A good foundation for number learning for five-year-olds? An evaluation of the English Early Learning ‘Numbers’ Goal in the light of research

This article sets out to evaluate the English Early Years Foundation Stage Goal for Numbers, in relation to research evidence. The Goal, which sets out to provide ‘a good foundation in mathematics’, has greater breadth of content and higher levels of difficulty than previous versions. Research suggests that the additional expectations are unachievable by the majority of rising fives, but that a more effective foundation to ensure progress and prevent later difficulties would prioritise number sense with numbers to 10, including subitising, numeral comprehension, relative number size, one more/less than and part–whole relationships.     

The existence of continuous invariants for linear difference equations

We study real continuous invariants for systems of linear difference equations. We shall prove a conjecture by Ladas about the existence of such invariants. In fact, necessary and sufficient conditions on existence of such invariants will be established. The invariants will be constructed when they exist.     

Seeing through Statistics

Abstract

Introduction to Graphical Modelling by David Edwards, New York: Springer-Verlag, 1995, 274 pp. Reviewed by Garry Robins

Introduction to Graphical Modelling by David Edwards, New York: Springer-Verlag, 1995, 274 pp. Reviewed by Garry Robins

Graphical Models by Steffen L. Lauritzen, Oxford, U.K.: Clarendon Press, 1996, 298 pp. Reviewed by Garry Robins     

Modulation versus Templating: Fine‐Tuning of Hierarchally Porous PCN‐250 Using Fatty Acids To Engineer Guest Adsorption

Modulation and templating are two synthetic techniques that have garnered significant attention over the last several years for the preparation of hierarchically porous metal–organic frameworks (HP‐MOFs). In this study, by using fatty acids with different lengths and concentrations as dual‐functional modulators/templates, we were able to obtain HP‐MOFs with tunable mesopores that exhibit different pore diameters and locations. We found that the length and concentration of the fatty acids can determine if micelle formation occurs, which in turn dictates the porosity of the resulting HP‐MOFs. The HP‐MOFs with different mesopores differed in their performance in gas uptake and dye adsorption, and the structure–performance relationships were ascribed to the pore diameters and locations. This approach could provide a potentially universal method to efficiently introduce hierarchal mesopores into existing microporous MOF adsorbents with tunable properties.     

Structural and NMR Studies of the Hexameric Copper(I) Complex with N‐n‐butyl‐N′‐ethoxycarbonyl‐thioureate

The copper sulfide mineral flotation collector, N‐n‐butyl‐N′‐ethoxycarbonyl‐thiourea (H₂bectu), and the 1:1 hexameric copper(I) thioureate complex, [Cu(Hbectu)]₆, have been characterized by single crystal X‐ray diffraction. H₂bectu crystallizes in the triclinic space group with a = 5.2754(4), b = 9.0042(7), c = 12.6030(9) Å, α = 80.528(6), β = 90.173(6), γ = 76.472(7)°. An intramolecular N‐H···O hydrogen bond between the thioamide proton and carbonyl oxygen forms a planar six‐membered ring in the central core of the molecule with C=O, C=S and C‐N bond lengths in accord with those reported for other N‐alkyl/aryl‐N′‐acyl‐thiourea compounds. [Cu(Hbectu)]₆ crystallizes in the monoclinic space group C2/c with a = 23.269(5), b = 13.243(4), c = 23.037(7) Å, β = 91.81(2)° as discrete hexameric clusters disposed about a crystallographic centre of symmetry with a Cu₆S₆ core consisting of two Cu₃S₃ chair‐shaped rings linked by coordination of the deprotonated amide nitrogen atom to a copper atom in the adjacent ring. The six ligands assemble as a paddlewheel structure with the ethoxy and n‐butyl substituents packing in an alternating head to tail arrangement. Temperature dependent solution ¹H NMR spectroscopic studies show that the hexameric structure of the complex is maintained in solution.     

Mass spectrometry analysis of terpene lactones in Ginkgo biloba

Terpene lactones are a family of compounds with unique chemical structures, first recognised in an extract of Ginkgo biloba. The discovery of terpene lactone derivatives has recently been reported in more and more plant extracts and even food products. In this study, mass spectrometric characteristics of the standard terpene lactones in Ginkgo biloba were comprehensively studied using both an ion trap and a quadrupole time-of-flight (QTOF) mass spectrometer. The mass spectral fragmentation data from both techniques was compared to obtain the mass spectrometric fragmentation pathways of the terpene lactones with high confidence. The data obtained will facilitate the analysis and identification of terpene lactones in future plant research via the fragmentation knowledge reported here.