An Investigation of Teachers’ Noticing, Cognitive Demand, and Mathematical Knowledge for Teaching: Video Reflections in an Elementary Mathematics Context

In the past decade, mathematics performance by all students, especially minority students in low socioeconomic schools, has shown limited improvement nationwide (NCES, 2011). Traditionally in the United States, mathematics has consisted of arithmetic and computational fluency; however, mathematics researchers widely believe that this method of instruction does not enhance the development of mathematical reasoning and ignores the research on students’ mathematical development (Blanton & Kaput, 2005; Stigler & Hiebert, 1999). Recommendations by the mathematics community are to broaden and strengthen teacher content knowledge in mathematics and to provide the pedagogical tools needed by teachers to extend their students’ thinking and reasoning (Darling-Hammond, Wei, Andree, Richardson, and Orphanos, 2009; Mewborn, 2003). The purpose of this quantitative study was to investigate the relationship between the teachers’ levels of noticing, the levels of cognitive demand in their enacted tasks, and their levels of mathematical knowledge for teaching in two urban high-need low performing elementary schools. The 54 elementary teachers participated in a long-term mathematics professional development program aimed at developing teachers’ mathematical knowledge for teaching and recognizing and fostering students’ early algebraic reasoning. The data for this dissertation included teachers’ self-selected video segments, written video reflections, and mathematical knowledge for teaching levels from the second year of the professional development. Relationships were explored between mathematical knowledge for teaching, teachers’ levels of noticing, and the levels of cognitive demand represented in mathematics lessons. The findings indicated shifts in teachers’ cognitive demand of enacted tasks and noticing over the course of the second year of professional development. Correlation results indicated significant relationships between teachers’ cognitive demand, teacher noticing, participation, and teachers’ mathematical knowledge for teaching. Moreover, the results showed that the teachers in the K-3 cohort benefited more from the professional development than their 4-6 cohort counterparts when it came to mathematical knowledge for teaching, noticing, and cognitive demand levels

Project for the analysis of technology transfer Quarterly evaluation report, 1 Jan. - 31 Mar. 1969

Technology transfer analysis project studying nonspace applications of NASA and AEC generated technolog

Nonequilibrium effects of anisotropic compression applied to vortex lattices in Bose-Einstein condensates

We have studied the dynamics of large vortex lattices in a dilute-gas Bose-Einstein condensate. While undisturbed lattices have a regular hexagonal structure, large-amplitude quadrupolar shape oscillations of the condensate are shown to induce a wealth of nonequilibrium lattice dynamics. When exciting an m = -2 mode, we observe shifting of lattice planes, changes of lattice structure, and sheet-like structures in which individual vortices appear to have merged. Excitation of an m = +2 mode dissolves the regular lattice, leading to randomly arranged but still strictly parallel vortex lines.Comment: 5 pages, 6 figure

Project for the analysis of technology transfer Quarterly report, 1 Jul. - 30 Sep. 1969

Research activities in technology transfer progra

Project for the analysis of technology transfer Quarterly report, 1 Oct. - 31 Dec. 1969

Analysis of Tech Brief-Technical Support Package progra

Potentials for which the Radial Schr\"odinger Equation can be solved

In a previous paper$^1$, submitted to Journal of Physics A -- we presented an infinite class of potentials for which the radial Schr\"odinger equation at zero energy can be solved explicitely. For part of them, the angular momentum must be zero, but for the other part (also infinite), one can have any angular momentum. In the present paper, we study a simple subclass (also infinite) of the whole class for which the solution of the Schr\"odinger equation is simpler than in the general case. This subclass is obtained by combining another approach together with the general approach of the previous paper. Once this is achieved, one can then see that one can in fact combine the two approaches in full generality, and obtain a much larger class of potentials than the class found in ref. $^1$ We mention here that our results are explicit, and when exhibited, one can check in a straightforward manner their validity

Gigantic transmission band edge resonance in periodic stacks of anisotropic layers

We consider Fabry-Perot cavity resonance in periodic stacks of anisotropic layers with misaligned in-plane anisotropy at the frequency close to a photonic band edge. We show that in-plane dielectric anisotropy can result in a dramatic increase in field intensity and group delay associated with the transmission resonance. The field enhancement appears to be proportional to forth degree of the number N of layers in the stack. By contrast, in common periodic stacks of isotropic layers, those effects are much weaker and proportional to N^2. Thus, the anisotropy allows to drastically reduce the size of the resonance cavity with similar performance. The key characteristic of the periodic arrays with the gigantic transmission resonance is that the dispersion curve omega(k)at the photonic band edge has the degenerate form Delta(omega) ~ Delta(k)^4, rather than the regular form Delta(omega) ~ Delta(k)^2. This can be realized in specially arranged stacks of misaligned anisotropic layers. The degenerate band edge cavity resonance with similar outstanding properties can also be realized in a waveguide environment, as well as in a linear array of coupled multimode resonators, provided that certain symmetry conditions are in place.Comment: To be submitted to Phys. Re