Designing and using professional development resources for inquiry-based learning

This paper describes an attempt to design, analyse and refine professional development (PD) resources that encourage the implementation of inquiry-based learning (IBL). We describe the iterative development of the resources in England with over 100 mathematics teachers from secondary, tertiary and adult education and then analyse the impact these resources had on teachers’ beliefs and practices and the issues arising. This evaluation revealed that teachers had moved away from transmission-based orientations, encouraged by the use of less structured tasks and sample lesson plans, but some found it difficult to adopt IBL pedagogies. The most significant issues for teachers may be summarised as: confusing IBL with ‘discovery’ learning; developing and managing collaborative cultures within the classroom; and planning lessons that adapt to the emerging needs of learners.     

Injective hulls are not natural

In a category with injective hulls and a cogenerator, the embeddings into injective hulls can never form a natural transformation, unless all objects are injective. In particular, assigning to a field its algebraic closure, to a poset or Boolean algebra its Mac-Neille completion, and to an R-module its injective envelope is not functorial, if one wants the respective embeddings to form a natural transformation. Received January 21, 2000; accepted in final form August 10, 2001. RID="h1" RID="h2" RID="h3" ID="h1"The hospitality of York University is gratefully acknowledged by the first author. ID="h2"Third author partially supported by the Grant Agency of the Czech Republic under Grant no. 201/99/0310, and the hospitality of York University is also acknowledged. ID="h3"Partial financial assistance by the Natural Sciences and Engineering Councel of Canada is acknowledged by the fourth author.     

Long-range magnetization transfer between uncoupled nuclei by dipole-dipole cross-correlated relaxation: a precise probe of beta-sheet geometry in proteins

Interference between dipolar interactions in covalently linked (13)C-(1)H and nonlinked (1)H-(1)H pairs can be used to generate antiphase magnetization between noncoupled spins. The buildup rate of such antiphase terms is highly sensitive to local geometry, in particular the interproton distance and the (13)C-(1)H-(1)H internuclear angle. These rates have been measured for opposing C(alpha)H(alpha) pairs in antiparallel beta-sheets in the third Igg-binding domain of protein G (GB3) and in HIV protease, complexed with the inhibitor DMP323. For GB3, good agreement with the 1.1-A crystal structure is found. However, this agreement rapidly deteriorates with decreasing resolution of the corresponding X-ray structure. For HIV protease, two separate crystal structures that differ by less than 0.2 A from one another exhibit lower agreement in their predicted cross-correlated relaxation rates relative to one another than is found between experimental rates and the average of the rates predicted for the two structures. These data indicate that quantitative measurement of these cross-correlated relaxation rates can provide highly accurate structural information in macromolecules.     

Broadband fivefold reduction of vacuum fluctuations probed by dyes in photonic crystals

We observed for the first time a strong angle-independent modification of spontaneous emission spectra from laser dyes in photonic crystals, made of inverse opals in titania. Comparison with spectra from such crystals with much smaller lattice spacing, for which emission is in the long wavelength limit, reveals inhibition of emission up to a factor approximately 5 over a large bandwidth of 13% of the first order Bragg resonance frequency. The center frequency and bandwidth of the inhibition agree with calculated total density of states, while the measured inhibition of vacuum fluctuations is much larger. Because of the specific location of the dye molecules, we likely probe the strongly modulated local photonic density of states.     

Influence of the coulomb interaction on the liquid-gas phase transition and nuclear multifragmentation

The liquid-gas phase transition is analyzed from the topologic properties of the event distribution in the observables space. A multicanonical formalism allows one to directly relate the standard phase transition with neutral particles to the case where the nonsaturating Coulomb interaction is present, and to interpret the Coulomb effect as a deformation of the probability distributions and a rotation of the order parameter. This formalism is applied to a statistical multifragmentation model and consequences for the nuclear multifragmentation phase transitions are drawn.     

Kochen-Specker theorem for a single qubit using positive operator-valued measures

A proof of the Kochen-Specker theorem for a single two-level system is presented. It employs five eight-element positive operator-valued measures and a simple algebraic reasoning based on the geometry of the dodecahedron.     

Multiple Li positions inside oxygen octahedra in lithiated TiO2 anatase

Intercalation of Li in TiO2 anatase results in a phase separation in a Li-poor and a Li-rich phase. The local lithium configuration in the coexisting crystallographic phases is resolved by detailed analysis of neutron diffraction data. In each of the phases, two distinct positions within the octahedral interstices are found, with a temperature-dependent occupancy. A combination of quasi-elastic neutron scattering and force field molecular dynamics simulations shows that Li is hopping on a picosecond time scale between the two sites in the octahedral interstices. The results also suggest a specific Li arrangement along the crystallographic a direction, albeit without long range order. It is likely that multiple discrete Li sites within a distorted oxygen octahedron occur not only in intercalated TiO2 anatase but also in other (transition metal) oxides.     

Morita Equivalence of Sketches

Equivalence of sketches S and T means the equivalence of their categories Mod_S and Mod_T of all Set-valued models. E. Vitale and the second author have characterized equivalence of limit-sketches by means of bimodels, where a bimodel for limit sketches S and T is a model of S in the category Mod_T. For general sketches, we show that an analogous result holds provided that Mod_T is substituted by a more complex category; e.g., in case of limit-coproduct sketches, that category is (Mod_T), the free product completion of Mod_T.