Connecting theories in mathematics education: challenges and possibilities

This paper is a commentary on the problem of networking theories. My commentary draws on the papers contained in this ZDM issue and is divided into three parts. In the first part, following semiotician Yuri Lotman, I suggest that a network of theories can be conceived of as a semiosphere, i.e., a space of encounter of various languages and intellectual traditions. I argue that such a networking space revolves around two different and complementary “themes”—integration and differentiation. In the second part, I advocate conceptualizing theories in mathematics education as triplets formed by a system of theoretical principles, a methodology, and templates of research questions, and attempt to show that this tripartite view of theories provides us with a morphology of theories for investigating differences and potential connections. In the third part of the article, I discuss some examples of networking theories. The investigation of limits of connectivity leads me to talk about the boundary of a theory, which I suggest defining as the “limit” of what a theory can legitimately predicate about its objects of discourse; beyond such an edge, the theory conflicts with its own principles. I conclude with some implications of networking theories for the advancement of mathematics education.     

Generalized double crossproducts associated with the quantized enveloping algebras

Abstract

In the present paper we consider a correspondence weaker than Galois connection and prove that this produces Kurosh–Amitsur radicals in a very general setting including all universal classes of Ω-groups. As a framework we introduce a simple combinatorial structure which uses mappings between complete lattices.     

Crystal blocking measurements of the time delay of fission induced by 32S, 48Ti, and 58Ni bombardment of W

The time delay in fission induced by bombardment of W with 180 MeV 32S, 240-255 MeV 48Ti, and 315-375 MeV 58Ni has been measured by observation of crystal blocking. There is a clear narrowing and a small increase in the minimum yield of the angular dips for fission compared with scaled dips for elastically scattered ions. This is interpreted as a fission delay of about 2 as, only weakly dependent on energy and atomic number. The delay is longer by 1 to 2 orders of magnitude than obtained from standard interpretations of measurements of prescission neutrons and giant-dipole-resonance gamma rays and from calculations of the nuclear dynamics in heavy-ion reactions.     

Quantifying heterogeneity and conformational dynamics from single molecule FRET of diffusing molecules: recurrence analysis of single particles (RASP)

Single molecule F?rster resonance energy transfer (FRET) experiments are a versatile method for investigating the conformational distributions and dynamics of biological macromolecules. In a common type of experiment, the fluorescence bursts from individual molecules freely diffusing in solution are detected as they pass through the observation volume of a confocal microscope. Correlation analysis of the fluorescence bursts shows that under typical experimental conditions, for time scales up to several tens of milliseconds, the probability for a molecule to return to the confocal volume is greater than the probability of a new molecule being detected. Here we present RASP (recurrence analysis of single particles), a method that is based on this recurrence behavior and allows us to significantly extend the information that can be extracted from single molecule FRET experiments. The number and peak shapes of subpopulations within the sample can be identified essentially in a model-free way by constructing recurrence FRET efficiency histograms. These are obtained by first selecting photon bursts from a small transfer efficiency range (initial bursts), and then building the FRET efficiency histogram only from bursts detected within a short time (the recurrence interval) after the initial bursts. Systematic variation of the recurrence interval allows the kinetics of interconversion between subpopulations to be determined on time scales from ~50 μs up to ~100 ms from equilibrium measurements. We demonstrate the applicability of the method on measurements of several peptides and proteins with different degrees of conformational heterogeneity and folding dynamics. The concepts presented here can be extended to other observables available from single molecule fluorescence experiments.     

Comparison of active and passive environmental sampling for safeguards applications

Towards development of electrical conductivity based sensors for online detection of formation of third phase during reprocessing of Pu rich spent nuclear fuels, laboratory studies were carried out using U⁴⁺ solutions in various experimental conditions. Third phases were generated by extracting U⁴⁺ from nitric acid medium by 1.09 M TBP at different A/O (A: aqueous, O: organic) ratios. The results of this study indicate that the third phase is nearly 100–300 times more conducting than lean organic phase and saturated phase. The higher conductivity of third phase as compared to that of other two phases is explained based on the principle of reverse micelle formation and charge movement between the micro emulsion globules by percolation phenomenon.     

