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2015
Rocha, H. Students' conceptions about the use of graphing calculators on tests. Proceedings of 8th Annual International Conference of Education, Research and Innovation. Seville, Spain: ICERI, 2015. Abstract

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.

2014
Rocha, H. Consensos e dilemas no ensino de matemática a alunos de um curso de Decoração e Pintura Cerâmica. Atas do VI Seminário Luso-Brasileiro Educação, Trabalho e Movimentos Sociais. Lisboa, Portugal: IE-UL, 2014. Abstractpaper.pdf

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.

Rocha, H. Different representations in mathematics teaching with technology. Proceedings of the 38th Conference of the International Group for the Psychology of Mathematics Education. Vancouver, Canada: PME, 2014. Abstract

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.

Rocha, H. Games and the learning of mathematics outside the classroom. Proceedings of the International Conference on Education and New Learning Technologies. Barcelona, Spain: EduLearn, 2014. Abstract

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.

Rocha, H. The influence of teacher’s knowledge for teaching mathematics with technology on the implementation of investigation tasks. Proceedings of 8th International Technology, Education and Development Conference. Valencia, Spain: INTED, 2014.
Rocha, H. Mathematics teaching in Education and Training Courses. Proceedings of the International Conference on Education and New Learning Technologies. Barcelona, Spain: EduLearn, 2014. Abstract

MATHEMATICS TEACHING IN EDUCATION AND TRAINING COURSES
H. Rocha

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

Education and Training Courses have been specifically designed to the high number of young people in a situation of school dropout and in transition to working life, particularly those who enter the labor market early with insufficient levels of schooling and professional training. Mathematics is one of the curriculum components of these courses, for its contribution to the exercise of citizenship in a democratic society. Being an important part of the cultural legacy of our society is too often seen by students as a source of exclusion. It is known that young people who enter these courses often had an experience of underachievement in the discipline, what justifies that motivating students is at once the great challenge faced by the teacher. The program suggests taking a more concrete and linked to reality approach, allowing students to learn to recognize the mathematics in the world around them and using technology to promote that learning. However, it is the teacher who is responsible for managing its implementation, shaping the learning situations and integrating them in a coherent and articulated way in the specific course that students attend. In what concerns to assessment, the program also takes into account the usual characteristics of the students. Thus, the assessment includes a strong appreciation of students’ work, its presentation and discussion and further improvement of that work. The directions given to the teacher diverge from the traditional option of the evaluation test, providing guidelines to the form that each evaluation can take depending on the contents in study. However, once again, the teacher's role in curriculum management is not neglected, being valued the adequacy of proposals to the characteristics of the students.
The study presented here had as its main goal to analyze and understand the choices made by the teacher during the different stages of his practice, giving attention to the dilemmas he faced and to the reasons he took into account when making decisions.

The study adopts a qualitative and interpretative methodological approach, undertaking one teacher case study. Data collection included semi-structured interviews, classroom observation and document collection. Data analysis was based on the evidence gathered in the light of the problem under study.

The conclusions of the study point to the important role of technology and suggest that the reduction of prerequisites, the intention of taking into account the students’ interests and the desire of improving students culture is central in what concerns to task selection; while the active involvement of students characterized the implementation of the classes. The dilemmas faced by the teacher focus mainly on the relative importance and on the demanding level that he should give to each content, as well as the articulation that he should promote between formal and intuitive knowledge. In what concerns to assessment, the results achieved highlight the impact that students ideas can have on teacher’s practice, conducting to the inclusion of tests as an assessment element, against the teacher’s intentions.

keywords: education and training courses, mathematics, innovation, technology.

Kahle, R., I. Oitavem, and H. Rocha. "Os problemas de Hilbert." Educação e Matemática. 130 (2014): 23-27.Website
Rocha, H. Práticas de avaliação sumativa na disciplina de Matemática Aplicada de um Curso de Educação e Formação. Atas do VI Seminário Luso-Brasileiro Educação, Trabalho e Movimentos Sociais. Lisboa, Portugal: IE-UL, 2014. Abstractpaper.pdf

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.

