Das Wissen um Unterschiede in den kognitiven Strukturen von Schülerinnen und Schülern als Erklärung von Unterrichtsbeiträgen

It is reported about a longitudinal study run at the Institute of Cognitive Mathematics of the University of Osnabrueck, in which pupils’ verbal and text productions from mathematics lessons at a grammar school are analysed by means of cognitive theoretical methods. First of all, a teaching scene from an instruction to probability calculus and further text productions from an introductory lesson about exponential functions are analysed, in which five pupils take part whose cognitive structures have been assessed and classified in individual examinations. The characteristics brought out according to these teaching scenes indicate different ideas and thinking processes of the pupils. The second part shows that the pupils’ behaviour described is not only to be regarded in isolation but it can also be found in longitudinal examinations and can therefore be considered as a stable, typical characteristic. These results lead to consequences for the planning and design of mathematics lessons based on a well-founded theory of cognition.     

The cochromatic index of a graph

We discuss partitions of the edge set of a graph into subsets which are uniform in their internal relationships; i.e., the edges are independent, they are incident with a common vertex (a star), or three edges meet in a triangle. We define the cochromatic index z′(G) of G to be the minimum number of subsets into which the edge set of G can be partitioned so that the edges in any subset are either mutually adjacent or independent.Several bounds for z′(G) are discussed. For example, it is shown that δ(G) - 1 ? z′(G)? Δ(G) + 1, with the lower bound being attained only for a complete graph. Here δ(G) and Δ(G) denote the minimum and maximum degrees of G, respectively. The cochromatic index is also found for complete n-partite graphs.Given a graph G define a sequence of graphs G₀, G₁,…, G_k, with G₀=G and

, with k being the first value of i for which G_i is regular. Let φ_i(G) = |V(G) – V(G_i| + Δ (G_i) and setφ(G) = min_0?i?kφ_i(G). We show that φ(G) ? 1 ?z′(G)?φ(G) + 1. We then s that a graph G is of class A, B or C, if z′(G) = φ(G) ? 1, φ(G), orφ(G) + 1, respectively. Examples of graphs of each class are presented; in particular, it is shown that any bipartite graph belongs to class B.Finally, we show that if a, b and c are positive integers with a?b?c + 1 and a?c, then unless a = c = b - 1 = 1, there exists a graph G having δ(G) = a, Δ(G) =c, and z′(G) = b.     

A Phosphorus Analogue of p-Quinodimethane with a Planar P4 Ring: A Metal-Free Diphosphorus Source

Para-quinodimethane (pQDM) is a fundamental structural component in many π-conjugated organic molecules and materials. The incorporation of phosphorus atom into π-conjugated frameworks offers unique opportunities for controlling the properties of derived species. A phosphorus analogue of p-quinodimethane (pQDM), (IPrC)₂P₄ [ 5 , IPr=C{N(Ar)CH₂}₂; Ar=2,6-iPr₂C₆H₃] featuring a planar P₄ ring, was readily accessible by KC₈-reduction of (IPrC)(PCl₂)₂ ( 2 ). Base-mediated C−H functionalization of IPrCH₂ ( 1 ) with PCl₃ afforded 2 . The formation of 5 was expected to occur through a dimerization of the transient 3H-diphosphirene (IPrC)P₂ ( 4 ), which was theoretically suggested to have an intermediate diradical character. Compound 5 underwent photo-induced ring-contraction reaction to form the singlet diradicaloid (IPrCP)₂ VI and white phosphorus (P₄). The formation of and VI and P₄ suggested the formal diphosphorus (P₂) elimination from 5 . Indeed, photolysis of a mixture of 1,3-cyclohexadiene (CHD) and 5 led to the formation of P₂-entrapped product (CHD)₂P₂ ( 6 ). The compound 5 represents the first organophosphorus species that functions as a P₂ source.     