Monitoring oligomer formation from self-aggregating amylin peptides using ESI-IMS-MS

Amyloid diseases are a serious cause for concern world-wide. To understand the mechanism of formation of the fibrillar structures associated with such disorders, it is necessary to study the progression from soluble protein or peptide monomer through an array of oligomers to the final, insoluble, fibrils. The protein IAPP is found in vivo in the form of insoluble amyloid deposits in the pancreatic islets of diabetes type II sufferers. Here, we have studied the in vitro self-aggregation of three fibril-forming peptides from the amyloidogenic core of IAPP. Using electrospray ionization—mass spectrometry coupled with ion mobility spectrometry, the mass and cross-sectional area of each oligomer present in the heterogeneous assembly mixtures can be determined individually in a single, rapid experiment over time. For the three peptides studied, oligomers ≤20-mer were characterized. Conversely, no oligomers higher than a dimer were detected for a non-assembling peptide control. The rate in which the cross-sectional area of the oligomers increases with increasing number of peptide sub-units indicates that assembly for the amyloid-forming peptides proceeds in a linear fashion until an oligomer of a certain size is attained. After this, a step increase in cross-sectional area occurs for the next higher-order oligomer. This behaviour can be explained by molecular modelling of singly, doubly, triply and quadruply stacked β-stranded structures. Using one peptide as an example, the cross-sectional areas of the lower order oligomers (dimer to pentamer) were found to be consistent with a single β-sheet model, whereas the higher order oligomers were consistent with double-stranded (hexamer to decamer oligomers), triply-stranded (11-mers to 15-mers) and quadruply-stranded (16-mers to 20-mers) β-sheet models.     

Towards an embodied,cultural, and material conception of mathematics cognition

In this paper I sketch an embodied, cultural, and material conception of cognition and discuss some of the implications for mathematics education. This approach, which I term sensuous cognition, rests on a cultural and historical dialectical materialist understanding of the senses, sensation, and the material and conceptual worlds. Sensation and matter are considered to be the substrate of mind, and of all psychic activity (cognitive, affective, volitional, etc.). I argue that human cognition can only be understood as a culturally and historically constituted multimodal sentient form of creatively responding, acting, feeling, transforming, and making sense of the world. To illustrate these ideas I briefly refer to a classroom episode involving 7- to 8-year-old students dealing with pattern generalization.     

The organization of aromatic side groups in an amyloid fibril probed by solid-state 2H and 19F NMR spectroscopy

Some 25 diseases are associated with proteins and peptides that assemble into amyloid fibrils composed of beta-strands connected by hydrogen bonds oriented parallel to the fiber long axis. There is mounting evidence that amyloid formation involves specific interactions between amino acid side groups, which bring together beta-sheets to form layers with buried and exposed faces. This work demonstrates how a combination of solid-state 2H and 19F NMR experiments can provide constraints on fibril architecture by probing the environment and spatial organisation of aromatic side groups. It is shown that phenylalanine rings within fibrils formed by a decapeptide fragment of the islet amyloid polypeptide, amylin, are highly motionally restrained and are situated within 6.5 A of one another. Taken together with existing structural constraints for this peptide, these results are consistent with a fibril architecture that comprises layers of two or more beta-sheets, with the aromatic residues facing into the inter-sheet space and possibly engaged in pi-pi interactions. The methods presented will be of general utility in exploring the architecture of fibrils of larger, full-length peptides and proteins, including amylin itself.     

Representations of pointed Hopf algebras and their Drinfel'd quantum doubles

We study representations of nontrivial liftings of nilpotent type of quantum linear spaces and their Drinfel'd quantum doubles. We construct a family of Verma-type modules in both cases and prove a parametrization theorem for simple modules. We compute the Loewy and socle series of Verma modules under a mild restriction on the datum of a lifting. We find bases and dimensions of simple modules.