Curricular orientations to real-world contexts in mathematics

A common claim about mathematics education is that it should equip students to use mathematics in the ‘real world’. In this paper, we examine how relationships between mathematics education and the real world are materialised in the curriculum across a sample of eleven jurisdictions. In particular, we address the orientation of the curriculum towards application of mathematics, the ways that real-world contexts are positioned within the curriculum content, the ways in which different groups of students are expected to engage with real-world contexts, and the extent to which high-stakes assessments include real-world problem solving. The analysis reveals variation across jurisdictions and some lack of coherence between official orientations towards use of mathematics in the real world and the ways that this is materialised in the organisation of the content for students

Very-high energy gamma-ray astronomy: A 23-year success story in high-energy astroparticle physics

Very-high energy (VHE) gamma quanta contribute only a minuscule fraction - below one per million - to the flux of cosmic rays. Nevertheless, being neutral particles they are currently the best "messengers" of processes from the relativistic/ultra-relativistic Universe because they can be extrapolated back to their origin. The window of VHE gamma rays was opened only in 1989 by the Whipple collaboration, reporting the observation of TeV gamma rays from the Crab nebula. After a slow start, this new field of research is now rapidly expanding with the discovery of more than 150 VHE gamma-ray emitting sources. Progress is intimately related with the steady improvement of detectors and rapidly increasing computing power. We give an overview of the early attempts before and around 1989 and the progress after the pioneering work of the Whipple collaboration. The main focus of this article is on the development of experimental techniques for Earth-bound gamma-ray detectors; consequently, more emphasis is given to those experiments that made an initial breakthrough rather than to the successors which often had and have a similar (sometimes even higher) scientific output as the pioneering experiments. The considered energy threshold is about 30 GeV. At lower energies, observations can presently only be performed with balloon or satellite-borne detectors. Irrespective of the stormy experimental progress, the success story could not have been called a success story without a broad scientific output. Therefore we conclude this article with a summary of the scientific rationales and main results achieved over the last two decades.Comment: 45 pages, 38 figures, review prepared for EPJ-H special issue "Cosmic rays, gamma rays and neutrinos: A survey of 100 years of research

Compressed Pairings

Pairing-based cryptosystems rely on bilinear non-degenerate maps called pairings, such as the Tate and Weil pairings defined over certain elliptic curve groups. In this paper we show how to compress pairing values, how to couple this technique with that of point compression, and how to benefit from the compressed representation to speed up exponentiations involving pairing values, as required in many pairing based protocols

Elliptic and Hyperelliptic Curves: A Practical Security Analysis

Motivated by the advantages of using elliptic curves for discrete logarithm-based public-key cryptography, there is an active research area investigating the potential of using hyperelliptic curves of genus 2. For both types of curves, the best known algorithms to solve the discrete logarithm problem are generic attacks such as Pollard rho, for which it is well-known that the algorithm can be sped up when the target curve comes equipped with an efficiently computable automorphism. In this paper we incorporate all of the known optimizations (including those relating to the automorphism group) in order to perform a systematic security assessment of two elliptic curves and two hyperelliptic curves of genus 2. We use our software framework to give concrete estimates on the number of core years required to solve the discrete logarithm problem on four curves that target the 128-bit security level: on the standardized NIST CurveP-256, on a popular curve from the Barreto-Naehrig family, and on their respective analogues in genus 2. © 2014 Springer-Verlag Berlin Heidelberg

The use of mathematical strategies: factors and features affecting performance

The study investigates the use of general mathematical strategies by pupils of junior high school age. The form of response required pupils to describe their reasoning and to articulate their reasons for the choice of particular strategies. Main effects were found for cognitive level, year level, and sex. The sex effects is in contrast with similar effects from most other mathematical studies in that girls outperformed boys at each year level. A second part of the study involves the characterization of responses by means of a qualitative hierarchy: The Solo Taxonomy. Features of responses representing different levels of functioning are discussed and used to suggest related implications for curriculum and teaching

System dynamics and university management

Contemporary management of the national university system in Australia is characterized by the pursuit of government goals through the linking of funding to expressed priorities. Competitive elements of policies at the national level have been translated by individual institutions into management strategies that promote competition between internal units such as faculties and schools. This is in addition to the competition that has been encouraged between the universities themselves in areas such as student enrolment and research productivity. Within these organizational contexts a variety of system archetypes can be identified "plying their trade". Efforts to contain debt by staff attrition, competition between institutions for students, setting of goals for enrolment levels, and distribution of scarce resources on the basis or research activity provide contexts for the manifestation of delayed feedback loops, escalation, sliding goals and 'tragedy' scenarios. Following illustration by means of example, some experiences of attempting to introduce system dynamic concepts into decision making discussions are shared

Aspects of proving: A clinical investigation of process

A clinical methodology was used to investigate the perceptions which pupils of secondary school age have concerning modes of mathematical argument which have an agreed status within the world of mathematics. The analysis of data obtained from three extended contexts led to the identification of clusters of characteristic response types. Differences were found to exist between the agreed meaning of some mathematical terms and procedures and the meaning ascribed to them by students. By considering levels of performance it was possible to identify particular components, the presence or absence of which consistently determined the capacity to structure or follow proofs and explanations

Attitudes to mathematics of beginning undergraduates and prospective teachers: Some implications for education

An acute shortage of mathematics teachers has emerged in the Western world. This article describes a study of the influence of tertiary education experiences on the attitudes of potential mathematics teachers towards their subject. Parallel questionnaires were given to groups of students who were respectively entering university mathematics courses as new undergraduates, and entering postgraduate teacher training courses. Both groups of students answered items pertaining to their experiences within secondary mathematics courses. The undergraduate students were also asked a series of questions about their expectations of tertiary mathematics courses. For the postgraduates a parallel set of questions was provided that required them to rate their actual tertiary experiences. An analysis of responses indicated that the postgraduate students were more positive about their recollection of secondary mathematics than were the undergraduate students with their more recent experience of it. They were also less positive about the reality of their tertiary courses than were the undergraduates about their expectations. Responses of the postgraduates suggested that tertiary mathematics is not merely an extension of secondary mathematics, but a subject with which distinctive and, in general, more negative reactions are associated. Mathematics emerges as a subject which progressively loses its appeal with further study and implications are drawn for both tertiary education and teacher supply

The mathematical vitality of secondary mathematics graduates and prospective teachers: A comparative study

This article links two areas of contemporary interest in mathematics education. These are respectively; mathematical characteristics of prospective teachers, and the notion of levels of understanding. The study reported here considered questions relating to the mathematical appreciation of successful students at both senior secondary (university entrance) and tertiary level. The tertiary students were graduates embarking upon a post-graduate course of teacher-training. It was found that response patterns were stable across a variety of institutions, geographical regions (Australian States), and education systems. Mathematical malfunctions present among secondary graduates were found to exist in substantially the same proportions among the graduate teacher trainees. It was found that the type of understanding sought in this study was not enhanced merely by taking more mathematics courses. The problem of recycling of attitudes and mathematical misconceptions within the secondary teaching structure is addressed, and the question of the approach to the study of mathematics at both secondary and tertiary level is raised as a matter of continuing concern