EXAMINING SECONDARY MATHEMATICS TEACHER CANDIDATES’ LEARNING AND ENACTMENT OF MATHEMATICS TEACHING PRACTICES: A MULTIPLE CASE STUDY

This qualitative case study explored three teacher candidates’ learning and enactment of discourse-focused mathematics teaching practices. Using audio and video recordings of their teaching practice this study aimed to identify the shifts in the way in which the teacher candidates enacted the following discourse practices: elicited and used evidence of student thinking, posed purposeful questions, and facilitated meaningful mathematical discourse. The teacher candidates’ written reflections from their practice-based coursework as well as interviews were examined to see how two mathematics methods courses influenced their learning and enactment of the three discourse focused mathematics teaching practices. These data sources were also used to identify tensions the teacher candidates encountered. All three candidates in the study were able to successfully enact and reflect on these discourse-focused mathematics teaching practices at various time points in their preparation programs. Consistency of use and areas of improvement differed, however, depending on various tensions experienced by each candidate. Access to quality curriculum materials as well as time to formulate and enact thoughtful lesson plans that supported classroom discourse were tensions for these teacher candidates. This study shows that teacher candidates are capable of enacting discourse-focused teaching practices early in their field placements and with the support of practice-based coursework they can analyze and reflect on their practice for improvement. This study also reveals the importance of assisting teacher candidates in accessing rich mathematical tasks and collaborating during lesson planning. More research needs to be explored to identify how specific aspects of the learning cycle impact individual teachers and how this can be used to improve practice-based teacher education courses.

The development of an eBook for teaching and learning mathematics

Mobile devices, smartphones, phablets and tablets, are widely avail‐ able. This is a generation of digital natives. We cannot ignore that they are no longer the same students for which the education system was designed tradition‐ ally. Studying math is many times a cumbersome task. But this can be changed if the teacher takes advantage of the technology that is currently available. We are working in the use of different tools to extend the classroom in a blended learning model. In this paper, it is presented the development of an eBook for teaching mathematics to secondary students. It is developed with the free and open standard EPUB 3 that is available for Android and iOS platforms. This specification supports video embedded in the eBook. In this paper it is shown how to take advantage of this feature, making videos available about lectures and problems resolutions, which is especially interesting for learning mathematics.

e-Math Workshops : a forum for exchanging experiences of mathematics e-learning at university level

We report here about a series of international workshops on e-learning of mathematics at university level, which have been jointly organized by the three publicly funded open universities in the Iberian Peninsula and which have taken place annually since 2009. The history, achievements and prospects for the future of this initiative will be addressed.

Contributions of Neuroscience to Develop Teaching Strategies and Learning of Mathematics

The goal of the present work is to develop some strategies based on research in neurosciences that contribute to the teaching and learning of mathematics. The interrelationship of education with the brain, as well as the relationship of cerebral structures with mathematical thinking was discussed. Strategies were developed taking into consideration levels that include cognitive, semiotic, language, affect and the overcoming of phobias to the subject. The fundamental conclusion was the imperative educational requirement in the near future of a new teacher, whose pedagogic formation must include the knowledge on the cerebral function, its structures and its implications to education, as well as a change in pedagogy and curricular structure in the teaching of mathematics.

Perceptions of online tutorials for distance learning in mathematics and computing

We report on student and staff perceptions of synchronous online teaching and learning sessions in mathematics and computing. The study is based on two surveys of students and tutors conducted 5 years apart, and focusses on the educational experience as well as societal and accessibility dimensions. Key conclusions are that both staff and students value online sessions, to supplement face-to-face sessions, mainly for their convenience, but interaction within the sessions is limited. Students find the recording of sessions particularly helpful in their studies.

Bayesian online algorithms for learning in discrete Hidden Markov Models

We propose and analyze two different Bayesian online algorithms for learning in discrete Hidden Markov Models and compare their performance with the already known Baldi-Chauvin Algorithm. Using the Kullback-Leibler divergence as a measure of generalization we draw learning curves in simplified situations for these algorithms and compare their performances.

