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

Confidence trick: the interpretation of confidence intervals

The frequent misinterpretation of the nature of confidence intervals by students has been well documented. This article examines the problem as an aspect of the learning of mathematical definitions and considers the tension between parroting mathematically rigorous, but essentially uninternalized, statements on the one hand and expressing imperfect but developing understandings on the other. A small-scale study among schoolteachers sought comments on four definitions expressing differing understandings of confidence intervals, and these are examined and discussed. The article concludes that some student wordings could be regarded as less inaccurate than they might seem at first sight and presents a case for accepting a wider range of more intuitive understandings as a work in progress

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

Immunological analysis of a Lactococcus lactis-based DNA vaccine expressing HIV gp120

For reasons of efficiency Escherichia coli is used today as the microbial factory for production of plasmid DNA vaccines. To avoid hazardous antibiotic resistance genes and endotoxins from plasmid systems used nowadays, we have developed a system based on the food-grade Lactococcus lactis and a plasmid without antibiotic resistance genes. We compared the L. lactis system to a traditional one in E. coli using identical vaccine constructs encoding the gp120 of HIV-1. Transfection studies showed comparable gp120 expression levels using both vector systems. Intramuscular immunization of mice with L. lactis vectors developed comparable gp120 antibody titers as mice receiving E. coli vectors. In contrast, the induction of the cytolytic response was lower using the L. lactis vector. Inclusion of CpG motifs in the plasmids increased T-cell activation more when the E. coli rather than the L. lactis vector was used. This could be due to the different DNA content of the vector backbones. Interestingly, stimulation of splenocytes showed higher adjuvant effect of the L. lactis plasmid. The study suggests the developed L. lactis plasmid system as new alternative DNA vaccine system with improved safety features. The different immune inducing properties using similar gene expression units, but different vector backbones and production hosts give information of the adjuvant role of the silent plasmid backbone. The results also show that correlation between the in vitro adjuvanticity of plasmid DNA and its capacity to induce cellular and humoral immune responses in mice is not straight forward

Teaching and Learning of Calculus

This survey focuses on the main trends in the field of calculus education. Despite their variety, the findings reveal a cornerstone issue that is strongly linked to the formalism of calculus concepts and to the difficulties it generates in the learning and teaching process. As a complement to the main text, an extended bibliography with some of the most important references on this topic is included. Since the diversity of the research in the field makes it difficult to produce an exhaustive state-of-the-art summary, the authors discuss recent developments that go beyond this survey and put forward new research questions

Doctoral students’ use of examples in evaluating and proving conjectures

The final publication is available at: www.springerlink.comThis paper discusses variation in reasoning strategies among expert mathematicians, with a particular focus on the degree to which they use examples to reason about general conjectures. We first discuss literature on the use of examples in understanding and reasoning about abstract mathematics, relating this to a conceptualisation of syntactic and semantic reasoning strategies relative to a representation system of proof. We then use this conceptualisation as a basis for contrasting the behaviour of two successful mathematics research students whilst they evaluated and proved number theory conjectures. We observe that the students exhibited strikingly different degrees of example use, and argue that previously observed individual differences in reasoning strategies may exist at the expert level. We conclude by discussing implications for pedagogy and for future research

At the intersection between the subject and the political: a contribution to an ongoing discussion

The issue of subjectivity has recently occasioned a lively discussion in this journal opposing socioculturalism and Lacanian psychoanalysis. By confronting Luis Radford’s cultural theory with Jacques Lacan’s psychoanalysis, Tony Brown sought to show the limitations of socioculturalism. This article takes advantage of that discussion to develop a critique of Radford’s theory of objectification, taken as an exemplary sociocultural theorization of the teaching and learning of mathematics. It does so by extending the criticism made by Brown at the level of the subject, namely by showing what is lost in socioculturalism when it reduces the Hegelian notion of dialectics to a relation between constituted entities; but mostly by exploring the possibility opened by contemporary theory to posit the discussion around subjectivity in the political. While socioculturalism assumes the possibility of a synthesis between person and culture thus making education possible, it will be argued that a theory which assumes the impossibility of education is in a better position to, on the one hand, conceptualize the resistance of many towards the learning of mathematics, and on the other hand, to address the ongoing political failure in achieving the desired goal of “mathematics for all”