Learning and Assessing Mathematics through Reading and Writing

Students learn mathematics more effectively and more deeply when reading and writing is directed at learning mathematics. Although reading and writing in mathematics may necessitate more skills and practice to master, the mathematical learning derived from reading and writing mathematics far outweighs the burden it places on teachers and students. This paper explores the vital role of purposeful reading and writing in the mathematics classroom and outlines some techniques to promote successful integration of reading and writing in mathematics.     

Ultrathin layer convolution

Ultrathin layers are defined as thin layers which are sufficiently thin that no concentration gradients are established within the layer on the time scale of a voltammetric measurement. Mathematically, ultrathin layers are characterized by ordinary differential equations in time. These equations are simpler to solve than the space and time dependent partial differential equations which describe all other electrode geometries. In this paper, a method is presented which capitalizes on the mathematical simplicity of the ultrathin layer to model any arbitrary, parameterizable electrode geometry. Laplace transforms are used to find an integral relationship between the current response of the modeled geometry and the ultrathin layer current. The integral relationship can be evaluated either analytically or numerically. Any voltammetric perturbation, under either Nernstian or mass transport-controlled conditions, can be modeled. The method is demonstrated for both planar and spherical electrodes. Cyclic voltammetric responses are modeled numerically and potential step responses are modeled analytically. It is also shown that for cyclic voltammetric perturbations, the current-voltage curves for the following systems have the same functional form. That is, the curves are identical within known multiplicative constants. The functionally equivalent responses are for (1) radial diffusion to a point electrode (the polarographic curve), (2) convective transport to a rotating disk, (3) the integral of the ultrathin layer response, and (4) the convolution or semi-integration of the response for linear diffusion to a planar electrode.     

The approximation of confluence by resonance in inhomogeneous plasmas

The degree to which a Budden-type equation describes the behaviour of the long wavelength branches of the modes of a higher order differential equation such as governs confluence in a plasma at ion-ion and upper hybrid resonance frequencies and at cyclotron harmonics is examined.     

The NCTM Process Standards and the Five Es of Science: Connecting Math and Science

This study investigates defining characteristics among the process standards of the Principles and Standards for School Mathematics and the 5 Es from the National Science Education Standards and the Inquiry and the National Science Education Standards. These characteristics are used to demonstrate similarities and differences between the learning of mathematics and science, discuss implications from such, and argue for the integration of mathematics and science curriculum and instruction.     

Kinetics of film-coated electrodes: Effect of a finite mass transfer rate of substrate across the film—solution interface at steady state

Previous models for describing the mediation kinetics of film-coated electrodes quantitatively are extended to account for a finite mass transfer rate of the substrate across the film—solution interface. Experimental data from polymer-coated rotating disk electrode experiments, analyzed by the extended model, provide evidence for finite interfacial mass transfer rates. Substrate size and charge contribute to this interfacial rate. Interfacial mass transfer resistance is more pronounced for polymerized films such as poly(vinyl ferrocene) and poly[Ru(vbpy)₃]²⁺ than for highly swollen polymers which bind electroactive species, e.g., poly(lysine).     

Polymer films on electrodes: Part XVIII. Determination of heterogeneous electron transfer kinetics at poly(vinylferrocene) and nafion/Ru(bpy)2+3 polymer-modified electrodes by convolution voltammetry

Convolution voltammetry was used to evaluate the rates of heterogeneous charge transfer to ferrocene groups in poly(vinylferrocene) and to Ru(bpy)²⁺₃ in Nafion-modified electrodes under semi-infinite conditions. This technique allows correction for uncompensated resistance and double layer capacitance, as well as detrmination of the diffusion coefficient, D, transfer coefficient, α, and half-wave potential, E_1/2, from a single cyclic voltammogram. Vinylferrocene in solution and a bound copolymer of vinylferrocene and styrene in a ratio of 58:42 were also examined. For the polymer films, the heterogeneous charge transfer rate constants, k°, are 10^?4 ≥ k° ≥ 10^?5 cm/s; these values are about two order of magnitude smaller than those for the similar species in homogeneous solution. The values of k°/D^1/2, however, are comparable to those in soluton; 10 > (k°/D^1/2) > 0.1 s^?1/2.     

Combating drug-resistant bacteria: small molecule mimics of plasmid incompatibility as antiplasmid compounds

A major mechanism for bacterial resistance to antibiotics is through the acquisition of a plasmid coding for resistance-mediating proteins. Described herein is a strategy to eliminate these plasmids from bacteria, thus resensitizing the bacteria to antibiotics. This approach involves mimicking a natural mechanism for plasmid elimination, known as plasmid incompatibility. The compound apramycin was identified as a tight binder to SLI RNA (Kd = 93 nM), the in vivo target of the plasmid incompatibility determinate RNA I, and footprinting/mutagenesis studies indicate apramycin binds SLI in the important regulatory region that dictates plasmid replication control and incompatibility. In vivo studies demonstrate that this compound causes significant plasmid loss and resensitizes bacteria to conventional antibiotics. The demonstration that a small molecule can mimic incompatibility, cause plasmid elimination, and resensitize bacteria to antibiotics opens up new targets for antibacterial research.