The structures of German mathematics textbooks

From a socio-cultural perspective it is argued that the modality of artefacts has, structuring effects on the activities in which the artefact is involved. The mathematics textbook is an artefact that has a major influence on the activity of learning mathematics. Against this setting, the structures of the units in German mathematics textbooks for different grades and ability levels have been analysed. Firstly, the different structural elements have been examined with regard to: characteristics in terms of content; linguistic characteristics; visual characteristics; their pedagogical functions within the learning process; and situative conditions. Secondly, the orders of the structural elements within the units of the different textbooks have been compared. The findings reveal that the structure of the units is very similar in different mathematics textbooks. The units are not only composed of analogous structural elements, but these elements are also arranged in almost the same sequence. In order to develop a deeper understanding of these findings the structure of the units has been compared to the influential learning theories of J. F. Herbart and H. Roth. On this basis it is argued that the structure of the units seems to reflect the phases of idealised learning processes in general. The issue is raised if this is an appropriate structure in order to provide opportunities to learn mathematics.     

Cylindrical spike model for the formation of diamondlike thin films by ion deposition

T /n_S of n_T rearrangements and n_S atoms in the spike volume as the crucial parameter characterizing the ability of a given ion–target combination to achieve complete rearrangement of the spike volume. n_T/n_S>1 is the optimum condition for diamondlike film growth. For aC films the ion energy dependence of n_T/n_S agrees well with the measured sp³ bond fraction. For Ar⁺-ion-assisted deposition of aC we find n_T/n_S>1 above 50 eV with no pronounced ion energy dependence. Furthermore, our model predicts optimum conditions for the formation of cubic boron nitride between 50 eV and 3 keV. Accepted: 14 October 1997     

Performance of nonrelativistic and quasi-relativistic hybrid DFT for the prediction of electric and magnetic hyperfine parameters in 57Fe Mössbauer spectra

57Fe electric and magnetic hyperfine parameters were calculated for a series of 10 iron model complexes, covering a wide range of oxidation and spin states. Employing the B3LYP hybrid method, results from nonrelativistic density functional theory (DFT) and quasi-relativistic DFT within the zero-order regular approximation (ZORA) were compared. Electron densities at the iron nuclei were calculated and correlated with experimental isomer shifts. It was shown that the fit parameters do not depend on a specific training set of iron complexes and are, therefore, more universal than might be expected. The nonrelativistic and quasi-relativistic electron densities gave fit parameters of similar quality; the ZORA densities are only shifted by a factor of 1.32, upward in the direction of the four-component Dirac-Fock value. From a correlation of calculated electric field gradients and experimental quadrupole splittings, the value of the 57Fe nuclear quadrupole moment was redetermined to a value of 0.16 barn, in good agreement with other studies. The ZORA approach gave no additional improvement of the calculated quadrupole splittings in comparison to the nonrelativistic approach. The comparison of the calculated and measured 57Fe isotropic hyperfine coupling constants (hfcc's) revealed that both the ZORA approach and the inclusion of spin-orbit contributions lead to better agreement between theory and experiment in comparison to the nonrelativistic results. For all iron complexes with small spin-orbit contributions (high-spin ferric and ferryl systems), a distinct underestimation of the isotropic hfcc's was found. Scaling factors of 1.81 (nonrelativistic DFT) and 1.69 (ZORA) are suggested. The calculated 57Fe isotropic hfcc's of the remaining model systems (low-spin ferric and high-spin ferrous systems) contain 10-50% second-order contributions and were found to be in reasonable agreement with the experimental results. This is assumed to be the consequence of error cancellation because g-tensor calculations for these systems are of poor quality with the existing DFT approaches. Excellent agreement between theory and experiment was found for the 57Fe anisotropic hfcc's. Finally, all of the obtained fit parameters were used for an application study of the [Fe(H2O)6]3+ ion. The calculated spectroscopic data are in good agreement with the Mossbauer and electron paramagnetic resonance results discussed in detail in a forthcoming paper.     

Electrospray ionization mass spectrometric studies of nucleophilic carbenes derived from pyrazolium and indazolium carboxylates

Pyrazolium-3-carboxylate and indazolium-3-carboxylate, which belong to the class of pseudo-cross-conjugated mesomeric betaines and which represent the electronically relevant partial structures of the betaine alkaloid Nigellicin, were examined by electrospray ionization mass spectrometry. These compounds decarboxylate to pyrazol-3-ylidene and indazol-3-ylidene. The formation of adducts of these new nucleophilic carbenes under the measurement conditions was examined.     

