The mathematical background of proving processes in discrete optimization -Exemplification with Research Situations for the Classrooms

Discrete mathematics brings interesting problems to teach and learn proof with accessible objects such as integers (arithmetic), graphs (modeling, order) or polyominoes (geometry). A lot of still open problems can be explained to a large public. The objects can be manipulated by simple dynamic operations (removing, adding, "gluing", contracting, splitting, decomposing, etc.). All these operations can be seen as tools for proving. This article particularly explores the field of "discrete optimization". A theoretical background is defined by taking two main axes into account: the epistemological analysis of discrete problems studied by contemporary researchers in discrete optimization and the design of adidactical situations for classrooms in the frame of the Theory of Didactical Situations. Two problems coming from ongoing research in discrete optimization (the Pentamino Exclusion and the Eight Queens problems) are developed. They underscore the learning potentialities of discrete mathematics and epistemological obstacles about proving processes. They emphasize the understanding of a necessary condition and a sufficient condition and problematize the difference between optimal and optimum. They provide proofs involving partitioning strategies, greedy algorithms but also primal-dual methods leading to the concept of duality. The way such problems can be implemented in the classrooms is described in a collaborative work between mathematicians and mathematics education researchers (Maths à Modeler Research Federation) through the Research Situations for the Classrooms

The mathematical background of proving processes in discrete optimization - Exemplification with Research Situations for the Classroom

International audienceDiscrete mathematics brings interesting problems for teaching and learning proof, with accessible objects such as integers (arithmetic), graphs (modeling, order) or polyominoes (geometry). Many problems that are still open can be explained to a large public. The objects can be manipulated by simple dynamic operations (removing, adding, ‘gluing’, contracting, splitting, decomposing, etc.). All these operations can be seen as tools for proving. In this paper we particularly explore the field of ‘discrete optimization’. A theoretical background is defined by taking two main axes into account, namely, the epistemological analysis of discrete problems studied by contemporary researchers in discrete optimization and the design of adidactical situations for classrooms in the frame of the Theory of Didactical Situations. Two problems coming from ongoing research in discrete optimization (the Pentamino Exclusion and the Eight Queens problems) are developed and transposed for the classroom. They underscore the learning potentialities of discrete mathematics and epistemological obstacles concerning proving processes. They emphasize the understanding of a necessary condition and a sufficient condition and problematize the difference between optimal and optimum. They provide proofs involving partitioning strategies, greedy algorithms but also primal–dual methods leading to the concept of duality. The way such problems can be implemented in the classroom is described in a collaborative study by mathematicians and mathematics education researchers (Maths à Modeler Research Federation) through the Research Situations for the Classroom project

The mathematical background of proving processes in discrete optimization -Exemplification with Research Situations for the Classrooms

Discrete mathematics brings interesting problems to teach and learn proof with accessible objects such as integers (arithmetic), graphs (modeling, order) or polyominoes (geometry). A lot of still open problems can be explained to a large public. The objects can be manipulated by simple dynamic operations (removing, adding, "gluing", contracting, splitting, decomposing, etc.). All these operations can be seen as tools for proving. This article particularly explores the field of "discrete optimization". A theoretical background is defined by taking two main axes into account: the epistemological analysis of discrete problems studied by contemporary researchers in discrete optimization and the design of adidactical situations for classrooms in the frame of the Theory of Didactical Situations. Two problems coming from ongoing research in discrete optimization (the Pentamino Exclusion and the Eight Queens problems) are developed. They underscore the learning potentialities of discrete mathematics and epistemological obstacles about proving processes. They emphasize the understanding of a necessary condition and a sufficient condition and problematize the difference between optimal and optimum. They provide proofs involving partitioning strategies, greedy algorithms but also primal-dual methods leading to the concept of duality. The way such problems can be implemented in the classrooms is described in a collaborative work between mathematicians and mathematics education researchers (Maths à Modeler Research Federation) through the Research Situations for the Classrooms

The mathematical background of proving processes in discrete optimization -Exemplification with Research Situations for the Classrooms

Discrete mathematics brings interesting problems to teach and learn proof with accessible objects such as integers (arithmetic), graphs (modeling, order) or polyominoes (geometry). A lot of still open problems can be explained to a large public. The objects can be manipulated by simple dynamic operations (removing, adding, "gluing", contracting, splitting, decomposing, etc.). All these operations can be seen as tools for proving. This article particularly explores the field of "discrete optimization". A theoretical background is defined by taking two main axes into account: the epistemological analysis of discrete problems studied by contemporary researchers in discrete optimization and the design of adidactical situations for classrooms in the frame of the Theory of Didactical Situations. Two problems coming from ongoing research in discrete optimization (the Pentamino Exclusion and the Eight Queens problems) are developed. They underscore the learning potentialities of discrete mathematics and epistemological obstacles about proving processes. They emphasize the understanding of a necessary condition and a sufficient condition and problematize the difference between optimal and optimum. They provide proofs involving partitioning strategies, greedy algorithms but also primal-dual methods leading to the concept of duality. The way such problems can be implemented in the classrooms is described in a collaborative work between mathematicians and mathematics education researchers (Maths à Modeler Research Federation) through the Research Situations for the Classrooms

MATHS À MODELER: RESEARCH-SITUATIONS FOR TEACHING MATHEMATICS

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Algorithmique et apprentissage de la preuve

