Bridging knowing and proving in mathematics An essay from a didactical perspective

Text of a talk at the conference "Explanation and Proof in Mathematics: Philosophical and Educational Perspective" held in Essen in November 2006International audienceThe learning of mathematics starts early but remains far from any theoretical considerations: pupils' mathematical knowledge is first rooted in pragmatic evidence or conforms to procedures taught. However, learners develop a knowledge which they can apply in significant problem situations, and which is amenable to falsification and argumentation. They can validate what they claim to be true but using means generally not conforming to mathematical standards. Here, I analyze how this situation underlies the epistemological and didactical complexities of teaching mathematical proof. I show that the evolution of the learners' understanding of what counts as proof in mathematics implies an evolution of their knowing of mathematical concepts. The key didactical point is not to persuade learners to accept a new formalism but to have them understand how mathematical proof and statements are tightly related within a common framework; that is, a mathematical theory. I address this aim by modeling the learners' way of knowing in terms of a dynamic, homeostatic system. I discuss the roles of different semiotic systems, of the types of actions the learners perform and of the controls they implement in constructing or validating knowledge. Particularly with modern technological aids, this model provides a basis designing didactical situations to help learners bridge the gap between pragmatics and theory

Crossover from spin accumulation into interface states to spin injection in the germanium conduction band

Electrical spin injection into semiconductors paves the way for exploring new phenomena in the area of spin physics and new generations of spintronic devices. However the exact role of interface states in spin injection mechanism from a magnetic tunnel junction into a semiconductor is still under debate. In this letter, we demonstrate a clear transition from spin accumulation into interface states to spin injection in the conduction band of $n$-Ge. We observe spin signal amplification at low temperature due to spin accumulation into interface states followed by a clear transition towards spin injection in the conduction band from 200 K up to room temperature. In this regime, the spin signal is reduced down to a value compatible with spin diffusion model. More interestingly, we demonstrate in this regime a significant modulation of the spin signal by spin pumping generated by ferromagnetic resonance and also by applying a back-gate voltage which are clear manifestations of spin current and accumulation in the germanium conduction band.Comment: 5 pages, 4 figure

On-line identification of temperature-dependent thermal conductivity

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MLVA-16 typing of 295 marine mammal <i>Brucella</i> isolates from different animal and geographic origins identifies 7 major groups within <i>Brucella ceti</i> and <i>Brucella pinnipedialis</i>

BackgroundSince 1994, Brucella strains have been isolated from a wide range of marine mammals. They are currently recognized as two new Brucella species, B. pinnipedialis for the pinniped isolates and B. ceti for the cetacean isolates in agreement with host preference and specific phenotypic and molecular markers. In order to investigate the genetic relationships within the marine mammal Brucella isolates and with reference to terrestrial mammal Brucella isolates, we applied in this study the Multiple Loci VNTR (Variable Number of Tandem Repeats) Analysis (MLVA) approach. A previously published assay comprising 16 loci (MLVA-16) that has been shown to be highly relevant and efficient for typing and clustering Brucella strains from animal and human origin was used.Results294 marine mammal Brucella strains collected in European waters from 173 animals and a human isolate from New Zealand presumably from marine origin were investigated by MLVA-16. Marine mammal Brucella isolates were shown to be different from the recognized terrestrial mammal Brucella species and biovars and corresponded to 3 major related groups, one specific of the B. ceti strains, one of the B. pinnipedialis strains and the last composed of the human isolate. In the B. ceti group, 3 subclusters were identified, distinguishing a cluster of dolphin, minke whale and porpoise isolates and two clusters mostly composed of dolphin isolates. These results were in accordance with published analyses using other phenotypic or molecular approaches, or different panels of VNTR loci. The B. pinnipedialis group could be similarly subdivided in 3 subclusters, one composed exclusively of isolates from hooded seals (Cystophora cristata) and the two others comprising other seal species isolates.ConclusionThe clustering analysis of a large collection of marine mammal Brucella isolates from European waters significantly strengthens the current view of the population structure of these two species, and their relative position with respect to the rest of the Brucella genus. MLVA-16 is confirmed as being a rapid, highly discriminatory and reproducible method to classify Brucella strains including the marine mammal isolates. The Brucella2009 MLVA-16 genotyping database available at http://mlva.u-psud.fr is providing a detailed coverage of all 9 currently recognized Brucella species

The French Didactic Tradition in Mathematics

This chapter presents the French didactic tradition. It first describes theemergence and development of this tradition according to four key features (role ofmathematics and mathematicians, role of theories, role of design of teaching andlearning environments, and role of empirical research), and illustrates it through two case studies respectively devoted to research carried out within this traditionon algebra and on line symmetry-reflection. It then questions the influence of thistradition through the contributions of four researchers from Germany, Italy, Mexicoand Tunisia, before ending with a short epilogue

Brucella spp. of amphibians comprise genomically diverse motile strains competent for replication in macrophages and survival in mammalian hosts

Twenty-one small Gram-negative motile coccobacilli were isolated from 15 systemically diseased African bullfrogs (Pyxicephalus edulis), and were initially identified as Ochrobactrum anthropi by standard microbiological identification systems. Phylogenetic reconstructions using combined molecular analyses and comparative whole genome analysis of the most diverse of the bullfrog strains verified affiliation with the genus Brucella and placed the isolates in a cluster containing B. inopinata and the other non-classical Brucella species but also revealed significant genetic differences within the group. Four representative but molecularly and phenotypically diverse strains were used for in vitro and in vivo infection experiments. All readily multiplied in macrophage-like murine J774-cells, and their overall intramacrophagic growth rate was comparable to that of B. inopinata BO1 and slightly higher than that of B. microti CCM 4915. In the BALB/c murine model of infection these strains replicated in both spleen and liver, but were less efficient than B. suis 1330. Some strains survived in the mammalian host for up to 12 weeks. The heterogeneity of these novel strains hampers a single species description but their phenotypic and genetic features suggest that they represent an evolutionary link between a soil-associated ancestor and the mammalian host-adapted pathogenic Brucella species