Theories of mathematics education: European perspectives, commentaries and viable research directions

We briefly comment on different perspectives on (1) the role of mathematics education theories, (2) the issue of plurality and healthy heterogeneity versus consolidation; (3) underlying inquiry systems or the implicit role of philosophy in theories of mathematics education. This paper also outlines developments within the European research scene on theory usage in mathematics education research, which complement the discussion at the 29th PME research forum in Melbourne.     

Solution and solid-state NMR characterization of poly(p-phenylene terephthalamide)

NMR spectroscopy has been used to characterize poly(p-phenylene terephthalamide) in the solid state and in solution in sulfuric acid. Solid-state ¹³C NMR spectra illustrate that the chain structure is highly ordered in the solid state and is of lower symmetry than in solution. Solid-state ¹³C and ¹H NMR results show that only very limited motion takes place over the temperature range of ?170 to +200°C. High-resolution NMR spectra can be observed only in very dilute isotropic solutions because it is the overall rotational motion of the polymer, not segmental motion, that averages the nuclear spin interactions to their isotropic values. These results demonstrate that previous solution NMR studies that were interpreted as reflecting the presence of isotropic and anisotropic high-molecular-weight polymer phases over a wide range of concentrations actually are representative of polymer degradation.     

Nonexponential Fluorescence Decay of 7-Azatryptophan Induced in a Peptide Environment

7-Azatryptophan is an alternative to tryptophan as an optical probe of protein structure and dynamics. 7-Azatryptophan is synthetically incorporated into an octapeptide (NAc-Lys-Ala-Cys-Pro-7-azatryptophan-Asn-Cys-Asp-NH₂) that mimics the active site of potato chymotrypsin inhibitor II, which is known to be a strong inhibitor of α-chymotrypsin. The synthetic octapeptide retains some of this inhibitory activity. This is the first compound containing the 7-azaindole chromophore to display a nonexponential fluorescence decay (well fit to two exponentials) in water when fluorescence is collected over the entire emission band. The effect of external quenchers on the fluorescence decay is monitored and seen to differ markedly for the two components. These results are discussed in terms of the solvation of the 7-azaindole chromophore itself, which promotes or impedes excited-state tautomerization. The fluorescence quenching of free indole and 7-azaindole are compared. The fluorescence quenching of octapeptides containing both chromophores is also compared. It is the thesis of this article that the nonexponential fluorescence decay of the 7-azatryptophan-containing octapeptide is a consequence of excited-state tautomerization of the 7-azaindole chromophore. This tautomerization is suggested to be promoted by solvent reorganization induced by the peptide backbone or by direct interactions of the 7-azaindole with neighboring amino acid side chains.     

The Role of Oxygen in the Antiviral Activity of Hypericin and Hypocrellin

The light-induced antiviral activity of hypericin and hypocrellin in the presence and absence of oxygen was examined under experimental conditions where the effect of oxygen depletion could be quantified. There was a significant reduction of light-induced antiviral activity of hypericin and hypocrellin under hypoxic conditions. Interestingly, antiviral activity of hypocrellin was not observed at low oxygen levels at which hypericin retained measurable virucidal activity. This suggests that additional pathways, such as the generation of protons from excited states of hypericin, may enhance the biological activity of activated oxygen species.     

Laser induced fluorencence matrix study of 1,3-difluorobenzene radical cation

Laser induced fluorescence excitation and resolved emission spectra of 1,3-C₆H₄F⁺₂ are obtained in a Ne matrix with high signal/noise despite the ion's low emission quantum yield. The ground state vibrational structure is mostly regular but that of the upper is very irregular suggesting nearly degenerate, mutually perturbing, excited states.     

Photolysis of pentacarbonyl(selenocarbonyl)chromium(0), Cr(CO)5(CSe), in an Argon matrix at 20 K

U. v. photolysis (254 and 366 nm) of Cr(CO)₅(CSe) isolated in an argon matrix at ca. 20 K produces a mixture of two isomeric forms of Cr(CO)₄(CSe), Cr(CO)₆ and molecular CO. The two Cr(CO)₄(CSe) isomers are assigned square pyramidal geometries (CSe group axial, C_4v symmetry; CSe group equatorial, C_s symmetry) by comparison with the results from a previous matrix isolation i.r. study of the products formed in the u.v. photolysis of the analogous thiocarbonyl complex, Cr(CO)₅(CS). On further irradiation of the Cr(CO)₄(CSe) species with visible light (λ > 425 nm), the parent Cr(CO)₅(CSe) molecule is regenerated.     

Wannier-Stark effect on surface states

A recursive Green-function (RGF) technique is used to study the surface states of an electrified solid. The concatenating nature of the method leads to a continued-fraction form of the RGF, which can be expressed as a ratio of Bessel functions. The surface density of states (SDOS), derived from the RGF, reveals a quasi-Stark-ladder distribution of the energy levels at the surface atom. The dependence of the SDOS on the applied field and surface perturbation parameters is discussed.