Rocha, H. Processos de mudança associados às práticas de avaliação nos Cursos de Educação e Formação de Adultos. Atas do XII Congresso da SPCE. Vila Real, Portugal: UTAD e SPCE, 2014. Abstractpaper.pdf

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.

Rocha, H. The teacher and the integration of the graphing calculator viewing window in the teaching of mathematics. Proceedings of 8th International Technology, Education and Development Conference. Valencia, Spain: INTED, 2014.
Rocha, H. Uma caracterização dos jogos com maior potencial para estimular a aprendizagem matemática. Atas do XII Congresso da SPCE. Vila Real, Portugal: UTAD e SPCE, 2014. Abstractpaper.pdf

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.

Rocha, H. Utilização, uso ou integração da tecnologia: contributo para a clarificação de um conceito. Actas do XXV SIEM. Braga: APM, 2014. Abstract

Abstract. The recognition of the potential of technology for the teaching
and learning of mathematics has encouraged many studies around
technology. In all these studies, the integration, the utilization or the use of
technology is (or should be) necessarily an important element. In this paper
I consider the most common terminologies present in research and the
meaning assigned to them, based on a research review and on the analysis
of the studies presented in SIEM over the last five years. The conclusions
reached suggest a diversity of understandings and a lack of explicitness of
these understandings. However, different types of technology use seem to be
recognized, usually associated with continuity or change of practices. The
teacher's role and a more directive or more student-centered approach,
associated with a change in the proposed tasks, are also mentioned. In what
concerns to the terminology adopted, there is great diversity, with cases of
differentiation in terms of some of the elements listed and cases of adoption
of multiple terms with apparently identical meanings.

Resumo. O reconhecimento das potencialidades da tecnologia para o
ensino e aprendizagem da Matemática tem motivado diversos estudos em
torno da tecnologia. Em todos eles a integração, a utilização ou o uso que é
feito da tecnologia é (ou deveria ser) necessariamente um elemento
importante. Neste artigo procuro ponderar as terminologias mais comuns
na investigação e o significado que lhes é atribuído, partindo de uma
revisão de literatura e analisando os estudos apresentados no SIEM nos
últimos cinco anos. As conclusões alcançadas apontam para uma
diversidade de entendimentos e para uma ausência de explicitação desses
entendimentos. Ainda assim, parecem ser reconhecidos diferentes tipos de
utilização da tecnologia, geralmente associados à manutenção ou alteração
das anteriores práticas. O papel do professor e o assumir de uma postura
mais diretiva ou mais centrada no aluno, associada a uma alteração
relativamente às tarefas propostas, são igualmente referidos. Quanto à
terminologia adotada, a diversidade é grande, com casos de diferenciação
em função de alguns dos elementos referidos e com casos de adoção de
múltiplos termos aparentemente com significados idênticos.

2013
Rocha, H. Aspectos da utilização da tecnologia no ensino da Matemática. Actas do XII Congresso Internacional Galego-Português de Psicopedagogia. Braga, Portugal: CIEd - IEUM, 2013. Abstract

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.

Rocha, H. Knowledge for Teaching Mathematics with Technology - a new framework of teacher knowledge. Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education. Kiel, Germany: PME, 2013. Abstract

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.

Rocha, H. Teacher knowledge and the implementation of investigation tasks. Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education. Kiel, Germany: PME, 2013.
2012
Rocha, H., and I. Oitavem. "A mudança que abala o mundo – Editorial." Educação e Matemática.116 (2012): 1.
Rocha, H. O recurso a diferentes representações no ensino das funções com o apoio da tecnologia. Actas do XXIII SIEM – Seminário de Investigação em Educação Matemática. Coimbra, Portugal: APM, 2012.
2011
Rocha, H. "A calculadora gráfica e a utilização que delas fazemos." Educação e Matemática.112 (2011): 41-42.
Rocha, H. A calculadora gráfica no ensino das funções: implicações sobre aspectos da prática de uma professora. Actas do EIEM – Encontro de Investigação em Educação Matemática. Póvoa do Varzim, Portugal: SPIEM, 2011.
Rocha, H. Teachers’ use of graphing calculators in high school mathematics classroom. Proceedings of CERME 7 – Seventh Congress of European Research in Mathematics Education. Rzezów, Poland: ERME, 2011.