Development of a robust technique for sampling volatile metal(loid)s in wetlands

The formation of volatile organic and inorganic metals and metalloids in aquatic environments is a known, but not very intensively investigated, process. Several techniques have been developed over the past 10 years to determine these trace components. These techniques are of limited use in wetland environments, where samples have to be taken from the soil-water interface, and require an immediate sample analysis due to thermodynamic instabilities of the volatile metal(loid)s. This paper presents an innovative sampling technique for total concentrations of volatile metal(loid)s in wetlands, based on an in situ gas-water separation via a porous PTFE membrane and stabilising the volatile metal(loid)s in a liquid sorbent (NaOCl solution). Samples may thus be collected even at remote sites, where longer storage times have to be accounted for. The sampling system was tested by means of a laboratory facility simulating the generation of arsine and dimethyl arsine under abiotic conditions as well as under field conditions. Results for sampling efficiency, reproducibility, and long-term storage are presented. Application of the sampling system in the field is shown.     

Complexes formed between nitrilotris(methylenephosphonic acid) and M(2+) transition metals: isostructural organic-inorganic hybrids

Nitrilotris(methylenephosphonic acid) (NTP, [N(CH(2)PO(3)H(2))(3)]) recently has been found to form three-dimensional porous structures with encapsulation of templates as well as layered and linear structures with template intercalation. It was, therefore, of interest to examine the type of organic-inorganic hybrids that would form with metal cations. Mn(II) was found to replace two of the six acid protons, while a third proton bonds to the nitrilo nitrogen, forming a zwitter ion. Two types of compounds were obtained. When the ratio of acid to Mn(II) was less than 10, a trihydrate, Mn[HN(CH(2)PO(3)H)(3)(H(2)O)(3)] (2) formed. Compound 2 is monoclinic P2(1)/c, with a = 9.283(2) A, b = 16.027(3) A, c = 9.7742(2) A, beta = 115.209(3) degrees, V = 1315.0(5) A(3), and Z = 4. The Mn atoms form zigzag chains bridged by two of the three phosphonate groups. The third phosphonate group is only involved in hydrogen bonding. The metal atoms are octahedrally coordinated with three of the sites occupied by water molecules. Adjacent chains are hydrogen-bonded to each other through POH and HN donors, and the additional participation of all the water hydrogens in H-bonding results in a corrugated sheet-like structure. Use of excess NTP at a ratio to metal of 10 to 1 yields an anhydrous compound Mn[HN(CH(2)PO(3)H)(3)] (1), P2(1)/n, a = 9.129(1) A, b = 8.408(1) A, c = 13.453(1) A, beta = 97.830(2) degrees, V = 1023.0(2) A(3), and Z = 4. Manganese is five coordinate forming a distorted square pyramid with oxygens from five different phosphonate groups. The sixth oxygen is 2.85 A from an adjacent Mn, preventing octahedral coordination. All the protonated atoms, three phosphonate oxygens and N, form moderately strong hydrogen bonds in a compact three-dimensional structure. The open-structured trihydrate forms a series of isostructural compounds with other divalent transition metal ions as well as with mixed-metal compositions. This is indicative that the hydrogen bonding controls the type of structure formed irrespective of the cation.     

Determination of estrogenic steroids in surface water and wastewater by liquid chromatography-electrospray tandem mass spectrometry

An analytical method is presented which permits trace level determination of 17alpha-ethynylestradiol (EE2), 17beta-estradiol (E2), and estrone (E1). Using this method, the estrogenic steroids were analyzed in drinking water, surface water, and wastewater (sewage influents and effluents) at concentrations down to 0.1 ng/L. Sample volumes between 100 and 500 mL are concentrated using automated solid-phase extraction. Analysis is performed by liquid chromatography with detection by tandem mass spectrometry. Applying simple clean-up procedures and internal standard calibration, recovery losses resulting from matrix-dependent ion suppression during electrospray ionization could be compensated for all of the investigated compounds. Recoveries around 100% were obtained for all analytes after correction using the internal standards. Limits of quantification (LOQ) were between 0.1 and 0.4 ng/L for purified sewage, surface, ground, and drinking water and between 1 and 2 ng/L in the case of raw sewage. Water treatment by wastewater treatment plants (WWTPs) or by a surface water treatment plant affected the removal of all estrogenic steroids. Thus, E1, E2, and EE2 were removed in the municipal WWTPs to the extent of 93%, 93%, and 80%, respectively. In the effluents of the WWTP in Ruhleben (Berlin, Germany), E1, E2, and EE2 were detected at the low ng/L level. E2 and EE2 were, however, not present in the Berlin surface water above the LOQ (0.2 ng/L). E1 was the only compound that could be detected in surface water samples. After additional surface water treatment it was still detectable but only at trace-level concentrations with a mean value of 0.16 ng/L.     

Measurement of γp→K + Λ and γp→K + Σ 0 at photon energies up to 1.47 GeV

The reactions pK ⁺ and pK ^{+ 0} have been measured with the multiparticle detector system SAPHIR at ELSA in Bonn. Besides the differential cross sections the polarization and, for the first time, the ⁰ polarization have been determined in a photon induced reaction. All data are presented as functions of the photon energy (from threshold up to 1.47 GeV) and of the kaon production angle (0°–180°). The polarization of both and ⁰ is substantial at all energies and varies strongly with the production angle.This work is supported by the Bundesminister für Forschung und Technologie (BMFT), FK 06 BN 621 I