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.     

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.     

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?     

Structural models in consumer credit

We propose a structural credit risk model for consumer lending using option theory and the concept of the value of the consumer’s reputation. Using Brazilian empirical data and a credit bureau score as proxy for creditworthiness we compare a number of alternative models before suggesting one that leads to a simple analytical solution for the probability of default. We apply the proposed model to portfolios of consumer loans introducing a factor to account for the mean influence of systemic economic factors on individuals. This results in a hybrid structural-reduced-form model. And comparisons are made with the Basel II approach. Our conclusions partially support that approach for modelling the credit risk of portfolios of retail credit.     

Sulfonyl fluorides as privileged warheads in chemical biology

Sulfonyl fluoride electrophiles have found significant utility as reactive probes in chemical biology and molecular pharmacology. As warheads they possess the right balance of biocompatibility (including aqueous stability) and protein reactivity. Their functionality is privileged in this regard as they are known to modify not only reactive serines (resulting in their common use as protease inhibitors), but also context-specific threonine, lysine, tyrosine, cysteine and histidine residues. This review describes the application of sulfonyl fluoride probes across various areas of research and explores new approaches that could further enhance the chemical biology toolkit. We believe that sulfonyl fluoride probes will find greater utility in areas such as covalent enzyme inhibition, target identification and validation, and the mapping of enzyme binding sites, substrates and protein–protein interactions.     

When to rebuild or when to adjust scorecards

Data-based scorecards, such as those used in credit scoring, age with time and need to be rebuilt or readjusted. Unlike the huge literature on modelling the replacement and maintenance of equipment there have been hardly any models that deal with this problem for scorecards. This paper identifies an effective way of describing the predictive ability of the scorecard and from this describes a simple model for how its predictive ability will develop. Using a dynamic programming approach one is then able to find when it is optimal to rebuild and when to readjust a scorecard. Failing to readjust or rebuild a scorecard when they aged was one of the defects in credit scoring identified in the investigations into the sub-prime mortgage crisis.