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.     

From dilute to dense self-avoiding walks on hypercubic lattices

A field-theoretic representation is presented to count the number of configurations of a single self-avoiding walk on a hypercubic lattice ind dimensions with periodic boundary conditions. We evaluate the connectivity constant as a function of the fractionf of sites occupied by the polymer chain. The meanfield approximation is exact in the limit of infinite dimensions, and corrections to it in powers ofd ^–1 can be systematically evaluated. The connectivity constant and the site entropy calculated throughout second order compare well with known results in two and three dimensions. We also find that the entropy per site develops a maximum atf1–(2d)^–1. Ford=2 (d=3), this maximum occurs atf~0.80 (f~0.86) and its value is about 50% (30%) higher than the entropy per site of a Hamiltonian walk (f=1).     

A new approach to micropatterning: application of potential-assisted ion transfer at the liquid-liquid interface for the local metal deposition

A new approach to micropatterning is demonstrated. The approach is based on driving an electrochemical process at the solid-liquid interface through the formation of a flux of ions from a micropipet that is held in close proximity to the surface. The flux of ions is generated by the so-called potential assisted ion transfer at the interface between two immiscible electrolyte solutions (ITIES). As a model system, the local deposition of silver was examined. Specifically, a constant potential, which was applied to a micropipet filled with an aqueous solution of silver ions, caused the transfer of Ag(+) into the outer nitrobenzene (NB) solution that consisted of an electrolyte, tetrabutylammonium tetrakis[4-chlorophenyl]borate (TBATPBCl). To facilitate the transfer of silver ions a macrocyclic ligand, that is, dibenzo-24-crown-8 (DB24C8), was added to the organic phase. The Faradaic current of this micro-ITIES was used as a means of controlling the tip-surface distance in scanning electrochemical microscopy (SECM) and depositing silver microstructures on a gold substrate.     

Hydro­gen‐bonded supramolecular lattice of the 1:3:4 complex between [5,10,15,20‐meso‐tetrakis(4‐hydroxy­phenyl)­porphyrinato‐κ4N]­zinc(II), dibenzo‐24‐crown‐8 and methanol

The title compound, [5,10,15,20‐meso‐tetrakis(4‐hydroxy­phenyl)­porphyrinato‐κ⁴N]­zinc(II) tris(dibenzo‐24‐crown‐8) methanol tetrasolvate, [Zn(C₄₄H₂₈N₄O₄)]·3C₂₄H₃₂O₈·4CH₄O, was synthesized and its molecular structure precisely charac­terized by low‐temperature single‐crystal analysis. All the components are involved in hydrogen bonding with each other, thus forming an extensively hydrogen‐bonded supramolecular lattice. The functionalized porphyrin moiety coordinates both equatorially and axially to the neighboring species.     

Palladium Complexes of Corroles and Sapphyrins

Palladium complexes of corrole and sapphyrin were prepared in high yield and fully characterized. The corrole provides a tetradentate/trianionic square planar coordination sphere for Pd^II, charge balanced by pyridinium. Both one and two Pd^II ions may be accommodated by the pentapyrrolic skeleton of the sapphyrin, and in each case the macrocycle acts as bidentate/monoanionic ligand and the inner-sphere square planar geometry is completed by allyl anions coordinated in an η³ fashion. NMR spectroscopy and X-ray crystallography data analyses uncovered the presence of interesting stereoisomers due to the flexibility of the ally ligands and also the pyrrole ring(s) that is/are not involved in metal binding.     

Kinetic energy of structural protons in silica xerogels

The kinetic energies of the protons in the silanol groups (Si–OH units) of silica xerogels were deduced by ab initio calculations using the basis set mp2/6-311G**. The silanol groups were simulated using the Si(OH)₄ unit. The calculated result of the H-kinetic energy was found to be 150 meV, which is ~ 50% smaller than a recently reported experimental value for porous silica xerogels. For comparison, the same calculations of the proton kinetic energies of other H-containing molecules such as H₂O and CH₄ (being also ~ 150 meV) were found to be in excellent agreement with measurements. Possible reasons for the huge deviations in the case of the silanols are discussed.     

High Charge Delocalization and Conjugation in Oligofuran Molecular Wires

The extent of charge delocalization and of conjugation in oligofurans and oligothiophenes was studied by using mixed valence systems comprising oligofurans and oligothiophenes capped at both ends by ferrocenyl redox units. Using electrochemical, spectral, and computational tools, we find strong charge delocalization in ferrocene‐capped oligofurans which was stronger than in the corresponding oligothiophene systems. Spectroscopic studies suggest that the electronic coupling integral (H_ab) is roughly 30–50 % greater for oligofuran‐bridged systems, indicating better energy matching between ferrocene units and oligofurans. The distance decay constant (damping factor), β, is similar for oligofurans (0.066 A^?1) and oligothiophenes (0.070 A^?1), which suggests a similar extent of delocalization in the bridge, despite the higher HOMO–LUMO gap in oligofurans. Computational studies indicate a slightly larger extent of delocalization in furan‐bridged systems compared with thiophene‐bridged systems, which is consistent with oligofurans being significantly more rigid and less aromatic than oligothiophenes. High charge delocalization in oligofurans, combined with the previously reported strong fluorescence, high mobility, and high rigidity of oligofuran‐based materials makes them attractive candidates for organic electronic applications.