Concept image and concept definition in mathematics with particular reference to limits and continuity

The concept image consists of all the cognitive structure in the individual's mind that is associated with a given concept. This may not be globally coherent and may have aspects which are quite different from the formal concept definition. The development of limits and continuity, as taught in secondary school and university, are considered. Various investigations are reported which demonstrate individual concept images differing from the formal theory and containing factors which cause cognitive conflict

Student perspectives on the relationship between a curve and its tangent in the transition from Euclidean Geometry to Analysis

The tangent line is a central concept in many mathematics and science courses. In this paper we describe a model of students’ thinking – concept images as well as ability in symbolic manipulation – about the tangent line of a curve as it has developed through students’ experiences in Euclidean Geometry and Analysis courses. Data was collected through a questionnaire administered to 196 Year 12 students. Through Latent Class Analysis, the participants were classified in three hierarchical groups representing the transition from a Geometrical Global perspective on the tangent line to an Analytical Local perspective. In the light of this classification, and through qualitative explanations of the students’ responses, we describe students’ thinking about tangents in terms of seven factors. We confirm the model constituted by these seven factors through Confirmatory Factor Analysis

Students’ Evolving Meaning About Tangent Line with the Mediation of a Dynamic Geometry Environment and an Instructional Example Space

In this paper I report a lengthy episode from a teaching experiment in which fifteen Year 12 Greek students negotiated their definitions of tangent line to a function graph. The experiment was designed for the purpose of introducing students to the notion of derivative and to the general case of tangent to a function graph. Its design was based on previous research results on students’ perspectives on tangency, especially in their transition from Geometry to Analysis. In this experiment an instructional example space of functions was used in an electronic environment utilising Dynamic Geometry software with Function Grapher tools. Following the Vygotskian approach according to which students’ knowledge develops in specific social and cultural contexts, students’ construction of the meaning of tangent line was observed in the classroom throughout the experiment. The analysis of the classroom data collected during the experiment focused on the evolution of students’ personal meanings about tangent line of function graph in relation to: the electronic environment; the pre-prepared as well as spontaneous examples; students’ engagement in classroom discussion; and, the role of researcher as a teacher. The analysis indicated that the evolution of students’ meanings towards a more sophisticated understanding of tangency was not linear. Also it was interrelated with the evolution of the meaning they had about the inscriptions in the electronic environment; the instructional example space; the classroom discussion; and, the role of the teacher

What Should We Expect from Somebody Who Teaches Mathematics in Elementary Schools?

Článek je rozšířenou verzí plenární přednášky na konferenci SEMT 11 (Vinner, 2011). Jsou zde rozpracovány některé otázky, kterým se autor ve své přednášce z časových důvodů nemohl věnovat. Autor doporučuje, aby kromě znalostí matematiky potřebných pro výuku matematiky na 1. stupni základní školy byla dostatečná pozornost věnována také dalším aspektům práce učitele. Doporučuje, aby důvody pro zařazení matematiky jako povinného předmětu pro celou populaci, stejně jako základní cíle vzdělávání byly diskutovány s učiteli. Doporučuje také, aby z přípravy učitelů byla vyřazena témata, která jsou vně ZPD (Zone of Proximal Development – zóny nejbližšího vývoje).This paper is an extension of my plenary talk in SEMT 11 (Vinner, 2011) in which I promised to elaborate on some issues that I could not discuss at the talk because of time restrictions. The paper recommends that, in addition to the mathematical knowledge needed for teaching in elementary schools, attention should be given to other aspects of the teacher’s work. It is suggested that the rationale of teaching mathematics as a compulsory subject to the entire population should be discussed with the teachers as well as the ultimate goals of education. The paper recommends also avoiding in teacher training topics which are beyond the zone of proximal development (ZPD) of the teachers

Terminal Pleistocene Alaskan genome reveals first founding population of Native Americans