Molecular all-photonic encoder-decoder

In data processing, an encoder can compress digital information for transmission or storage, whereas a decoder recovers the information in its original form. We report a molecular triad consisting of a dithienylethene covalently linked to two fulgimide photochromes that performs as an all-photonic single-bit 4-to-2 encoder and 2-to-4 decoder. The encoder compresses the information contained in the four inputs into two outputs. The inputs are light of four different wavelengths that photoisomerize the fulgimide, dithienylethene, or both. The outputs are absorbance at two wavelengths. The two decoder inputs are excitation at two wavelengths, whereas the four outputs, which recover the information compressed into the inputs, are absorbance at two wavelengths, transmittance at one wavelength, and fluorescence emission. The molecule can be cycled through numerous encoder and decoder functions without significant photodecomposition. Molecular photonic encoders and decoders could potentially be used for labeling and tracking of nano- and microscale objects as well as for data manipulation.     

Hydrothermal Synthesis and Crystal Structure of Novel Alkali Trimolybdates

Novel alkali trimolybdates of the triclinic (M, M′)₂Mo₃O₁₀ (M = Rb; M′ = K, Cs) type were obtained through a systematic hydrothermal approach based on the reaction of MoO₃ with alkali halide solutions at 180 °C. The crystal structures were determined from X‐ray single crystal data. The alkali trimolybdates extend the family of known alkali trimolybdates in an unexpected fashion, because they contain a distorted variation of [Mo₃O₁₀]^2? chains as a key structural motif that has only been found in a single compound before, namely ethylenediammonium trimolybdate, (C₂H₁₀N₂)[Mo₃O₁₀]. The applied hydrothermal strategy is discussed in the general context of systematic pathways to polyoxomolybdates. Furthermore, the templating role of the alkali cations and their interaction with the polyoxomolybdate surroundings is compared to (C₂H₁₀N₂)[Mo₃O₁₀] in terms of electrostatic calculations.     

Structural Investigations and Thermal Behaviour of Mercury(II) Sulfito Complexes

The crystal structures of (NH₄)[HgSO₃Cl] ( 1 ) and of (NH₄)₂[Hg(SO₃)₂] ( 2 ) were determined from single crystal diffractometer data sets. 1 : 22 °C, Pnma, Z = 4, a = 15.430(3), b = 5.525(1), c = 6.679(1) Å, R(F) = 0.0256, R_w(F²) = 0.0642 (all 1056 unique reflections). 2 : ?108 °C, P2₁2₁2₁, Z = 4, a = 6.2240(4), b = 9.3908(6), c = 13.6110(8) Å, R(F) = 0.0179, R_w(F²) = 0.0493 (all 2699 unique reflections). The structure of 1 contains bent Cl‐Hg‐SO₃ entities (site symmetry m; d(Hg‐Cl) = 2.3403(13) Å, d(Hg‐S) = 2.3636(12) Å, ∠(Cl‐Hg‐S) = 164.51(5)°, d(S‐O) 2×1.458(3) Å, 1.468(4) Å, = 1.461Å) linked to undulated ribbons parallel to the b ‐axis by intermolecular secondary bonds SO···Hg (d(O···Hg) = 2×2.595(3) Å). These ribbons in turn aggregate to layers around the bc ‐plane. The layers are stacked along the a ‐axis with interlayer distances of a /2. The structure of 2 is made up of O₃S‐Hg‐SO₃ moieties (d(Hg‐S) = 2.3935(7), 2.3935(8) Å; ∠(Hg‐S‐Hg) = 174.41(3)°; = 1.474Å), that are linked to ribbons parallel to the a axis by coordination of Hg to three remote O atoms (2.801(4) < d(Hg‐O) < 2.844(3) Å). Adjacent ribbons are joined together by an additional Hg‐O contact of 2.733(3) Å, leading to a three‐dimensional anionic framework. Both crystal structures are stabilised by disordered NH₄⁺ cations, placed between the anionic layers or in the vacancies of the framework, via moderate hydrogen bonding interactions N‐H···O with donor‐acceptor distances ranging from 2.8 to 3.2Å. 1 and 2 were further characterised by thermal analysis (TG, DSC). They start to decompose at temperatures above 130 °C.     

