Learning to See: Making Sense of the Mathematics of Change in Middle School

In this article, we will describe the results of a study of 6th grade students learning about the mathematics of change. The students in this study worked with software environments for the computer and the graphing calculator that included a simulation of a moving elevator, linked to a graph of its velocity vs. time. We will describe how the students and their teacher negotiated the mathematical meanings of these representations, in interaction with the software and other representational tools available in the classroom. The class developed ways of selectively attending to specific features of stacks of centimeter cubes, hand-drawn graphs, and graphs (labeled velocity vs. time) on the computer screen. In addition, the class became adept at imagining the motions that corresponded to various velocity vs. time graphs. In this article, we describe this development as a process of learning to see mathematical representations of motion. The main question this article addresses is: How do students learn to see mathematical representations in ways that are consistent with the discipline of mathematics? This revised version was published online in July 2006 with corrections to the Cover Date.     

Formation of tetra(ethylene oxide) terminated Si-C linked monolayers and their derivatization with glycine: an example of a generic strategy for the immobilization of biomolecules on silicon

Surface modification with oligo(ethylene oxide) functionalized monolayers terminated with reactive headgroups constitutes a powerful strategy to provide specific coupling of biomolecules with simultaneous protection from nonspecific adsorption on surfaces for the preparation of biorecognition interfaces. To date, oligo(ethylene oxide) functionalized monolayer-forming molecules which can be activated for attachment of biomolecules but which can selectively form monolayers onto hydrogen terminated silicon have yet to be developed. Here, self-assembled monolayers (SAMs) containing tetra(ethylene oxide) moieties protected with tert-butyl dimethylsilyl groups were formed by thermal hydrosilylation of alkenes with single-crystal Si(111)-H. The protection group was used to avoid side reactions with the hydride terminated silicon surface. Monolayer formation was carried out using solutions of the alkene in the high-boiling-point solvent 1,3,5-triethylbenzene. The protecting group was removed under very mild acidic conditions to yield a free hydroxyl functionality, a convenient surface moiety for coupling of biological entities via carbamate bond formation. The chemical composition and structure of the monolayers before and after deprotection were characterized by X-ray photoelectron spectroscopy (XPS) and X-ray reflectometry. To demonstrate the utility of this surface for covalent modification, two reagents were compared and contrasted for their ability to activate the surface hydroxyl groups for coupling of free amines, carbonyl diimidazole (CDI), and disuccinimidyl carbonate (DSC). Analysis of XP spectra before and after activation by CDI or DSC, and after subsequent reaction with glycine, provided quantitative information on the extent of activation and overall coupling efficiencies. CDI activated surfaces gave poor coupling yields under various conditions, whereas DSC mediated activation followed by aminolysis at neutral pH was found to be an efficient method for the immobilization of amines on tetra(ethylene oxide) modified surfaces.     

The Bond-Rotational Mobility of Guanidinium Ion

The NMR. spectrum of guanidinium ion 1 is studied in anisotropic liquid crystalline nematic solution. Assuming an HNH-angle of 120°, the distance ratio NH /NC = 0.784 is obtained, from which using NC = 1.330 Å (from X-ray data) NH = 1.043 Å results. An upper bound for the free energy of activation for bond rotation of ΔG⁺ ≤ 13 kcal/mol is deduced. The bondrotational mobility of 1 is also investigated using the MINDO/3-SCF-procedure. The results obtained for the three conceivable consecutive activation energies for bond-rotation indicate that the observed bond-rotational mobility of 1 does not involve cooperative two- or three-bond rotations. The ‘conjugative stabilization’ of 1 has been estimated to be of the order of 24–26 kcal/mol.     

A Product Formed from Glycylglycine in the Presence of Vanadate and Hydrogen Peroxide: The (Glycylde-N-hydroglycinato-kappa(3)N(2),N(N)(),O(1))oxoperoxovanadate(V) Anion

A crystalline glycylglycine complex of monoperoxovanadate has been obtained and its X-ray structure determined. The coordination is pentagonal bipyramidal with the peroxo group and a tridentate glycylglycine occupying the equatorial positions. The axial positions of the anion are occupied by the oxo ligand and by one oxygen of the peroxo group of the adjacent anion. The latter interaction establishes the seventh bond and produces a dimeric structure in the crystalline material. NMR studies of its dissolution in water combined with previously reported results from equilibrium measurements show that the dimer dissociates in water to the monomeric precursor. It is proposed that this monomer corresponds to the complex responsible for the inhibition of the vanadium-catalyzed decomposition of hydrogen peroxide by glycylglycine. Crystal structure of [NEt(4)][VO(O(2))(GlyGly)].1.58H(2)O: monoclinic, space group P2(1); Z = 4; a = 10.618(2) ?; b = 14.803(2) ?; c = 11.809(2) ?; beta = 101.37(2) degrees; V = 1819.7 ?(3); T = 198 K; R(F)() = 0.029 for 2664 data (I(o) >/= 2.5sigma(I(o))) and 431 variables.     

Synthesis of 4-Alkyl-4-alkoxybutenolides Having Unsaturated Side Chains via Chromium Carbene Complex Photochemistry: (+)-Cerulenin

The photochemical reaction between optically active ene carbamates and chromium alkoxycarbene complexes containing unsaturated aliphatic side chains was further developed. Although remote olefinic groups, including conjugated dienes, were tolerated, a homoallylic side chain underwent intramolecular reaction to give a strained cyclobutanone. (+)-Cerulenin was synthesized utilizing the photochemical reaction of an alkynylcarbene complex with an optically active ene carbamate and the bis(pi-crotyl)nickel halide alkylation of a vinyl bromide as key steps.     