Despite broad agreement that the Americas were initially populated via Beringia, the land bridge that connected far northeast Asia with northwestern North America during the Pleistocene epoch, when and how the peopling of the Americas occurred remains unresolved. Analyses of human remains from Late Pleistocene Alaska are important to resolving the timing and dispersal of these populations. The remains of two infants were recovered at Upward Sun River (USR), and have been dated to around 11.5 thousand years ago (ka). Here, by sequencing the USR1 genome to an average coverage of approximately 17 times, we show that USR1 is most closely related to Native Americans, but falls basal to all previously sequenced contemporary and ancient Native Americans. As such, USR1 represents a distinct Ancient Beringian population. Using demographic modelling, we infer that the Ancient Beringian population and ancestors of other Native Americans descended from a single founding population that initially split from East Asians around 36 ± 1.5 ka, with gene flow persisting until around 25 ± 1.1 ka. Gene flow from ancient north Eurasians into all Native Americans took place 25–20 ka, with Ancient Beringians branching off around 22–18.1 ka. Our findings support a long-term genetic structure in ancestral Native Americans, consistent with the Beringian ‘standstill model’. We show that the basal northern and southern Native American branches, to which all other Native Americans belong, diverged around 17.5–14.6 ka, and that this probably occurred south of the North American ice sheets. We also show that after 11.5 ka, some of the northern Native American populations received gene flow from a Siberian population most closely related to Koryaks, but not Palaeo-Eskimos, Inuits or Kets, and that Native American gene flow into Inuits was through northern and not southern Native American groups. Our findings further suggest that the far-northern North American presence of northern Native Americans is from a back migration that replaced or absorbed the initial founding population of Ancient Beringians

Engineering bacteriophage encapsulation processes to improve stability and controlled release using pH responsive formulations

Enteric pathogens form a large part of infectious diseases which contribute to a bulk of the healthcare costs. Enteric infections are usually contracted via the faecal-oral route or through contact with contaminated surfaces. Treatment by antibiotics is becoming increasingly ineffective due to the growing number of antibiotic resistant strains. Anti-microbial resistance poses a serious threat to the future of healthcare worldwide and necessitates the search for alternate forms of therapy. Bacteriophages (phages), are viruses which specifically infect and lyse bacteria. To introduce phages as a viable form of therapy, route of administration needs to be considered carefully. Model phages with broad host ranges are ideal for therapy however oral delivery to the lower gastro-intestinal (GI) poses several challenges. The acidic stomach environment can be detrimental to phages, rendering them inactive during passage. To overcome this challenge and improve the stability of phage during encapsulation and storage, this PhD research has been conducted. pH responsive polymers, Eudragit and alginate were used to develop composite microparticles which protected phage from acidic pH (pH 1-3). A novel method of acidifying oil was developed for crosslinking droplets in vitro to avoid the use of harsh solvent systems that can cause phage inactivation. Platform microfluidic technology was employed for phage encapsulation for the first time. Monodispersed droplets and particles were produced, offering fine-tuning of droplet diameter to tailor the release and pH protection of encapsulated phage. Process scale-up was attempted using membrane emulsification (ME) to produce larger volumes of encapsulated phage. In vitro and in-situ models investigated the efficacy of encapsulated phage-bacterial killing. Industrial scale method of spray drying, and electrospinning were also used to demonstrate the versatility of the formulation. Tableting dry powder phage, showed an effective method for producing solid dosage forms for therapy. Additionally, electrospun phage fibres also showed the potential use of pH responsive formulations in addressing wound infections. Improvement in encapsulated phage storage stability was observed with the addition of trehalose in the formulation. This research underpins the need for testing phage encapsulation for site-specific delivery and offers insight into the potential use of commercially available technologies

AMERICA, ARGUS og Niels

Max Vinner er velkendt blandt træskibsejere blandt andet gennem sin virksomhed som redaktør for T.S.-bladet. Max Vinner har desuden i mange år været en af hovedmændene i bådlauget, der sejler med Nordlandsbåden »Rana«, hvor man gennem sejladsforsøg har indvundet nye erfaringer vedrørende sejlegenskaberne hos vikingetidens fartøjer. Christian Nielsen var meget interesseret i de 3 omtalte fartøjers fælles træk, og Max Vinner har i sin artikel samlet og udnyttet, Christians notater, optegnelser, skitser og tegninger