Existence of multiple solutions for second-order discrete boundary value problems

We give conditions on f involving pairs of discrete lower and discrete upper solutions which lead to the existence of at least three solutions of the discrete two-point boundary value problem yk+1 - 2yk + yk-1 + f (k, yk, vk) = 0, for k = 1,..., n - 1, y0 = 0 = yn,, where f is continuous and vk = yk - yk-1, for k = 1,..., n. In the special case f (k, t, p) = f (t) greater than or equal to 0, we give growth conditions on f and apply our general result to show the existence of three positive solutions. We give an example showing this latter result is sharp. Our results extend those of Avery and Peterson and are in the spirit of our results for the continuous analogue. (C) 2002 Elsevier Science Ltd. All rights reserved.

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.

The nonexistence of spurious solutions to discrete, two-point boundary value problems

We investigate difference equations which arise as discrete approximations to two-point boundary value problems for systems of second-order, ordinary differential equations. We formulate conditions under which all solutions to the discrete problem satisfy certain a priori bounds which axe independent of the step-size. As a result, the nonexistence of spurious solutions are guaranteed. Some existence and convergence theorems for solutions to the discrete problem are also presented. (C) 2002 Elsevier Science Ltd. All rights reserved.

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.

Contextual intelligent load management with ANN adaptive learning module

With the current increase of energy resources prices and environmental concerns intelligent load management systems are gaining more and more importance. This paper concerns a SCADA House Intelligent Management (SHIM) system that includes an optimization module using deterministic and genetic algorithm approaches. SHIM undertakes contextual load management based on the characterization of each situation. SHIM considers available generation resources, load demand, supplier/market electricity price, and consumers’ constraints and preferences. The paper focus on the recently developed learning module which is based on artificial neural networks (ANN). The learning module allows the adjustment of users’ profiles along SHIM lifetime. A case study considering a system with fourteen discrete and four variable loads managed by a SHIM system during five consecutive similar weekends is presented.

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.

PCMAT, an adaptive learning platform

This paper is about PCMAT, an adaptive learning platform for Mathematics in Basic Education schools. Based on a constructivist approach, PCMAT aims at verifying how techniques from adaptive hypermedia systems can improve e-learning based systems. To achieve this goal, PCMAT includes a Pedagogical Model that contains a set of adaptation rules that influence the student-platform interaction. PCMAT was subject to a preliminary testing with students aged between 12 and 14 years old on the subject of direct proportionality. The results from this preliminary test are quite promising as they seem to demonstrate the validity of our proposal.

Modellus: Learning Physics with Mathematical Modelling

Tese de doutoramento em Ciências da Educação, área de Teoria Curricular e Ensino das Ciências

A new horizon for online teaching and learning

The Polytechnic Institute of Oporto (IPP), which has a solid history of online education and innovation through the use of technology, has been particularly interested and focused on Massive Open Online Courses (MOOC) developments. The aim of this paper is to present the whole process from initial discussions to completion of the “Mathematics Without Limits” MOOC Project that exists in IPP and also to contribute for a change in the way as teaching and learning Mathematics is seen and practiced nowadays. In 2013, IPP developed its own platform, which gave us the opportunity to explore new educational techniques as a pedagogical resource as well as to enhance students’ motivation, through a set of interactive materials at their disposal, totally adapted to their needs. Students lack of motivation is mainly justified by their weak Math preparation, poor consolidated basis on the subject and different backgrounds of the students. To tackle this issue and based on our Math online courses teaching experience, we decided to create short duration MOOC, expecting to aid retention of students and also to reverse the path of students giving up on Math by giving them a friendly way of managing their own learning commitment. We also think that this MOOC will be a good approach to level out some math skills among freshmen.

A new horizon for online teaching and learning

The Polytechnic Institute of Oporto (IPP), which has a solid history of online education and innovation through the use of technology, has been particularly interested and focused on Massive Open Online Courses (MOOC) developments. The aim of this paper is to present the whole process from initial discussions to completion of the “Mathematics Without Limits” MOOC Project that exists in IPP and also to contribute for a change in the way as teaching and learning Mathematics is seen and practiced nowadays. In 2013, IPP developed its own platform, which gave us the opportunity to explore new educational techniques as a pedagogical resource as well as to enhance students’ motivation, through a set of interactive materials at their disposal, totally adapted to their needs. Students lack of motivation is mainly justified by their weak Math preparation, poor consolidated basis on the subject and different backgrounds of the students. To tackle this issue and based on our Math online courses teaching experience, we decided to create short duration MOOC, expecting to aid retention of students and also to reverse the path of students giving up on Math by giving them a friendly way of managing their own learning commitment. We also think that this MOOC will be a good approach to level out some math skills among freshmen.