Helena Rocha

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.

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.

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.

Games are commonly appointed as a methodological tool capable of promoting students’ effective learning. In this context, this study intends to analyze the impact of mathematical discussions developed while
playing a polynomial game. Namely it intends to analyze the impact on the consolidation of mathematical concepts previously worked in the classroom and on the communications skills. Two case studies where developed involving 10th grade students. Data gathering was based on direct observation and an inquiry. The main conclusions suggest that the game encouraged the discussion about the mathematical contents and therefore promoted the development of the mathematical discourse. Besides that, it allowed a deeper apprehension of mathematical concepts, and the overcome of some difficulties.

A formação e o desenvolvimento profissional do professor são determinantes para as opções que este assume na sala de aula. É o seu conhecimento, aquilo que valoriza e o contexto onde se encontra inserido que determinam as experiências de aprendizagem que proporciona aos seus alunos. Mas esse conhecimento profissional envolve uma multiplicidade de dimensões que decorrem da sua formação inicial e contínua, mas também das experiências que teve ocasião de vivenciar e de processos de socialização, onde a interação com os pares e as oportunidades de desenvolver trabalho colaborativo são elementos importantes. A aula de matemática surge assim como o campo aglutinador do trabalho do professor numa dupla vertente que se une num ciclo único: por um lado a aula de Matemática é o foco do trabalho do professor, onde as opções previamente assumidas são implementadas; e, por outro lado, é um ponto de partida para a reflexão e o desenvolvimento profissional do professor.

Da planificação da aula, onde a escolha das tarefas e a forma de as implementar são aspetos centrais e onde a vertente histórica não deixará de estar presente; à sua implementação, operacionalizando diferentes recursos (nomeadamente os tecnológicos) e assumindo dinâmicas de aula diferenciadas; até à fase de reflexão entre pares, que termina e reinicia um novo ciclo – estas são as grandes etapas em torno das quais este texto se organiza e onde a formação inicial e contínua não deixarão de estar presentes.

The teacher's knowledge of the mathematical fidelity of technology and the impact it has on the teacher’s practice is the focus of this article. Based on the conceptualization of Knowledge for Teaching Mathematics with Technology (KTMT), and involving the teaching of Functions at the 10th grade, we analyze: the situations of lack of mathematical fidelity considered by the teacher in the classes, the way how the teacher manages students' contact with this kind of situations, and how the teacher supports students when they are faced with a lack of mathematical fidelity. The conclusions reached point to: some devaluation of the situations of lack of mathematical fidelity, with only one type of situation being explicitly addressed; a careful selection of tasks, in order to ensure that these situations do not occur too soon; a focus on the identification by the students of this type of situation, suggesting what they can do to confirm the suspicion but without effective implementation of the process. As a consequence, knowledge of mathematical fidelity does not necessarily have a relevant impact on teacher’s practice and it is not easily transformed into a deep teacher’s KTMT.

Deductive reasoning, being central in mathematics, is also usually a source of difficulties for students, more used to the empirical approaches. In this study we focus on mathematical proof and we try to give attention to how this kind of reasoning is envisaged by the students, to the options they assume when asked to develop a deductive reasoning and to the factors affecting the implementation of this kind of reasoning. The study follows a qualitative and interpretative methodological approach, including the completion of two case studies of students of the 10th grade. Data were collected in work sessions and through interviews. The main findings point to a devaluation of mathematical proof and a strong preference for empirical approaches. Yet students show ability to develop different approaches. The preference for the mathematical subject and the attention given in class to the deduction work, appears to be relevant factors when considering the students' ability to develop a deductive reasoning when involved on a mathematical proof.

This article focuses the choice and use of examples by two students of the 11th grade to prove or refute a set of statements. The use of representations of sequences and functions is also considered. The study adopts a qualitative approach and data were collected by interviews and documental gathering. The conclusions suggest most of the examples used were well-known sequences or functions. However, the students sought different purposes for the use of examples, such as understanding the conjecture, demonstrate the falsity or truthfulness of the statement and conveying a general argument. The students made a satisfactory articulation between the various types of representations but relied mostly in the cartesian graph.

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 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.

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.

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.

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.

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.