Preservice Teachers’ Understandings of the Relation Between a Fraction or Integer and Its Decimal Expansion: Strength and Stability of Belief

Abstract

In an earlier study of preservice elementary and middle school teachers’ beliefs about repeating decimals, Weller, Arnon, and Dubinsky (2009) reported on a comparison of the mathematical performance of 77 preservice teachers who completed an APOS-based instructional unit with 127 preservice teachers who completed traditional instruction. The purpose of the current study, based on 47 interviews conducted 4 months after the instruction, during which time there was no further instruction on this topic, is to determine the strength and stability (over time) of the students’ beliefs, to uncover thinking that did not arise in the earlier study, and to see whether the interviews yield similar comparative results. The interviews did uncover changes in student thinking. The students who received the APOS-based instruction developed stronger and more stable (over time) beliefs that a repeating decimal is a number; a repeating decimal has a fraction or integer to which it corresponds; a repeating decimal in general equals its corresponding fraction or integer; and, in particular, In addition, a number of indices and categories were developed that may prove useful in other studies consisting of the comparison of interview and questionnaire data involving a large number of interview subjects.     

A Demonstration of High-Throughput Electrophoresis

Automating electrophoresis significantly reduces the time required for loading a large number of samples, increases the speed of electrophoresis analysis, and maximizes the resolution power (clear separation of fragments) of this technique. In addition, automation increases the precision of electrophoresis analysis. Here we demonstrate an automated, high-throughput method of loading 96 samples simultaneously onto an electrophoresis gel, using the Apogent Discoveries Tango™ system and the Invitrogen™ E-Gel® 96 system.     

Particle rearrangement during sintering of heterogeneous powder mixtures: A combined experimental and theoretical study

We studied the morphology evolution of individual three-particle W-Cu-W agglomerates during sintering anneals at 1000 °C. The angle between the lines connecting the centers of W and Cu particles decreased during annealing, provided its initial value was different from 180°. While the W particles retained their spherical shape during sintering, the shape of Cu particles changed significantly after long sintering times. Both the rate of the particles’ rearrangement and the linear shrinkage of the particle pairs determined in the experiments increased with decreasing initial angle between the particles. We proposed an analytical model of sintering of cylindrical three-wire agglomerates controlled by self-diffusion of Cu atoms along the surface of Cu wire and along the W-Cu interphase boundary. The proposed model incorporates a thermodynamic method for calculating the distribution of chemical potential of metal atoms moving along the rigid cylindrical interphase boundary. The predictions of the model were in good agreement with the experimental data, underlying an important role played by capillary-driven diffusion in rearrangement of particles during sintering.     

Quorum-sensing Cascades Governing Bacterial Multicellular Communities

Quorum sensing (QS) is an efficient mode of intercellular communication between bacteria. This mode is regulated by self-produced small chemical signals, activating dedicated receptors once accumulated. Numerous architecturally complex QS cascades are cardinal for governing bacterial behaviors, such as pathogenicity, luminescence, and bacterial competence. Importantly, QS cascades are essential for the formation of bacterial multicellular communities. Once informed via QS cascades, motile cells often organize themselves into conspicuous multicellular structures that carry out specialized tasks. This review focuses on the major QS systems, playing an active role in the rise of complex bacterial communities in different bacterial models.     

Capillary-driven motion of nanoparticles attached to curved rigid substrates

We considered the shape and crystallographic orientation evolution of nanoparticles attached to a curved rigid substrate at elevated temperatures at which the particle atoms move by diffusion along the particle surface and along the particle–substrate interface. We employed a variational method for calculating the chemical potential of the atoms of nanoparticles located at the particle–substrate interface. This chemical potential can be determined by calculating the change of total surface and interface energy of the system caused by two infinitesimal orthogonal translations and one rigid rotation of the particle. Once the distribution of the chemical potential along the particle–substrate interface is known, the kinetics of particle movements can be determined by coupling diffusion fluxes along the particle surface and particle–substrate interface. We illustrate the general theory by analyzing the particle movements on the surface step, on the sinusoidally patterned substrate, and within the corner formed by two intersecting planar surfaces. We also show that the developed theory can be applied to the analysis of burrowing of rigid ceramic nanoparticles into the metal substrate.     

