Visualisation and proof: a brief survey of philosophical perspectives

The contribution of visualisation to mathematics and to mathematics education raises a number of questions of an epistemological nature. This paper is a brief survey of the ways in which visualisation is discussed in the literature on the philosophy of mathematics. The survey is not exhaustive, but pays special attention to the ways in which visualisation is thought to be useful to some aspects of mathematical proof, in particular the ones connected with explanation and justification.     

Probing interactions in exploratory teaching: a case study

This study investigates an exploratory teaching style used in an undergraduate geometry course to help students identify an ellipse. We attempt to probe beneath the surface of exploration to understand how the actions of teachers can contribute to developing students’ competence in justifying an ellipse. We analyse the complex interactions between student, content, and teacher, and discuss explicit pedagogical strategies that help students develop a higher level of geometric reasoning. The findings indicate that students engaged in guided explorations by the teacher and in group discussions with peers were able to identify an ellipse and justify their reasoning.     

Molecular organization in MAPLE‐deposited conjugated polymer thin films and the implications for carrier transport characteristics

The morphological structure of poly(3‐hexylthiophene) (P3HT) thin films deposited by both Matrix Assisted Pulsed Laser Evaporation (MAPLE) and solution spin‐casting methods are investigated. The MAPLE samples possessed a higher degree of disorder, with random orientations of polymer crystallites along the side‐chain stacking, π–π stacking, and conjugated backbone directions. Moreover, the average molecular orientations and relative degrees of crystallinity of MAPLE‐deposited polymer films are insensitive to the chemistries of the substrates onto which they were deposited; this is in stark contrast to the films prepared by the conventional spin‐casting technique. Despite the seemingly unfavorable molecular orientations and the highly disordered morphologies, the in‐plane charge carrier transport characteristics of the MAPLE samples are comparable to those of spin‐cast samples, exhibiting similar transport activation energies (56 vs. 54 meV) to those reported in the literature for high mobility polymers. © 2016 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2017 , 55, 39–48     

Interfacial differences in enhanced schottky barrier height Au/n-GaAs diodes deposited at 77 K

The use of cryogenic temperatures (∼77 K) during Au Schottky contact deposition onto n-GaAs produces an increase in barrier height from 0.73 eV for room temperature diodes to 0.82 eV. Not all Schottky metals show this enhancement—for example Pt and Ti do not show any significant change in barrier height whereas Au, Pd and Ni show increases between 7 and 18%. We used X-ray reflectivity to show that the main difference between Au deposited at 77 K and room temperature is a decreased metal roughness while the interfacial roughness between the Au and GaAs is basically the same. As the diodes are annealed to 300 °C both the difference in barrier height and interfacial roughness is lost. This is a simple method with potential for improving the performance of GaAs metal-semiconductor-field-effect-transistors (MESFETs).     

Teaching proof in the context of physics

This paper describes an application of statics to geometrical proofs in the classroom. The aim of the study was to find out whether the use of concepts and arguments from statics can help students understand and produce proofs of geometrical theorems. The two theorems studied were (1) that the medians in a triangle meet at a single point which is the centre of gravity of the triangle, and (2) the Varignon theorem, that the lines joining the midpoints of successive sides of a quadrilateral form a parallelogram. The classroom experiment showed that most students were successful in using arguments from statics in their proofs, and that they gained a better understanding of the theorems. These findings lend support to the claim that the introduction of statics helps students produce proofs and grasp their meaning.     

Intermediate coupling model description of55Fe and57Fe

The properties of the negative parity states of⁵⁵Fe and⁵⁷Fe are investigated in the framework of the intermediate coupling model. In the model, a neutron or a quasineutron is coupled to anharmonic vibrations of the core. Anharmonicities of the vibrations are estimated through the observed properties of the core. Energy levels, spectroscopic factors and electromagnetic properties have been calculated. The results of the present calculations are also compared with available experimental results and other theoretical results. The model reasonably accounts for many of the properties of the low-lying states.