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Here we provide an in-depth structural characterization of the amorphous ionic glasses LiPON and LiSiPON with high Li content. Based on ab-initio molecular dynamics simulations, the structure of these materials is an inverted structure with either isolated polyanion tetrahedra or polyanion dimers suspended in a Li+ matrix. Based on neutron scattering data, this type of inverted structure leads to a significant amount of medium-range ordering in the structure, as demonstrated by two sharp diffraction peaks and a periodic structural oscillation in the density function G(r). On a local scale, adding N and Si increases the number of anion bridges and polyanion dimer structures, leading to higher ionic conductivity. In the medium range ordering, the addition of Si leads to more disorder in the polyanion substructure but a significant increase in the ordering of the O substructure. Finally, we demonstrate that this inverted structure with medium range ordering results in a glassy material that is both mechanically stiff and ductile on the nanoscale. 

We study the ideas about teaching and learning mathematics that undergraduate students generate when they encounter tasks designed to embed approximations of teaching practice in mathematics courses taken by a general population of students. These tasks attend to the dual goals of developing an understanding of mathematics content and an understanding of how teachers provide classroom experiences that foster mathematics learning. The study employs a qualitative, multiple-case study methodology, with four cases bounded by the content areas of abstract algebra, single variable calculus, discrete mathematics, and introductory statistics. The data for the study come from undergraduate students’ written work on mathematical tasks, interviews with a subset of students from each course, and interviews with each instructor throughout the term during which they implemented the tasks. Our findings indicate that students identified the broad applicability of teaching skills (discussed by 32 of the 61 interviewed students), recognized the value of examining hypothetical learners’ mathematical work (discussed by 59 of the 61 interviewed students), and reported empathy for hypothetical learners (discussed by 38 of the 61 interviewed students). These findings persisted across the course content and course levels we studied, leading us to conclude that our findings can transfer to additional mathematics courses in secondary mathematics teacher preparation.

 

The Mathematical Education of Teachers as an Application of Undergraduate Mathematics project provides lessons integrated into various mathematics major courses that incorporate mathematics teaching connections as a legitimate application area of undergraduate mathematics. One feature of the lessons involves posing tasks that require undergraduates to interpret or analyze the work of another student. This paper reports on thematic analysis of hour-long interviews for eight participants enrolled in an undergraduate abstract algebra course from two different implementation sites. We focus on student work and reactions to these interpreting or analyzing student thinking (AST) applications as they relate to their perceptions regarding the use of AST applications as a mechanism to both deepen their content knowledge and improve their skills for communicating mathematics. Several participants identify positive benefits, but more research is needed to determine the how to incorporate AST applications to accommodate some participants’ reluctance to engage in new mathematical contexts.