Helena Rocha

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.

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.

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.

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

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.

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.

GAMES AND THE LEARNING OF MATHEMATICS OUTSIDE THE CLASSROOM
H. Rocha

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

Playing games is a recreational activity that is also highly recognized as a potentially rich activity for the teaching and learning. It is an activity that involves the recognition and observance of rules, as well as the development of strategies to achieve victory. It is thus an activity that encourages compliance with rules but also the development of learning and therefore has a socializing character while stimulating critical thinking and analysis of situations. This is why many authors think about playing games as a problem-solving activity with great potential for the learning of mathematics. However, a review of the literature suggests that mathematical learning does not always occur, pointing to the relevance of the specific features of the game and the circumstances in which it is used. Looking to contribute to a better understanding of these issues, the project that was the basis of this study focuses on the use of games by middle school students, intending to promote their mathematical learning in a voluntary and informal context, outside the classroom. The games were available in MatLab, a room of the school supervised by mathematics teachers, which students could visit in their leisure time. In this communication I intend to analyze how the visits to MatLab contributed to the mathematical learning of students, considering the influence of specific characteristics of the games and the atmosphere created in MatLab, given the students’ previous mathematical knowledge.

The study adopts a qualitative and interpretative methodological approach, undertaking two student case studies. Data collection was completed over three months and included observation of twenty visits of these students to MatLab. Data collection was made through the development of a logbook, audio record of the students’ visits and two interviews to the students and to their teacher. Data analysis was based on the evidence gathered in the light of the problem under study.

The conclusions reached stress the importance of certain features of the games to promote student engagement, leading to a desire for self-improvement, very important for the development of sustained learning. Computer games have proven to have a stronger potential to engage students than board games. Nevertheless, the most important characteristics of a game seem to be related to the possibility of playing at different mathematical levels (without getting blocked by lack of knowledge) and to the possibility of keep getting better marks (without the existence of a maximum level from which evolution is not possible). In what concerns to achievement in mathematics’ classes, the students’ teacher reports an improvement in mathematics knowledge (more evident in the average achiever student) as well as an increase in students’ involvement in class work (more evident in the low achiever student).

keywords: game-based learning, mathematics, informal learning.

The main goal of this work is to characterize how the knowledge of pre-service teachers about the characteristics of the tasks and the role of technology evolves. Based on a case study carried out around a pair of pre-service teachers, the main conclusions point to the contribution of the reflection around a set of six tasks on Functions selected by the pre-service teachers. Central to this reflection was an analyze of the role technology can play in tasks, the comments made by the colleagues to their tasks and some experiences on modeling and open-ended tasks. These elements provided the development of a greater awareness regarding aspects such as the level of structuring of the task and its degree of challenge. And this was determinant for an appropriation of the different characteristics of the tasks and to the development of the pre-service teachers’ knowledge.

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

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.

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 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.

THE IMPACT OF THE CULTURAL CONTEXT ON THE PROFESSIONAL PRACTICE OF THE TEACHER

H. Rocha

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

The professional knowledge is a key element of the teacher’s practice. This knowledge is naturally influenced by the teacher’s beliefs and conceptions and by his training, but the context where he develops his practice is perhaps the most decisive influence. At this level, the school where the teacher works and his colleagues are a powerful influence, but the characteristics of his students are even a stronger influence. The cultural diversity of the students and specifically the linguistic diversity are highly relevant elements. A classroom where different languages converge is always a complex context which requires a deeper professional knowledge with inevitable repercussions over the teacher’s practice.

This study focuses on a teacher working with a mathematics’ class of foreign students with heavy linguistic limitations on the language of instruction and it intends to analyze the impact of this context on the teacher’s practice. In particular, it intends to analyze how this context interferes with the characteristics of the tasks proposed by the teacher and with the way how mathematical concepts are presented to the students.

The study adopts a qualitative and interpretative methodological approach, undertaking one teacher case study. Data were collected during one school year by semi-structured interviews, class observation, and documental data gathering. All interviews and classes observed were audio taped and transcribed. Data analysis was conducted in an interpretative way.

The conclusions reached point to an increase on the appreciation of mechanization, to a large reduction in the use of problematic situations and to a presentation of Mathematics as calculation, disconnected from any application, and where reasoning appears as a marginal element or is even missing. The use of several examples becomes a key element of the practice of this teacher. The main finding of this study suggests that language limitations caused a strong impact on the practice of a teacher who considers the understanding and the development of reasoning from the discussion around mathematical ideas as central to the teaching of this subject. It was also possible to identify that the need to find a way to communicate reinforced the formalism of the mathematical language, placing it in the center of the learning process.

Keywords: cultural context, teacher’s practice, mathematics.

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.

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 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.

Knowledge for Teaching Mathematics with Technology (KTMT) is a theoretical model that seeks to articulate previously existing models on professional knowledge and the conclusions that the investigation around the integration of technology has achieved. KTMT is a dynamic knowledge, informed by the practice, that develops from the knowledge on the base domains (Mathematics, Teaching and Learning, Technology and Curriculum), evolving as knowledge in the base domains interacts and as this promotes the development of inter-domain knowledge, which continue to interact, strengthening relations and leading to the development of an integrated knowledge, where knowledge on the base domains and on the two sets of inter-domains appears deeply integrated into a global knowledge.