On the relations between historical epistemology and students’ conceptual developments in mathematics

There is an ongoing discussion within the research field of mathematics education regarding the utilization of the history of mathematics within mathematics education. In this paper we consider problems that may emerge when the historical epistemology of mathematics is paralleled to students’ conceptual developments in mathematics. We problematize this attempt to link the two fields on the basis of Grattan-Guinness’ distinction between “history” and “heritage”. We argue that when parallelism claims are made, history and heritage are often mixed up, which is problematic since historical mathematical definitions must be interpreted in its proper historical context and conceptual framework. Furthermore, we argue that cultural and local ideas vary at different time periods, influencing conceptual developments in different directions regardless of whether historical or individual developments are considered, and thus it may be problematic to uncritically assume a platonic perspective. Also, we have to take into consideration that an average student of today and great mathematicians of the past are at different cognitive levels

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

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

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

Notes for a history of the teaching of algebra

Abundant literature is available on the history of algebra. However, the history of the teaching of algebra is largely unwritten, and as such, this chapter essentially constitutes some notes that are intended to be useful for future research on this subject. As well as the scarcity of the works published on the topic, there is the added difficulty of drawing the line between the teaching of algebra and the teaching of arithmetic—two branches of knowledge whose borders have varied over time (today one can consider the arithmetic with the four operations and their algorithms and properties taught in schools as nothing more than a small chapter of algebra). As such, we will be very brief in talking about the more distant epochs, from which we have some mathematics documents but little information on how they were used in teaching. We aim to be more explicit as we travel forwards into the different epochs until modern times. We finish, naturally, with some reflections on the present-day and future situation regarding the teaching of algebra.info:eu-repo/semantics/publishedVersio

Catch-Up Growth Following Fetal Growth Restriction Promotes Rapid Restoration of Fat Mass but Without Metabolic Consequences at One Year of Age

BACKGROUND: Fetal growth restriction (FGR) followed by rapid weight gain during early life has been suggested to be the initial sequence promoting central adiposity and insulin resistance. However, the link between fetal and early postnatal growth and the associated anthropometric and metabolic changes have been poorly studied. METHODOLOGY/PRINCIPAL FINDINGS: Over the first year of post-natal life, changes in body mass index, skinfold thickness and hormonal concentrations were prospectively monitored in 94 infants in whom the fetal growth velocity had previously been measured using a repeated standardized procedure of ultrasound fetal measurements. 45 infants, thinner at birth, had experienced previous FGR (FGR+) regardless of birth weight. Growth pattern in the first four months of life was characterized by greater change in BMI z-score in FGR+ (+1.26+/-1.2 vs +0.58 +/-1.17 SD in FGR-) resulting in the restoration of BMI and of fat mass to values similar to FGR-, independently of caloric intakes. Growth velocity after 4 months was similar and BMI z-score and fat mass remained similar at 12 months of age. At both time-points, fetal growth velocity was an independent predictor of fat mass in FGR+. At one year, fasting insulin levels were not different but leptin was significantly higher in the FGR+ (4.43+/-1.41 vs 2.63+/-1 ng/ml in FGR-). CONCLUSION: Early catch-up growth is related to the fetal growth pattern itself, irrespective of birth weight, and is associated with higher insulin sensitivity and lower leptin levels after birth. Catch-up growth promotes the restoration of body size and fat stores without detrimental consequences at one year of age on body composition or metabolic profile. The higher leptin concentration at one year may reflect a positive energy balance in children who previously faced fetal growth restriction

Effects of rapid urbanisation on the urban thermal environment between 1990 and 2011 in Dhaka Megacity, Bangladesh

This study investigates the influence of land-use/land-cover (LULC) change on land surface temperature (LST) in Dhaka Megacity, Bangladesh during a period of rapid urbanisation. LST was derived from Landsat 5 TM scenes captured in 1990, 2000 and 2011 and compared to contemporaneous LULC maps. We compared index-based and linear spectral mixture analysis (LSMA) techniques for modelling LST. LSMA derived biophysical parameters corresponded more strongly to LST than those produced using index-based parameters. Results indicated that vegetation and water surfaces had relatively stable LST but it increased by around 2 °C when these surfaces were converted to built-up areas with extensive impervious surfaces. Knowledge of the expected change in LST when one land-cover is converted to another can inform land planners of the potential impact of future changes and urges the development of better management strategies

Hepatic alterations are accompanied by changes to bile acid transporter-expressing neurons in the hypothalamus after traumatic brain injury

Annually, there are over 2 million incidents of traumatic brain injury (TBI) and treatment options are non-existent. While many TBI studies have focused on the brain, peripheral contributions involving the digestive and immune systems are emerging as factors involved in the various symptomology associated with TBI. We hypothesized that TBI would alter hepatic function, including bile acid system machinery in the liver and brain. The results show activation of the hepatic acute phase response by 2 hours after TBI, hepatic inflammation by 6 hours after TBI and a decrease in hepatic transcription factors, Gli 1, Gli 2, Gli 3 at 2 and 24 hrs after TBI. Bile acid receptors and transporters were decreased as early as 2 hrs after TBI until at least 24 hrs after TBI. Quantification of bile acid transporter, ASBT-expressing neurons in the hypothalamus, revealed a significant decrease following TBI. These results are the first to show such changes following a TBI, and are compatible with previous studies of the bile acid system in stroke models. The data support the emerging idea of a systemic influence to neurological disorders and point to the need for future studies to better define specific mechanisms of action