Diagnosing students' difficulties in learning mathematics

This study considers the results of a diagnostic test of student difficulty and contrasts the difference in performance between the lower attaining quartile and the higher quartile. It illustrates a difference in qualitative thinking between those who succeed and those who fail in mathematics, illustrating a theory that those who fail are performing a more difficult type of mathematics (coordinating procedures) than those who succeed (manipulating concepts). Students who have to coordinate or reverse processes in time will encounter far greater difficulty than those who can manipulate symbols in a flexible way. The consequences of such a dichotomy and implications for remediation are then considered

The fundamental cycle of concept construction underlying various theoretical frameworks

In this paper, the development of mathematical concepts over time is considered. Particular reference is given to the shifting of attention from step-by-step procedures that are performed in time, to symbolism that can be manipulated as mental entities on paper and in the mind. The development is analysed using different theoretical perspectives, including the SOLO model and various theories of concept construction to reveal a fundamental cycle underlying the building of concepts that features widely in different ways of thinking that occurs throughout mathematical learning

Stevin numbers and reality

We explore the potential of Simon Stevin's numbers, obscured by shifting foundational biases and by 19th century developments in the arithmetisation of analysis.Comment: 22 pages, 4 figures. arXiv admin note: text overlap with arXiv:1104.0375, arXiv:1108.2885, arXiv:1108.420

The cetaceans of Guinea, a first check-list of documented species. Scientific Committee document SC/58/O15, International Whaling Commission, May-June 2006, St. Kitts

A CMS workshop on West African Cetacea (Conakry, May 2000), called for i.a. ‘carrying out .. inventory of cetacean species; collection, treatment and compilation of data for each state.’ The present paper is a preliminary faunal checklist of cetaceans occurring in Guinea’s EEZ. Information was gleaned from strandings, bycatches, scientific and opportunistic sightings and a literature review. Ten species are included for which supporting voucher material and data were available for examination. These are, three baleen whales: Balaenoptera brydei, Balaenoptera acutorostrata and Megaptera novaeangliae; and seven species of odontocetes: Kogia breviceps, Tursiops truncatus, Sousa teuszii, Stenella frontalis, Delphinus delphis, Steno bredanensis and Globicephala macrorhynchus. Another two species, Physeter macrocephalus and Stenella attenuate were sighted off Guinea but no photographic evidence was obtained. The current account is thought to reflect an incomplete picture of Guinea’s cetacean biodiversity. Future surveys are expected to update and investigate spatial and temporal distribution patterns for each species along Guinea’s coast. A few bycatches landed by artisanal fishers were utilised locally, but there are no signs of any substantial captures. Nonetheless, monitoring should be continued. The set-up of a national reference collection and database is recommended. The population identities of the encountered Atlantic humpback dolphin, minke whale and humpback whale are of particular interest

Genome sequence of an Enterobacter helveticus strain, 1159/04 (= LMG 23733), isolated from fruit powder

We report the draft genome sequence of Enterobacter helveticus strain LMG 23733, isolated from fruit powder. The draft genome assembly for E. helveticus strain LMG 23733 has a size of 4,635,476 bp and a G+C content of 55.9%

Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond

Many historians of the calculus deny significant continuity between infinitesimal calculus of the 17th century and 20th century developments such as Robinson's theory. Robinson's hyperreals, while providing a consistent theory of infinitesimals, require the resources of modern logic; thus many commentators are comfortable denying a historical continuity. A notable exception is Robinson himself, whose identification with the Leibnizian tradition inspired Lakatos, Laugwitz, and others to consider the history of the infinitesimal in a more favorable light. Inspite of his Leibnizian sympathies, Robinson regards Berkeley's criticisms of the infinitesimal calculus as aptly demonstrating the inconsistency of reasoning with historical infinitesimal magnitudes. We argue that Robinson, among others, overestimates the force of Berkeley's criticisms, by underestimating the mathematical and philosophical resources available to Leibniz. Leibniz's infinitesimals are fictions, not logical fictions, as Ishiguro proposed, but rather pure fictions, like imaginaries, which are not eliminable by some syncategorematic paraphrase. We argue that Leibniz's defense of infinitesimals is more firmly grounded than Berkeley's criticism thereof. We show, moreover, that Leibniz's system for differential calculus was free of logical fallacies. Our argument strengthens the conception of modern infinitesimals as a development of Leibniz's strategy of relating inassignable to assignable quantities by means of his transcendental law of homogeneity.Comment: 69 pages, 3 figure

Clinical and pathological kidney aspects of sickle cell anemia at Dakar: study of 11 cases of renal biopsies

Few studies are devoted to the practice of renal biopsy in sickle cell nephropathy; our objective was to determine the histological and evolutionary patterns of renal lesions in sickle cell patients who underwent renal biopsy in Dakar.Methods:This was a retrospective multicentric study (conducted from December 2009 to August 2011) on renal biopsies performed on sickle cell anaemic patients at the Nephrology Department of Teaching Hospital Aristide Le Dantec and the Albert Royer Childrens Hospital. The histological, therapeutic and evolutionary data were analysed.From the 292 total renal biopsies, 11 (3.80%) were performed on sickle cell patients (6SS, 1SBth + 4 AS) with a mean age of 23.1 [13-51 years]. Nephrotic syndrome was the indication of renal biopsy in all cases. Focal segmental glomerulosclerosis was the most frequent histological finding (five cases), followed by a combination of various specific lesions (hypertrophy of glomerular and peritubular capillaries), minimal glomerular lesions (three cases), membranoproliferative glomerulonephritis (two cases) and extra-membranous glomerulonephritis (one case). Complete remission after treatment was achieved in seven cases and one patient expired. Three patients did not continue with follow-up appointments.Conclusions:Renal biopsy is not very frequent in the course of sickle cell anaemia and in most cases it is performed because of nephrotic syndrome. The histological findings are diverse with a predominance of focal segmental glomerulosclerosis

A Burgessian critique of nominalistic tendencies in contemporary mathematics and its historiography

We analyze the developments in mathematical rigor from the viewpoint of a Burgessian critique of nominalistic reconstructions. We apply such a critique to the reconstruction of infinitesimal analysis accomplished through the efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy's foundational work associated with the work of Boyer and Grabiner; and to Bishop's constructivist reconstruction of classical analysis. We examine the effects of a nominalist disposition on historiography, teaching, and research.Comment: 57 pages; 3 figures. Corrected misprint

A Cauchy-Dirac delta function

The Dirac delta function has solid roots in 19th century work in Fourier analysis and singular integrals by Cauchy and others, anticipating Dirac's discovery by over a century, and illuminating the nature of Cauchy's infinitesimals and his infinitesimal definition of delta.Comment: 24 pages, 2 figures; Foundations of Science, 201

Ten Misconceptions from the History of Analysis and Their Debunking

The widespread idea that infinitesimals were "eliminated" by the "great triumvirate" of Cantor, Dedekind, and Weierstrass is refuted by an uninterrupted chain of work on infinitesimal-enriched number systems. The elimination claim is an oversimplification created by triumvirate followers, who tend to view the history of analysis as a pre-ordained march toward the radiant future of Weierstrassian epsilontics. In the present text, we document distortions of the history of analysis stemming from the triumvirate ideology of ontological minimalism, which identified the continuum with a single number system. Such anachronistic distortions characterize the received interpretation of Stevin, Leibniz, d'Alembert, Cauchy, and others.Comment: 46 pages, 4 figures; Foundations of Science (2012). arXiv admin note: text overlap with arXiv:1108.2885 and arXiv:1110.545