A study of early career teachers’ practices related to language and language diversity during mathematics instruction

The role of language in mathematics teaching and learning is increasingly highlighted by standards and reform movements in the US. However, little is known about teachers’, and especially early career teachers’ (ECTs) practices and understandings related to language in mathematics instruction. This multiple case study explored the language-related understandings and practices of six ECTs in diverse elementary classrooms. Using iterative cycles of analysis, we found that all ECTs regularly attended to students’ mathematical vocabulary use and development. Yet, there was variability in ECTs’ focus on how to teach mathematical vocabulary, expectations for students’ precise use of mathematical terminology, and the use of multiple languages during instruction. These findings indicate that ECTs need more targeted support during teacher preparation and early career teaching in order to better support all students’ language development in the mathematics classroom.     

Quantum Mechanical Second Virial Coefficient for Cutoff Potentials

The high- and low temperature approximations of the second virial coefficient for cutoff potentials are discussed and compared with numerical results.     

Dielectronic recombination rate for the Be-sequence

The dielectronic recombination (DR) rate coefficient α^DR is explicitly calculated for the Mo, Fe, Ar, and Ox target ions of the Be-sequence with four electrons, in the isolated resonance approximation. This work extends the previous study of the Mo³⁸⁺ ions at 1.4, 2.8, and 5.6 keV electron temperatures. Both Δn≠0 and Δn = 0 transitions are considered in detail. The Δn≠0 contribution still dominates, but the Δn = 0 effect becomes quite large for heavy ions. An explicit LS coupling scheme is employed throughout for the dominant transitions calculated here, and contributions from many other intermediate states and cascade transitions are included by comparing the dominant contribution with the more complete Mo³⁸⁺ case and proportionately scaling their effect. Nonrelativistic Hartree-Fock wave functions are used in the evaluation of the Auger and radiative amplitudes, and the continuum wave functions are calculated using the Hartree-Fock direct potential and explicit nonlocal exchange potential. The scaling property of α^DR and its breakdown are examined, and an improved form of the phenomenological formula is proposed.     

Inward solidification of a superheated liquid in a cooled horizontal tube

Inward solidification has been studied experimentally and analytically under conditions where the liquid phase is above the fusion temperature (i.e., superheated). The liquid was housed in a horizontal circular tube in which the surface was maintained at a uniform, time-invariant temperature during test runs. Three phase change materials (n-heptadecane,n-octadecane, and water) were used in the tests. Both analysis and experiments have established that for inward solidification, natural convection in a superheated liquid is not important in controlling the solidliquid interface motion for Stefan numbers less than unity. The interface velocity is determined primarily by the thermal resistance across the solid layer. Good agreement has been obtained between experimentally measured and analytically predicted solid-liquid interface positions when the density differences between the phases were accounted for.     

Numerical ranges of reducible companion matrices

In this paper, we show that a reducible companion matrix is completely determined by its numerical range, that is, if two reducible companion matrices have the same numerical range, then they must equal to each other. We also obtain a criterion for a reducible companion matrix to have an elliptic numerical range, put more precisely, we show that the numerical range of an n-by-n reducible companion matrix C is an elliptic disc if and only if C is unitarily equivalent to A⊕B, where A∈M_n-2, B∈M₂ with σ(B)={aω₁,aω₂}, , ω₁≠ω₂, and .     

Lucas' theorem refined ∗

We prove a refined version of the classical Lucas' theorem: if p is a polynomial with zeros a ₁,…,a _n+1 all having modulus one and φis the Blaschke product whose zeros are those of the derivative p ¹, then the compression of the shift S(φ) has its numerical range circumscribed about by the (n+ l)-gon a ₁…a _n+1 with tangent points the midpoints of then+ l sides of the polygon. This is proved via a special matrix representation of S(φ) and is a generalization of the known case for n= 2.     

A new strategy for designing non-C2-symmetric monometallic bifunctional catalysts and their application in enantioselective cyanation of aldehydes

A monometallic bifunctional catalyst, in which only one imidazolyl moiety is directly attached at the 3-position of a binaphthol moiety, has been developed. The ligand (R)-1, which lacks C2-symmetry and flexible linkers, in combination with Ti(OiPr)4, has been demonstrated to promote the enantioselective cyanation of aldehydes with trimethylsilylcyanide (TMSCN), giving excellent enantioselectivities of up to 98 % ee and high yields of up to 99 %. The use of this bifunctional catalytic system obviates the need for additives and is extremely simple as the reagents are added in one portion at the beginning of the reaction. The protocol has been found to tolerate a relatively wide range of aldehydes when 10 mol % of the (R)-1/Ti(OiPr)4 complex is deployed in CH2Cl2 at -40 degrees C, the conditions which proved most practical and effective. The asymmetric cyanations also proceeded with lower catalyst loadings (5 mol %, or even 2 mol %), still giving satisfactory enantiomeric excesses and yields. Interestingly, the use of freshly distilled TMSCN dried over CaH2 gave a low enantioselectivity and only a moderate yield of the adduct as compared with direct use of the commercial reagent. The results of 13C NMR spectroscopic studies implicate HCN as the actual reactive nucleophile.     

Surface functionalization of polymer latex particles. II. Catalytic oxidation of poly(methylstyrene) latexes in the presence of cetyltrimethylammonium bromide

Surface-functionalized cationic poly(methylstyrene) (PMS) latex particles containing aldehyde and carboxylic acid groups were successfully achieved via an emulsion polymerization of 3(4)-methylstyrene in the presence of cetyltrimethylammonium bromide, followed by an in-situ oxidation catalyzed by copper chloride and tert-butyl hydroperoxide (t-BuOOH). Factors such as the type of metal catalyst, oxidant, and their concentration strongly affected the rate of oxidation. Step addition of t-BuOOH resulted in both a higher degree of oxidation and a more uniform distribution of particle size of the functionalized PMS as compared to the batch addition method. The effect of organic solvent was found to be insignificant, and the oxidation could still proceed in its absence. The particle sizes increased significantly during the oxidation but could be controlled by using crosslinked PMS latexes. Finally, the versatility of this oxidation process was demonstrated by oxidation of the polymer with a solid loading as high as 28%. © 1997 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 35 : 3585–3593, 1997     

Crystal structure and ionic conductivity of the soft solid crystal: isoquinoline<Subscript>3</Subscript>•(LiCl)<Subscript>2</Subscript>

A soft solid crystal composed of isoquinoline (IQ) and LiCl was prepared based on the concept of Pearson’s hard-soft acid-base (HSAB) theory. Single-crystal X-ray diffraction best described the isoquinoline₃?(LiCl)₂ as consisting of molecular Li₄Cl₄(isoquinoline)₆ units, where the LiCl cluster is an array of edge-fused Li₂Cl₂ rhombs. Electrochemical impedance measurements on pressed pellets showed that conductivity occurs mainly through the bulk via a hopping mechanism, with a calculated activation energy of E_a = 67 kJ/mol. The high value of the activation energy was due to Li₄Cl₄ clusters that were well separated by intervening IQ ligands in the crystal structure, requiring large hops for ions to migrate through the lattice.