Future teachers’ competence to plan a lesson: first results of a six-country study on the efficiency of teacher education

The study "Mathematics Teaching in the 21st Century (MT21)" focuses beyond others on the measurement of teachers’ general pedagogical knowledge (GPK). GPK is regarded as a latent construct embedded in a larger theory of teachers’ professional competence. It is laid out how GPK was defined and operationalized. As part of an international comparison GPK was measured with several complex vignettes. In the present paper, the results of future mathematics teachers’ knowledge from four countries (Germany, South Korea, Taiwan, and the US) with very different teacher-education systems are presented. Significant and relevant differences between the four countries as well as between future teachers at the beginning and at the end of teacher education were found. The results are discussed with reference to cultural discourses about teacher education.     

On the combinatorics of the Pfaff identity

Recently, there has been a revival of interest in the Pfaff identity on hypergeometric series because of the specialization of Simons and a generalization of Munarini. We present combinatorial settings and interpretations of the specialization and the generalization; one is based on free Dyck paths and free Schröder paths, and the other relies on a correspondence of Foata and Labelle between the Meixner endofunctions and bicolored permutations, and an extension of the technique developed by Labelle and Yeh for the Pfaff identity. Applying the involution on weighted Schröder paths, we derive a formula for the Narayana numbers as an alternating sum of the Catalan numbers.     

Observations on the regio- and stereoselective trapping of organozinc carbenoids by acetoxybutadienes: a useful route to functionalised vinylcyclopropanes

Regiospecific trapping of organozinc carbenoids generated from carbonyl compounds with Z-1-acetoxybutadiene provides a useful preparative route to functionalised vinyl and divinylcyclopropanes.     

Singular finite elements for the sudden-expansion and the die-swell problems

The singular finite element method is used to solve the sudden-expansion and the die-swell problems in order to improve the accuracy of the solution in the vicinity of the singularity and to speed up the convergence. The method requires minor modifications to standard finite element schemes, and even coarse meshes give more accurate results than refined ordinary finite element meshes. Improved normal stress results for the sudden-expansion problem have been obtained for various Reynolds numbers up to 100 using the singular elements constructed for the creeping flow problem. In addition, the normal stresses at the walls appear to be insensitive to the singularity powers used in the construction of the singular basis functions. The die-swell problem is solved using the singular elements constructed for the stick–slip problem. The singular elements accelerate the convergence of the free surface dramatically.     

Synchronisation of canonical measures for hyperbolic attractors

Under suitable conditions it is shown how to change the velocity of aC ² AxiomA attractor so that the Sinai-Ruelle-Bowen measure coincides with the measure of maximal entropy. These measures are obtained as limits of certain closed orbital measures.     

Limits of minimal surfaces and Scherk's Fifth Surface

Communicated by J. C. C. Nitsche     

Synthesis and acylation of 2-nitro-11H-dibenzo[b,e][1,4]dioxepin

A one-pot synthesis of the 11H-dibenzo[b,e]1,4]dioxepin ring system from catechol and an o-chlorobenzyl chloride is described. Friedel-Crafts acylation occurs at the 7-position as shown by X-ray analysis.     

Effect of acidification treatment and morphological stability of sulfonated poly(arylene ether sulfone) copolymer proton‐exchange membranes for fuel‐cell use above 100 °C

Directly copolymerized wholly aromatic sulfonated poly(arylene ether sulfone) copolymers derived from 4,4′‐biphenol, 4,4′‐dichlorodiphenyl sulfone, 3,3′‐disulfonated, and 4,4′‐dichlorodiphenyl sulfone (BPSH) were evaluated as proton‐exchange membranes for elevated temperature operation (100–140 °C). Acidification of the copolymer from the sulfonated form after the nucleophilic step (condensation) copolymerization involved either immersing the solvent‐cast membrane in sulfuric acid at 30 °C for 24 h and washing with water at 30 °C for 24 h (method 1) or immersion in sulfuric acid at 100 °C for 2 h followed by similar water treatment at 100 °C for 2 h (method 2). The fully hydrated BPSH membranes treated by method 2 exhibited higher proton conductivity, greater water absorption, and less temperature dependence on proton conductivity as compared with the membranes acidified at 30 °C. In contrast, the conductivity and water absorption of a control perfluorosulfonic acid copolymer (Nafion 1135) were invariant with treatment temperature; however, the conductivity of the Nafion membranes at elevated temperature was strongly dependent on heating rate or temperature. Tapping‐mode atomic force microscope results demonstrated that all of the membranes exposed to high‐temperature conditions underwent an irreversible change of the ionic domain microstructure, the extent of which depended on the concentration of sulfonic acid sites in the BPSH system. The effect of aging membranes based on BPSH and Nafion at elevated temperature on proton conductivity is also discussed. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 2816–2828, 2003     

Composition constants for raising the orders of unconventional schemes for ordinary differential equations

Many models of physical and chemical processes give rise to ordinary differential equations with special structural properties that go unexploited by general-purpose software designed to solve numerically a wide range of differential equations. If those properties are to be exploited fully for the sake of better numerical stability, accuracy and/or speed, the differential equations may have to be solved by unconventional methods. This short paper is to publish composition constants obtained by the authors to increase efficiency of a family of mostly unconventional methods, called reflexive.

     

Preparation and measurement in quantum physics

To honor Henry Margenau on the occasion of his 90th birthday, we attempt in this essay to integrate certain aspects of the physics, philosophy, and pedagogy of quantum mechanics in a manner very much inspired by Margenau's idealist scientific epistemology. Over half a century ago, Margenau was perhaps the first philosopher of science to recognize and elaborate upon the essential distinction between thepreparation of a quantum state and themeasurement of an observable associated with a system in that state; yet in contemporary quantum texts that distinction rarely receives adequate emphasis even though, as we demonstrate, it may be explicated through a series of simple illustrations.