The effect of evaporation on size and shape evolution of faceted gold nanoparticles on sapphire

We studied the size and shape evolution of about 180 faceted gold nanoparticles attached to a sapphire substrate during annealing at 950 °C in air. We employed the scanning force microscopy and interrupted annealing techniques to track the changes in size and shape of individual nanoparticles. The height of all single-crystalline nanoparticles was constant up to the longest cumulative annealing time of 65 h. The lateral dimensions of ∼20% of all nanoparticles shrunk during anneals, while all three dimensions of the remaining 80% of nanoparticles remained constant. Only the nanoparticles with the height below the average (for all particles) were laterally shrinking. We formulated a kinetic model relating the lateral shrinkage of the nanoparticles to the evaporation of Au atoms adsorbed on sapphire. We also assumed that the process controlling particles shrinkage is the slow self-diffusion of Au atoms along the lateral facets of the nanoparticles toward the substrate. The model predicted a power law dependence of the shrinkage rate on the particle height, with the exponent n = 3. The corresponding exponent determined from the experimental data was n = −2.9 ± 0.3, in excellent agreement with the theory. The low value of the effective self-diffusion coefficient along the lateral facets determined with the aid of our model (3.2 ± 0.2 × 10⁻¹⁷ m² s⁻¹) was attributed to the difficulties of step nucleation on atomically flat facets.     

Pseudoelasticity of Metal Nanoparticles Is Caused by Their Ultrahigh Strength

An assembly of hemispherical Ag nanoparticles is prepared by solid‐state dewetting of thin Ag film deposited on the sapphire substrate. The in situ nanomechanical compression testing of the particles with a flat diamond punch inside the scanning electron microscope demonstrates the deformation behavior typical for the nucleation‐controlled plasticity: high elastic deformation followed by an abrupt particles collapse. The latter is associated with the dislocations nucleation in otherwise pristine particle. The average contact pressure in the contact zone at the onset of dislocation‐controlled plasticity is about 8 GPa, and does not depend on particle size. This observation supports the hypothesis that the pseudoelasticity of much smaller Ag nanoparticles is intrinsically related to their ultrahigh strength. A stress‐induced diffusion along the particle–substrate and particle–punch interfaces is identified as a factor controlling the pseudoelastic deformation. The corresponding diffusion model allows estimating the room‐temperature self‐diffusion coefficient of Ag along the Ag–W and Ag–zirconia interfaces, which is quite close to the estimated value of the grain boundary self‐diffusion coefficient in Ag. Based on this finding, the map of pseudoelastic deformation of crystalline materials is proposed.     

Mg3Cd: A model alloy for studying the destabilization of magnesium hydride

We prepared an ordered Mg₃Cd alloy by high energy ball milling of elemental powders. The synthesized alloy exhibited good hydrogenation kinetics and reversibly absorbed about 2.8 wt. % of hydrogen. The temperature dependence of hydrogenation kinetics of the alloy measured in the range of temperatures covering the order-disorder phase transformations in the Mg₃Cd and MgCd phases did not exhibit any anomalies and could be fitted with a single Arrhenius line. The measured apparent activation energy (69 ± 2 kJ/mol) hinted that hydrogenation process was controlled by diffusion of Cd in metallic phase. The pressure-composition isotherms exhibited negligible pressure hysteresis and sloping pressure plateau. Based on microstructural evidence obtained with the aid of X-ray diffraction and scanning electron microscopy, we built a thermodynamic model predicting the plateau hydrogen pressure for partially hydrogenated alloy. The predictions of the model were in a good agreement with the experimental data. Finally, we discussed the origins and the growth mechanisms of Cd whiskers observed in the alloys after full hydrogenation cycle.     

A bacteriophage reagent for Salmonella: molecular studies on Felix 01

Felix 01 (F01) is a bacteriophage originally isolated by Felix and Callow which lyses almost all Salmonella strains and has been widely used as a diagnostic test for this genus. Molecular information about this phage is entirely lacking. In the present study, the DNA of the phage was found to be a double-stranded linear molecule of about 80 kb. 11.5 kb has been sequenced and in this region A + T content is 60%. There are relatively few restriction endonuclease cleavage sites in the native genome and clones show this is due to their absence rather than modification. A restriction map of the genome has been constructed. The ends of the molecule cannot be ligated although they contain 5' phosphates. At least 60% of the genome must encode proteins. In the sequenced portion, many open reading frames exist and these are tightly packed together. These have been examined for homology to published proteins but only 1 to 17 shows similarity to known proteins. F01 is therefore the prototype of a new phage family. On the basis of restriction sites, codon usage and the distribution of nonsense codons in the unused reading frames, a strong case can be made for natural selection that reacts to mRNA structure and function.