A copper-catalyzed C-N bond formation involving sp-hybridized carbons. A direct entry to chiral ynamides via N-alkynylation of amides

A copper-catalyzed new C-N bond formation involving a sp-hybridized carbon is described here leading to a facile entry for syntheses of chiral ynamides. This direct N-alkynylation of amides should have a significant impact on the future development of synthetic methodologies employing ynamides.     

Characterization of the oxidation products of styryl-substituted terthiophenes and sexithiophenes using electronic absorption spectroscopy and time-dependent DFT

The electronic absorption spectra of a series of alkoxy-styryl substituted terthiophenes, their corresponding sexithiophenes, and the oxidation products of both have been measured. The terthiophenes studied sigma-dimerize to sexithiophenes during the oxidation process and there is clear evidence of sexithiophene radical cations, dications, and pi-dimers in the electronic absorption spectra. The oxidation of concentrated solutions produces predominantly pi-dimer bands, as expected. The absorption spectrum of the styryl-functionalized sexithiophene dication without alkoxy substitution closely resembles that of unsubstituted sexithiophene, while alkoxy substitution induces changes in the wavelength of the dication band maximum and the overall band shape. Time-dependent density functional theory (TDDFT) calculations have shown that styryl-based molecular orbitals are important in the transitions of the neutral molecules as well as the charged species, the dication in particular. Kinetics analyses confirm the stabilization effect induced by the alkoxy substituents. The presence of a reversible pi-dimer equilibrium was verified by cyclic voltammetry. It is clear from the experimental observations and the theoretical calculations that both the styryl and alkoxy groups are influencing the electronic properties of this class of molecules.     

Palladium salicylaldimine complexes containing boronate esters

Reactions of salicylaldehydes with boronate ester derivatives of aniline have been examined. Addition of these Schiff base ligands to palladium acetate or Na₂PdCl₄ afforded novel boron-containing trans-bis(N-arylsalicylaldiminato) palladium complexes.Condensation of salicylaldehyde (2-HOC₆H₄C(O)H) with H₂NC₆H₄Bpin (pin=1,2-O₂C₂Me₄) afforded the boron-containing Schiff bases, 2-HOC₆H₄C(H)=NC₆H₄Bpin (1–3a). Similar reactivity with 2-hydroxy-5-nitrobenzaldehyde and 2-hydroxy-1-naphthaldehyde gave the corresponding Schiff bases (1-3b) and (1-3c), respectively. Reaction of Schiff bases (2) and (3) with palladium acetate or Na₂PdCl₄ afforded complexes of the type PdL₂ (4,5), where L=deprotonated Schiff base. The molecular structure of the nitro-salicylaldehyde 4-Bpin palladium complex (5b) was characterized by an X-ray diffraction study. All new palladium compounds have been characterized fully and tested for their antifungal activity against Aspergillus niger and Aspergillus flavus.     

Quantum‐Chemical and Electrochemical Investigation of the Electrochemical Windows of Halogenated Carborate Anions

The range of electrochemical stability of a series of weakly coordinating halogenated (Hal=F, Cl, Br, I) 1‐carba‐closo‐dodecaborate anions, [1‐R‐CB₁₁X₅Y₆]^? (R=H, Me; X=H, Hal, Me; Y=Hal), has been established by using quantum chemical calculations and electrochemical methods. The structures of the neutral and dianionic radicals, as well as the anions, have been optimized by using DFT calculations at the PBE0/def2‐TZVPP level. The calculated structures are in good agreement with existing experimental data and with previous calculations. Their gas‐phase ionization energies and electron affinities were calculated based on their optimized structures and were compared with experimental (cyclic and square‐wave) voltammetry data. Electrochemical oxidation was performed in MeCN at room temperature and in liquid sulfur dioxide at lower temperatures. All of the anions show a very high resistance to the onset of oxidation (2.15–2.85 V versus Fc^0/+), with only a minor dependence of the oxidation potential on the different halogen substituents. In contrast, the reduction potentials in MeCN are strongly substituent dependent (?1.93 to ?3.32 V versus Fc^0/+). The calculated ionization energies and electron affinities correlate well with the experimental redox potentials, which provide important verification of the thermodynamic validity of the mostly irreversible redox processes that are observed for this series. The large electrochemical windows that are afforded by these anions indicate their suitability for electrochemical applications, for example, as supporting electrolytes.     

Development and validation of an ELISA kit for the detection of ricin toxins from biological specimens and environmental samples

Ricin is a toxin that can be easily extracted from seeds of Ricinus communis plants. Ricin is considered to be a major bio-threat as it can be freely and easily acquired in large quantities. A deliberate release of such toxin in civilian populations would very likely overwhelm existing public health systems, resulting in public fear and social unrest. There is currently no commercially available or FDA-approved prophylaxis such as vaccines, or therapeutic antitoxins or antidotes, available for ricin intoxication. Patient treatment is typically supportive care based on symptoms, often designed to reinforce the body’s natural response. This paper describes the development and validation of a robust ELISA test kit, which can be used to screen for ricin in biological specimens such as whole blood and faeces. Faecal specimens are shown in this study to have better diagnostic sensitivity and a wider diagnostic window compared to whole blood. From these results, it is concluded that faeces is the most suitable clinical specimen for diagnosis of ricin poisoning via the oral route. The ELISA test kit can also detect ricin in environmental samples. An advantage of this ELISA kit over other commercial off-the-shelf (COTS) detection kits currently on the market that are developed to screen environmental samples only is its ability to diagnose ricin poisoning from clinical specimens as well as detect ricin from environmental samples.