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.     

The mechanism of the Fischer-Hepp rearrangement of aromatic N-nitroso-amines

Rate equations have been deduced for two possible mechanisms for the Fischer-Hepp rearrangement of aromatic N-nitroso-amines in acid solution: (a) for the commonly assumed intermolecular process involving de-nitrosation to the secondary amine and a free nitrosating agent, followed by a direct C-nitrosation of the secondary amine by this nitrosating agent, and (b) for a mechanism whereby rearrangement and de-nitrosation occur concurrently, by two separate reactions of the protonated N-nitroso-amine. All the experimental evidence supports mechanism (b), whilst most of it is incompatible with (a). Particularly diagnostic of the mechanism are (1) the observed rearrangement: de-nitrosation product ratios under certain limiting conditions, (2) the question of halide ion catalysis, and (3) the rate form found under the limiting condition of a large excess of added secondary amine.     

TDPAC studies on181Hf implanted into diamond

¹⁸¹Hf ions were implanted at (900±10)K into (100) cleaved single crystal diamonds. After subsequent annealing residence site parameters were determined for the implanted ions employing the (133 keV)–(482 keV) cascade in¹⁸¹Ta in TDPAC technique. Fractions of 5% each were determined that experience axially symmetric electric field gradients (EFG) V_zz(1)=5.5×10¹⁷ V cm^–2 and V_zz(2)=9.6×10¹⁷ V cm^–2, respectively. The bulk of the ions are strongly disturbed by still higher electric field gradients.     

Preparation and reactions of 3-oxa- and 4-oxa-5-hexenyllithiums

Low-temperature lithium–iodine exchange between tert-butyllithium and the appropriate iodide has been used to generate 3-oxa-5-hexenyllithium and a variety of 4-oxa-5-hexenyllithiums. The 3-oxa system is inherently unstable and fragments via facile β-elimination to give the anion of allyl alcohol and ethylene. The 4-oxa-5-hexenyllithiums, in contrast, are stable at low temperatures but undergo novel isomerization upon warming to deliver the lithium salt of a 4-alken-1-ol in the formal equivalent of an unprecedented [1,4]-Wittig rearrangement. The rearrangement is most likely mediated by 5-exo-trig ring closure of the 4-oxa-5-hexenyllithium to a (2-tetrahydrofuranyl)methyllithium followed by rapid opening to the alkoxide.     

Temperature dependence of the electric field gradient in CdSe and HfO2

The time differential perturbed angular correlation technique was applied to study the electric field gradient in the semiconductor CdSe and the insulator HfO₂ at different temperatures. Whereas the semiconductor CdSe shows an increasing quadrupole coupling constant with increasing temperature, the insulator HfO₂ exhibits no temperature dependence of the electric field gradient over the whole temperature range investigated.     

Indications about the shape of the universe from the Wilkinson microwave anisotropy probe data

It is shown that the recent observations of NASA's Explorer mission, "Wilkinson Microwave Anisotropy Probe," hint that our Universe may possess a nontrivial topology. As an example we discuss the Picard space which is stretched out into an infinitely long horn but with finite volume.     

To ask or not to ask,that is the question

Applicants for credit have to provide information for the risk assessment process. In the current conditions of a saturated consumer lending market, and hence falling “take” rates, can such information be used to assess the probability of a customer accepting the offer?