Using transactional distance theory to redesign an online mathematics education course for pre-service primary teachers

This paper examines the impact of a series of design changes to an online mathematics education course in terms of transactional distance between learner and teachers, pre-service education students' attitudes towards mathematics, and their development of mathematical pedagogical knowledge. Transactional distance theory (TDT) was utilised to investigate and describe the interactions among course structure, course dialogue and student autonomy in an online course over a two-year period. Findings indicate that Web 2.0 technologies, when used thoughtfully by teachers, can afford high levels of structure and dialogue. Feedback from pre-service teachers indicated an improved attitude towards mathematics and an increase in their mathematical pedagogical content knowledge. These findings have implications for universities moving towards the delivery of teacher education courses entirely online

The optimality of attaching unlinked labels to unlinked meanings

Vocabulary learning by children can be characterized by many biases. When encountering a new word, children as well as adults, are biased towards assuming that it means something totally different from the words that they already know. To the best of our knowledge, the 1st mathematical proof of the optimality of this bias is presented here. First, it is shown that this bias is a particular case of the maximization of mutual information between words and meanings. Second, the optimality is proven within a more general information theoretic framework where mutual information maximization competes with other information theoretic principles. The bias is a prediction from modern information theory. The relationship between information theoretic principles and the principles of contrast and mutual exclusivity is also shown.Peer ReviewedPostprint (published version

Scientific Knowledge in Aristotle’s Biology

Aristotle was the first thinker to articulate a taxonomy of scientific knowledge, which he set out in Posterior Analytics. Furthermore, the “special sciences”, i.e., biology, zoology and the natural sciences in general, originated with Aristotle. A classical question is whether the mathematical axiomatic method proposed by Aristotle in the Analytics is independent of the special sciences. If so, Aristotle would have been unable to match the natural sciences with the scientific patterns he established in the Analytics. In this paper, I reject this pessimistic approach towards the scientific value of natural sciences. I believe that there are traces of biology in the Analytics as well as traces of the Analytics’ theory in zoological treatises. Moreover, for a lack of chronological clarity, I think it’s better to unify Aristotle’s model of scientific research, which includes Analytics and the natural sciences together

The Micro-Evolution of Mathematical Knowledge: The Case of Randomness

In this paper we explore the growth of mathematical knowledge and in particular, seek to clarify the relationship between abstraction and context. Our method is to gain a deeper appreciation of the process by which mathematical abstraction is achieved and the nature of abstraction itself, by connecting our analysis at the level of observation with a corresponding theoretical analysis at an appropriate grain size. In this paper we build on previous work to take a further step towards constructing a viable model of the micro-evolution of mathematical knowledge in context. The theoretical model elaborated here is grounded in data drawn from a study of 10-11 year olds’ construction of meanings for randomness in the context of a carefully designed computational microworld, whose central feature was the visibility of its mechanisms-how the random behavior of objects actually worked. In this paper, we illustrate the theory by reference to a single case study chosen to illuminate the relationship between the situation (including, crucially, its tools and tasks) and the emergence of new knowledge. Our explanation will employ the notion of situated abstraction as an explanatory device that attempts to synthesize existing micro- and macro-level descriptions of knowledge construction. One implication will be that the apparent dichotomy between mathematical knowledge as de-contextualized or highly situated can be usefully resolved as affording different perspectives on a broadening of contextual neighborhood over which a network of knowledge elements applies

Applications of Mathematical Models in Engineering

The most influential research topic in the twenty-first century seems to be mathematics, as it generates innovation in a wide range of research fields. It supports all engineering fields, but also areas such as medicine, healthcare, business, etc. Therefore, the intention of this Special Issue is to deal with mathematical works related to engineering and multidisciplinary problems. Modern developments in theoretical and applied science have widely depended our knowledge of the derivatives and integrals of the fractional order appearing in engineering practices. Therefore, one goal of this Special Issue is to focus on recent achievements and future challenges in the theory and applications of fractional calculus in engineering sciences. The special issue included some original research articles that address significant issues and contribute towards the development of new concepts, methodologies, applications, trends and knowledge in mathematics. Potential topics include, but are not limited to, the following: Fractional mathematical models; Computational methods for the fractional PDEs in engineering; New mathematical approaches, innovations and challenges in biotechnologies and biomedicine; Applied mathematics; Engineering research based on advanced mathematical tools

Revisiting the Majorana Relativistic Theory of Particles with Arbitrary Spin

In 1932 Ettore Majorana published an article proving that relativity allows any value for the spin of a quantum particle and that there is no privilege for the half integer spin. The Majorana idea was so innovative for the time that the scientific community understood its importance only towards the end of the thirties. This paper aims to highlight the depth of the scientific thought of Majorana that, well in advance of its time, opened the way for modern particle physics and introduced for the first time the idea of a universal quantum equation, able to explain the behavior of particles with arbitrary spin and of any nature, regardless the value of their speed. It will be analyzed in detail and made explicit all the steps that lead to the physical mathematical formulation of the Majorana theory. A part of these steps require basic knowledge of quantum physics but not for this should be regarded as trivial since they show the physical meaning hidden into the structure of the equation. Moreover, the explicit method for the construction of the infinite matrices will be given, by which the infinite components of the wave functions representing the fundamental and excited states of the particle are calculated.Comment: Paper revised after publication on "Advances in Physics Theories and Applications", Vol. 48 (2015) - ISSN (Paper)2224-719X ISSN (Online)2225-063