The residual effects of cooperative-learning experiences: A two-year follow-up

The author investigated how training in small-group and interpersonal behaviors affected children's behavior and interactions as they worked in small groups 2 years later. The authors assigned 52 fifth graders, who had been trained 2 years previously in cooperative group behaviors, to the trained condition and 36 fifth graders, who had not previously been trained, to the untrained condition. Both were reconstituted from the pool of students who had participated previously in group activities. The results showed a residual training effect, with the children in the trained groups being more cooperative and helpful than their untrained peers.

Systems thinking: learning or driving?

The ability to foresee how behaviour of a system arises from the interaction of its components over time - i.e. its dynamic complexity – is seen an important ability to take effective decisions in our turbulent world. Dynamic complexity emerges frequently from interrelated simple structures, such as stocks and flows, feedbacks and delays (Forrester, 1961). Common sense assumes an intuitive understanding of their dynamic behaviour. However, recent researches have pointed to a persistent and systematic error in people understanding of those building blocks of complex systems. This paper describes an empirical study concerning the native ability to understand systems thinking concepts. Two different groups - one, academic, the other, professional – submitted to four tasks, proposed by Sweeney and Sterman (2000) and Sterman (2002). The results confirm a poor intuitive understanding of the basic systems concepts, even when subjects have background in mathematics and sciences.

Systems thinking: learning or driving?

The ability to foresee how behaviour of a system arises from the interaction of its components over time - i.e. its dynamic complexity – is seen an important ability to take effective decisions in our turbulent world. Dynamic complexity emerges frequently from interrelated simple structures, such as stocks and flows, feedbacks and delays (Forrester, 1961). Common sense assumes an intuitive understanding of their dynamic behaviour. However, recent researches have pointed to a persistent and systematic error in people understanding of those building blocks of complex systems. This paper describes an empirical study concerning the native ability to understand systems thinking concepts. Two different groups - one, academic, the other, professional – submitted to four tasks, proposed by Sweeney and Sterman (2000) and Sterman (2002). The results confirm a poor intuitive understanding of the basic systems concepts, even when subjects have background in mathematics and sciences.

How to teach mathematics with MOODLE

The aim of this article is to present a Project in the Oporto’s Institute of Accounting and Administration, which pretends to contribute for a change in the way of teaching and learning Mathematics. One of the main objectives of this project is to innovate the teaching and learning processes, exploring technologies as a pedagogical resource and to induce higher motivation to students, improve the rate of success and make available to students a set of materials adapted to their needs. This concern is justified due to the fact that students have a weak preparation, without consolidated basis. Since the year 2007/2008 the courses were adjusted to the Bologna process, which requires several changes in teacher’s and student’s roles, methodologies and assessment. The number of weekly classes has been reduced, so it was necessary to develop new strategies and methodologies to support the student. With the implementation of the Bologna Process in the Accounting degree, we felt a great need to provide other types of activities to students. To complement our theoretical and practical classes we have developed a project called MatActiva based on the Moodle platform offered by PAOL - Projecto de Apoio On-Line (Online Support Project). Moodle allows us to use the language TEX to create materials that use mathematical symbols. Using this functionality, we created a set of easy to use interactive resources. In MatActiva project, the students have access to a variety of different materials. We have followed a strategy that makes the project compatible with the theoretical and practical subjects/classes, complementing them. To do so, we created some resources, for instance multiple-choice tests, which are the most accessed by the students. These tests can be realized and corrected on-line and for each wrong answer there is a feedback with the resolution. We can find other types of resources: diagnostic tests, theoretical notes. There are not only the pre-requirements for subjects mathematics, but also materials to help students follow up the programs. We also developed several lessons. This activity consists of a number of pages, where each page has contents and leads to other pages, based on the student's progress. The teacher creates the choices and determines the next page that the student will see, based upon their knowledge. There is also an area of doubts, where the students can place all the mathematical doubts they have, and a teacher gives the answers or clues to help them in their work. MatActiva also offers an area where we can find some humour, curiosities, contests and games including mathematical contents to test the math skills, as well as links to pages about mathematical contents that could be useful for the study. Since ISCAP receives ERASMUS students and some of them attend mathematics, we developed some materials in English, so they can also use MatActiva. The main objectives of our project are not only to bring success in the subjects of mathematics, but also to motivate the students, encourage them to overcome theirs difficulties through an auto-study giving them more confidence and improve their relationship with the mathematics as well as the communication between students and teachers and among students.