International audienceDans cet article, les auteurs membres de l'équipe "Maths à Modeler", présentent tout d'abord une étude épistémologique sur la place et le rôle de l'algorithme dans la science mathématique. Ils étudient les différents aspects de l'algorithme suivant une dichotomie outil-objet, puis développent le lien privilégié qu'il entretient avec la preuve. En s'appuyant sur cette étude, une analyse des programmes du lycée et des manuels sont proposées.Dans un troisième temps, une situation de recherche en classe mettant en jeu l'algorithme est présentée. Les résultats d'expérimentations de cette situation montrent comment la construction d'algorithmes, leur preuve et l'analyse de leur complexité peuvent être questionnées en classe

Common occurrence of antibacterial agents in human intestinal microbiota

International audienceLaboratory experiments have revealed many active mechanisms by which bacteria can inhibit the growth of other organisms. Bacteriocins are a diverse group of natural ribosomally synthesized antimicrobial peptides produced by a wide range of bacteria and which seem to play an important role in mediating competition within bacterial communities. In this study, we have identified and established the structural classification of putative bacteriocins encoded by 317 microbial genomes in the human intestine. On the basis of homologies to available bacteriocin sequences, mainly from lactic acid bacteria, we report the widespread occurrence of bacteriocins across the gut microbiota: 175 bacteriocins were found to be encoded in Firmicutes, 79 in Proteobacteria, 34 in Bacteroidetes, and 25 in Actinobacteria. Bacteriocins from gut bacteria displayed wide differences among phyla with regard to class distribution, net positive charge, hydrophobicity and secondary structure, but the α-helix was the most abundant structure. The peptide structures and physiochemical properties of bacteriocins produced by the most abundant bacteria in the gut, the Firmicutes and the Bacteroidetes, seem to ensure low antibiotic activity and participate in permanent intestinal host defense against the proliferation of harmful bacteria. Meanwhile, the potentially harmful bacteria, including the Proteobacteria, displayed highly effective bacteriocins, probably supporting the virulent character of diseases. These findings highlight the eventual role played by bacteriocins in gut microbial competition and their potential place in antibiotic therapy

Intrafamilial Circulation of Tropheryma whipplei, France

has been detected in 4 % of fecal samples from the general adult population of France. To identify T. whipplei within families, we conducted serologic and molecular studies, including genotyping, on saliva, feces, and serum from 74 relatives of 13 patients with classic Whipple disease, 5 with localized chronic T. whipplei infection, and 3 carriers. Seroprevalence was determined by Western blot and compared with 300 persons from the general population. We detected T. whipplei in 24 (38%) of 64 fecal samples and 7 (10%) of 70 saliva samples from relatives but found no difference between persons related by genetics and marriage. The same circulating genotype occurred signifi cantly more often in families than in other persons. Seroprevalence was higher among relatives (23 [77%] of 30) than in the general population (143 [48%] of 300). The high prevalence of T. whipplei within families suggests intrafamilial circulation. Whipple disease, a rare sporadic disease, was first considered a metabolic disease (1) and later suspected to be an infectious disease caused by a rare bacterium, Tropheryma whipplei (2). However, the causative bacterium is common (3–5), and the well-known and classic form of Whipple disease (characterized by periodic acid–Schiff-stained bacilli in infected smallbowel macrophages) represents only 1 rare clinical form of infection caused by T. whipplei (6,7). In the absence of intestinal lesions, T. whipplei is involved in subacute or chronic infections, such as endocarditis (8), encephalitis (2), uveitis (9,10), adenopathy (2), and osteoarticular infections (2,11). Recently, T. whipplei was reported to cause acute infections, such as pneumonia (12,13), gastroenteritis (14,15), and bacteremia (16). Asymptomatic carriers have been identified for whom T. whipplei prevalence varied Author affi litaion: Université de la Méditerranée, Marseille, France DOI

Relationships between empathy and creativity in collective games: a comparison between handball and sitting ball

In collective motor situations, creativity and empathy are central and strongly connected to cognitive and affective processes. Indeed, in the environment of high social uncertainty of games and sports, empathy would allow the player to anticipate motor behaviors in order to promote creative decision-making, i.e., to destabilize his opponents. On this basis, this study pursues two objectives. The first is to propose indicators to question the links between sociomotor empathy and motor creativity in an ecological situation. The second is to investigate the potential influence of the internal logic of two very different collective games (handball and Sitting ball) on the type of links that are woven between empathy and creativity. Two groups of students were recruited (n = 22 and 23) and participated in each of the games mentioned. The games were video recorded. The praxical communications made by each player were recorded and sorted by two trained observers. The results revealed major differences between the two studied collective games. In handball, there was a correlation between instrumental empathy (valuing cognitive aspects) and indicators of motor creativity (p  < 0.05). The more creative the players are (quantity, diversity and quality of performance), the more they manage to accurately anticipate the behavior of other players. In Sitting Ball, there was no correlation between creativity indicators and instrumental empathy. On the other hand, it is noticed that instrumental empathy was correlated with socio-affective empathy (p  < 0.001). To make their motor decisions, the players do not rely exclusively on the decoding of behaviors but significantly mobilize the feelings that they ascribe to the other co-participants. The results of this work invite reflection on the diversity of playful reading grids to be offered to students in order to develop their motor adaptability