Mathematical working space and paradigms as an analysis tool for the teaching and learning of analysis

ZDM – Mathematics Education - Mathematical working space (MWS) is a model that is used in research in mathematics education, particularly in the field of geometry. Some MWS elements are...     

Mathematical modelling and the learning trajectory: tools to support the teaching of linear algebra

In this article we present a didactic proposal for teaching linear algebra based on two compatible theoretical models: emergent models and mathematical modelling. This proposal begins with a problematic situation related to the creation and use of secure passwords, which leads students toward the construction of the concepts of spanning set and span. The objective is to evaluate this didactic proposal by determining the level of match between the hypothetical learning trajectory (HLT) designed in this study with the actual learning trajectory in the second experimental cycle of an investigation design-based research more extensive. The results show a high level of match between the trajectories in more than half of the conjectures, which gives evidence that the HLT has supported, in many cases, the achievement of the learning objective, and that additionally mathematical modelling contributes to the construction of these linear algebra concepts.     

From the principle of bijection to the isomorphism of structures: an analysis of some teaching paradigms in discrete mathematics

This paper is concerned with the teaching of Discrete Mathematics to university undergraduate students. Two to three decades ago this course became a requirement for math and computer science students in most universities world wide. Today this course is taken by students in many other disciplines as well. The paper begins with a discussion of a few topics that we feel should be included in the syllabus for any course in Discrete Mathematics, independent of the audience. We then discuss several potential models for teaching the course, depending upon the interests and mathematical background of the audience. We also investigate various educational links with other components of the curriculum, consider pedagogical issues associated with the teaching of discrete mathematics, and discuss some logistical and psychological difficulties that must be overcome. A special emphasis is placed on the role of textbooks.     

Mathematical modelling as a tool for the connection of school mathematics

We start introducing some aspects of the theoretical framework: the Anthropological Theory of Didactics (ATD). Then, we consider on the research domain commonly known as “modelling and applications” and briefly describe its evolution using the ATD as an analytical tool. We propose a reformulation of the modelling processes from the point of view of the ATD, which is useful to identify new educational phenomena and to propose and tackle new research problems. Finally, we focus on the problem of the connection of school mathematics. The reformulation of the modelling processes emerges as a didactic tool to tackle this research problem. We work on the problem of the articulation of the study of functional relationships in Secondary Education and present a teaching proposal designed to reduce the disconnection in the study of functional relationships in Spanish Secondary Education.     

Connected functional working spaces: a framework for the teaching and learning of functions at upper secondary level

ZDM – Mathematics Education - This paper aims at contributing to remedy the narrow treatment of functions at upper secondary level. Assuming that students make sense of functions by working...     

Designing an intelligent teaching simulator for learning to teach by practicing

Learning to teach is difficult for prospective teachers because of the complex nature of the work of teaching. Practicing (Lampert in J Teach Educ 61(1–2):21–34, 2010), interacting with the practice of teaching from a first-person perspective, may give them a unique experience in learning to teach. Computer-based simulators in which the apprentice teacher can interact with virtual students may be used to create that kind of experience. In this paper, we show how to apply techniques in artificial intelligence to design an intelligent learning environment. We show how to model the apprentice’s decision making and resources that can help him or her improve the practice of teaching.     

Games as an educational resource in the teaching and learning of mathematics: an educational experiment in Portuguese middle schools

This article is based on an experiment using the game ‘Caminhando e Calculando’ (Moving and Calculating) in order to analyse the potential of the game as an educational resource for the teaching and learning of mathematics in Portuguese middle schools, where most students are 10 or 11 years old. Students' data obtained during the games will be used to analyse the different options used for solving the game, identifying its potential and its weaknesses. We start with a theoretical analysis of games as an inherent element of human culture. Combining our innate desire for fun with the different types of teaching and learning styles allows for fun and knowledge to be combined into more efficient and meaningful types of knowledge. Playing games are a primordial aspect of what it means to be a child and they develop within a motivating environment; therefore, not to take advantage of games as a learning resource would be to neglect an important asset. With regard to mathematics, emphasis will be given to the advantages that this teaching and learning tool provides for certain mathematical processes, such as problem-solving.     

Error linearization as an effective tool for experimental analysis of the numerical stability of algorithms

After introducing briefly the principles of theerror linearization method, which is able to determine the coefficients of the first order error approximation, a collection of examples is presented to demonstrate its efficiency as a test bench for analyzing numerical algorithms. These examples illustrate the propagation of initial errors, the effect of cancellation, the easy location of the most unstable parts of an algorithm, calculation of condition numbers, approximating the statistical behavior of accumulated errors and the convergence of iterative methods.     

Mathematical problems of stochastic quantum mechanics: Harmonic analysis on phase space and quantum geometry

In this paper we review the mathematical methods and problems that are specific to the programme of stochastic quantum mechanics and quantum spacetime. The physical origin of these problems is explained, and then the mathematical models are developed. Three notions emerge as central to the programme: positive operator-valued (POV) measures on a Hilbert space, reproducing kernel Hilbert spaces, and fibre bundle formulations of quantum geometries. A close connection between the first two notions is shown to exist, which provides a natural setting for introducing a fibration on the associated overcomplete family of vectors. The introduction of group covariance leads to an extended version of harmonic analysis on phase space. It also yields a theory of induced group representations, which extends the results of Mackey on imprimitivity systems for locally compact groups to the more general case of systems of covariance. Quantum geometries emerge as fibre bundles whose base spaces are manifolds of mean stochastic locations for quantum test particles (i.e., spacetime excitons) that display a phase space structure, and whose fibres and structure groups contain, respectively, the aforementioned overcomplete families of vectors and unitary group representations of phase space systems of covariance.Work supported in part by the Natural Science and Engineering Research Council of Canada (NSERC) grants.     

Working with tasks for the learning of problem solving in maths teaching as an issue of the first teacher training phase

This article describes learning goals of teacher training for the working with tasks in maths lessons. Selected common and different features of tasks intended for the learning and performing are especially referred to     

Dimensional analysis: an elegant technique for facilitating the teaching of mathematical modelling

Dimension analysis is promoted as a technique that promotes better understanding of the role of units and dimensions in mathematical modelling problems. The authors' student base consists of undergraduate students from the Science and Engineering Faculties who generally have one or two semesters of calculus and some linear algebra as part of their curriculum. Because of ‘In Service Training’ which is an integral part of their education, they have a reasonable understanding of the link between theory and practice in their particular industry, but manipulating mathematical formulae is not necessarily a strong point. Dimensional analysis involves both dimensionless products and linear algebra and, because of the latter, this branch of mathematical modelling was, until recently, beyond the reach of most undergraduates. However, it has been found that the skills of a good technologist can be blended with the use of computer algebra systems to successfully teach dimensional analysis to these undergraduates. This note illustrates the concept of dimensional analysis by examining the simple pendulum problem and shows how dimensionless products can lead to the discovery of the connection between the period of the pendulum swing and its length. Dimensional analysis is shown to lead to interesting systems of linear equations to solve, and can point the way to more quantitative analysis, and two student problems are discussed. It is the authors' experience that dimensional analysis broadens a student's viewpoint to include units and dimensions as an integral part of any physical problem. With this approach coupled with a computer algebra systems such as DERIVE, students can concentrate on understanding the model and the modelling process rather than the solution technique. Finally, it has been observed that students find dimensional analysis fun to do.