Molecular and Spectroscopic Characterization of Green and Red Cyanine Fluorophores from the Alexa Fluor and AF Series**

The use of fluorescence techniques has an enormous impact on various research fields including imaging, biochemical assays, DNA-sequencing and medical technologies. This has been facilitated by the development of numerous commercial dyes with optimized photophysical and chemical properties. Often, however, information about the chemical structures of dyes and the attached linkers used for bioconjugation remain a well-kept secret. This can lead to problems for research applications where knowledge of the dye structure is necessary to predict or understand (unwanted) dye-target interactions, or to establish structural models of the dye-target complex. Using a combination of optical spectroscopy, mass spectrometry, NMR spectroscopy and molecular dynamics simulations, we here investigate the molecular structures and spectroscopic properties of dyes from the Alexa Fluor (Alexa Fluor 555 and 647) and AF series (AF555, AF647, AFD647). Based on available data and published structures of the AF and Cy dyes, we propose a structure for Alexa Fluor 555 and refine that of AF555. We also resolve conflicting reports on the linker composition of Alexa Fluor 647 maleimide. We also conducted a comprehensive comparison between Alexa Fluor and AF dyes by continuous-wave absorption and emission spectroscopy, quantum yield determination, fluorescence lifetime and anisotropy spectroscopy of free and protein-attached dyes. All these data support the idea that Alexa Fluor and AF dyes have a cyanine core and are a derivative of Cy3 and Cy5. In addition, we compared Alexa Fluor 555 and Alexa Fluor 647 to their structural homologs AF555 and AF(D)647 in single-molecule FRET applications. Both pairs showed excellent performance in solution-based smFRET experiments using alternating laser excitation. Minor differences in apparent dye-protein interactions were investigated by molecular dynamics simulations. Our findings clearly demonstrate that the AF-fluorophores are an attractive alternative to Alexa- and Cy-dyes in smFRET studies or other fluorescence applications.     

An All‐Photonic Molecular Keypad Lock

Off and on : A molecular triad, consisting of a porphyrin linked to two different, independently addressable photochromic moieties, functions as a molecular keypad lock with all‐photonic inputs and output. The porphyrin correlates the responses of the two inputs to light of different wavelengths and provides an appropriate output as fluorescence, which results only when one of eight possible input combinations has been applied (see figure).

     

Halogenomercury Salts of Sterically Crowded Triazenides – Convenient Starting Materials for Redox‐Transmetallation Reactions

Diaryl‐substituted triazenides Ar(Ar′)N₃HgX [Ar/Ar′ = Dmp/Mph, X = Cl ( 2a ), Br ( 3a ), I ( 4a ); Ar/Ar′ = Dmp/Tph, X = Cl ( 2b ), I ( 4b ) with Mph = 2‐MesC₆H₄, Mes = 2,4,6‐Me₃C₆H₂, Tph = 2′,4′,6′‐triisopropylbiphenyl‐2‐yl and Dmp = 2,6‐Mes₂C₆H₃] were synthesized by salt‐metathesis reactions in ethyl ether from the readily available starting materials Ar(Ar′)N₃Li and HgX₂. These compounds may be used for redox‐transmetallation reactions with rare‐earth or alkaline earth metals. Thus, reaction of 4b or 2b with magnesium or ytterbium in tetrahydrofuran afforded the triazenides Dmp(Tph)N₃MX(thf) ( 5b : M = Mg, X = I; 6b : M = Yb, X = Cl) in good yield. All new compounds were characterized by melting point, ¹H and ¹³C NMR spectroscopy and for selected species by IR spectroscopy or mass spectrometry. In addition, the solid‐state structures of triazenides 2a , 2b , 3a , 4b , 5b and 6b were investigated by single‐crystal X‐ray diffraction.