Planning District‐Wide Professional Development: Insights Gained From Teachers and Students Regarding Mathematics Teaching in a Large Urban District

This paper describes the mechanism used to gain insights into the state of the art of mathematics instruction in a large urban district in order to design meaningful professional development for the teachers in the district. Surveys of close to 2,000 elementary, middle school, and high school students were collected in order to assess the instructional practices used in mathematics classes across the district. Students were questioned about the frequency of use of various instructional practices that support the meaningful learning of mathematics. These included practices such as problem solving, use of calculators and computers, group work, homework, discussions, and projects, among others. Responses were analyzed and comparisons were drawn between elementary and middle school students' responses and between middle school and high school responses. Finally, fifth‐grade student responses were compared to those of their teachers. Student responses indicated that they had fewer inquiry‐based experiences, fewer student‐to‐student interactions, and fewer opportunities to defend their answers and justify their thinking as they moved from elementary to middle school to high school. In the elementary grades students reported an overemphasis on the use of memorization of facts and procedures and sparse use of calculators. Results were interpreted and specific directions for professional development, as reported in this paper, were drawn from these data. The paper illustrates how student surveys can inform the design of professional development experiences for the teachers in a district.     

A characterization of adding machine maps

Let f :X→X be a continuous map of a compact metric space to itself. We prove that f is topologically conjugate to an adding machine map if and only if X is an infinite minimal set for f and each point of X is regularly recurrent. Moreover, if X is an infinite minimal set for f and one point of X is regularly recurrent, then f is semiconjugate to an adding machine map.     

Matrix extensions and eigenvalue completions, the generic case

In this paper we provide new necessary and sufficient conditions for the so-called eigenvalue completion problem.

     

Fluorination of diamond — C4F9I and CF3I photochemistry on diamond (100)

The radiation-induced decomposition of C₄F₉I and CF₃I overlayers at 119 K on diamond (100) surfaces has been shown to be an efficient route to fluorination of the diamond surface. X-ray photoelectron spectroscopy has been used for photoactivation as well as for studying the photodecomposition of the fluoroalkyl iodide molecules, the attachment of the photofragments to the diamond surface, and the thermal decomposition of the fluoroalkyl ligands. Measured chemical shifts agree well with ab initio calculations of both C 1s and F 1s binding energies. It is found that chemisorbed CF₃ groups on diamond (100) decompose by 300 K whereas C₄F₉ groups decompose over the range 300 to 700 K and this reactivity difference is rationalized on steric grounds. Both of these thermal decomposition processes produce surface C---F bonds on the diamond. The surface C---F species thermally decompose over a wide temperature range extending up to 1500 K. Hydrogen passivation of the diamond surface is ineffective in preventing free radical attack from the photodissociated products of the fluoroalkyl iodides; I atoms produced photolytically abstract H from surface C---H bonds to yield hydrogen iodide at 119 K allowing diamond fluorination. The attachment of chemisorbed F species to the diamond (100) surface causes band bending as the surface states are occupied as a result of chemisorption. This results in a shift to higher binding energy of the diamond-related C 1s levels present in the surface and subsurface regions which are sampled by XPS on the diamond. The use of photoactivation of fluoroalkyl iodides for the fluorination of diamond surfaces provides a convenient route compared to other methods involving the action of atomic F, molecular F₂, XeF₂ and F-containing plasmas.     

Stereospecific solid-state NaBH4-promoted reduction of 1-methylpentacyclo[5.4.0.02,6.03,10.05,9]undecane-8,11-dione

NaBH₄ reduction of a cage dione proceeds in a stereospecific fashion to give theendo,endo-diol. This reactivity is related to the crystal structure.     

The Livsic Cocycle Equation for Compact Lie Group Extensions of Hyperbolic Systems

In this article we extend well-known results of Livsic on theregularity of measurable solutions to the cocycle equation.Livsic proved a regularity result for real valued cocycles,but for compact Lie group valued cocycles his result was restrictedto coboundaries (that is, trivial or constant cocycles). Inthis article we prove the complete result. We present several useful applications of these results.