The active mathematics project at ISCAP

This paper summarizes a project that is contributing to a change in the way of teaching and learning Mathematics. Mathematics is a subject of the Accounting and Administration course. In this subject we teach: Functions and Algebra. The aim is that the students understand the basic concepts and is able to apply them in other issues, when possible, establishing a bridge between the issues that they have studied and their application in Accounting. As from this year, the Accounting course falls under in Bologna Process. The teacher and the student roles have changed. The time for theoretical and practical classes has been reduced, so it was necessary to modify the way of teaching and learning. In the theoretical classes we use systems of multimedia projection to present the concepts, and in the practical classes we solve exercises. To supplement our theoretical and practical classes we have developed an active mathematics project called MatActiva based on the Moodle platform offered by PAOL - Projecto de Apoio Online (Online Support Project). In the last versions of Moodle, it is possible use the TeX language to create math questions. Using this tool we created a set of interactive materials. With the creation of this new project we wanted to take advantage already obtained results with the previous experiences, giving to the students opportunities to complement their study in Mathematics. One of the great objectives is to motivate students, encourage them to overcome theirs difficulties through an auto-study, giving them more confidence and the opportunity to seeing others perspectives of the mathematics subjects. In the MatActiva project the students have a big collection of information about the way of the subject works, which includes the objectives, the program, recommended bibliography, evaluation method and summaries. It works as material support for the practical and theoretical classes, the slides of the theoretical classes are available, the sheets with exercises for the students to do in the classroom and complementary exercises, as well as the exams of previous years. Students can also do diagnostic tests and evaluation tests online. Our approach is a reflexive one, based on the professional experience of the teachers that explore and incorporate new tools of Moodle with their students and coordinate the project MatActiva.

New challenges in mathematics for the european higher education

In this paper we will talk about a math project submitted to the Lifelong Learning Programme. European higher education needs a reform in order to play its full role in the Europe of Knowledge. Modernisation of higher education is necessary in the areas of curricula (Bologna process), funding and governance so that higher education institutions can face the challenges posed by globalisation and contribute more effectively to the training and retraining of the European workforce. On the other hand Mathematics is an essential component of all educational systems. Mathematical literacy is being scrutinized in assessment efforts such as the OCDE Programme for International Student Assessment (PISA). This showed a low level in Europe. Due to the Bologna Process, which brought several didactical implications for Higher Education (HE) institutions, there is the need of lifelong learning. This evolution is in conflict with the earlier mentioned lack of competencies on basic sciences, such as Mathematics. Forced by this duality, efforts are combined to share expertise in the Math field and the integration of pedagogical methodologies becomes a necessity. Thus, several European countries have proposed an International Project to the Lifelong Learning Programme, Action ERASMUS Modernisation of Higher Education, to make institutions more attractive and more responsive to the needs of the labour market, citizens and society at large. One of the main goals of the project is to attract students to math through high-quality instructional units in an understandable, exciting and attractive way.

Learning objects recommendation in an adaptive educational hypermedia system

The aim of this paper is presenting the modules of the Adaptive Educational Hypermedia System PCMAT, responsible for the recommendation of learning objects. PCMAT is an online collaborative learning platform with a constructivist approach, which assesses the user’s knowledge and presents contents and activities adapted to the characteristics and learning style of students of mathematics in basic schools. The recommendation module and search and retrieval module choose the most adequate learning object, based on the user's characteristics and performance, and in this way contribute to the system’s adaptability.