguma diversidade tanto nas características das tarefas em que as professoras recorrem à tecnologia (exercícios, explorações, problemas, modelação de situações reais), como no papel que lhe é atribuído (obter informação, efectuar cálculos, experimentar), mas apontam também para uma diversidade com características que se traduzem em perfis de utilização da tecnologia distintos: um mais exploratório e outro mais prescritivo.

The usual difficulties of students regarding the choice of an appropriate window when using the graphing calculator in the study of functions and the importance of the teachers’ knowledge to overcoming them, led to this study. The main goal was to characterize the way teachers address the viewing window in the classroom, trying to infer aspects of the Knowledge for Teaching Mathematics with Technology that can justify that practice. The conclusions reached point to the importance of a set of specific knowledge where I highlight the knowledge of the students’ difficulties, the knowledge of mathematical content necessary to understand the impact of the viewing window on the graphic, and the knowledge of teaching strategies that address both the students’ difficulties and the relevant mathematical knowledge.

Neste estudo analisamos as perceções que professores do 3.º ciclo e do ensino secundário têm da sua prática no âmbito do ensino de Funções, com o objetivo de as caracterizar e de identificar as diferenças existentes entre estes dois grupos de professores. Um aspeto particularmente relevante se tivermos em conta que se tratam de dois grupos de professores com formações iniciais idênticas. Adotamos uma metodologia mista, com uma vertente quantitativa apoiada na aplicação de questionários e uma vertente qualitativa baseada na realização de entrevistas. As principais conclusões alcançadas apontam para semelhanças nas perceções dos professores, mas também para algumas diferenças em função do ciclo de ensino. Na planificação das aulas os manuais são amplamente utilizados, mas de forma diferente consoante o ciclo de ensino do professor. Os professores de ambos os ciclos de ensino estabelecem conexões entre diferentes representações, mas valorizam de diferentes formas as representações disponíveis. O envolvimento dos alunos nas atividades da aula é outro aspeto destacado pelos professores, mas uma vez mais existem diferenças. Na avaliação o recurso ao teste é enfatizado pelos dois grupos de professores, mas já existem diferenças quanto à importância atribuída ao trabalho de grupo.