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 spectral Paley-Wiener theorem

The Fourier inversion formula in polar form is \(f(x) = \int_0^\infty {P_\lambda } f(x)d\lambda \) for suitable functionsf on ? ⁿ , whereP _λ f(x) is given by convolution off with a multiple of the usual spherical function associated with the Euclidean motion group. In this form, Fourier inversion is essentially a statement of the spectral theorem for the Laplacian and the key question is: how are the properties off andP _λ f related? This paper provides a Paley-Wiener theorem within this avenue of thought generalizing a result due to Strichartz and provides a spectral reformulation of a Paley-Wiener theorem for the Fourier transform due to Helgason. As an application we prove support theorems for certain functions of the Laplacian.