The mathematical knowledge and beliefs of elementary mathematics specialist-coaches

This paper reports on one aspect of a larger research project conducted in the United States that designed and implemented an elementary mathematics, specialist-coach preparation program and evaluated the effect of qualified specialist-coaches on student achievement. The paper discusses a conceptual framework for coaching in which a specialist-coach is to serve as a “more knowledgeable other” for a community of practice in a school, and ultimately to impact both the knowledge and professional practice of teachers and the school’s mathematics program as a whole. Specialist-coaches have unique opportunities and challenges in this daunting task, and the paper discusses one program designed to prepare well-respected teachers for the transition to the role and responsibilities of a specialist-coach. The reported analyses document changes in specialist-coaches’ mathematical content knowledge, mathematical knowledge for teaching, and beliefs regarding mathematics teaching and learning over the preparation program and during the specialist-coaches’ first years of service in a school. These specialist-coaches’ mathematical content knowledge grew and their beliefs became more aligned with a Making Sense perspective during the preparation program, and their changed state persisted throughout 2–3 years of service as specialist-coaches. Evidence addressing the specialist-coaches’ mathematical knowledge for teaching was mixed, but suggested that growth occurred both during the preparation program and in their first year of coaching, stabilizing in the years following.     

Reaction of acetylated carbohydrates with trimethylaluminum: concise synthesis of 1,2-O-isopropylidene d-ribofuranose

Treatment of β-d-ribose tetraacetate with trimethylaluminum gives α-3,5-O-acetyl-1,2-O-isopropylidene-d-ribofuranoside in excellent yield. This reaction allows for efficient and high-yielding installation of the 1,2-isopropylidene acetal (acetonide), which is difficult to prepare using more traditional acid-catalyzed methods. The reaction of trimethylaluminum with other per-acetylated carbohydrates is also described.     

Controlling influences in bisspiroketal formation: synthesis of the ABC ring system of azaspiracid

Substitution effects on the stereochemical outcome of bisspiroketalization on the C₁-C₁₇ carbon backbone of azaspiracid is presented. A possible explanation is offered to explain the observed stereochemical outcome.     

GMRES and the minimal polynomial

We present a qualitative model for the convergence behaviour of the Generalised Minimal Residual (GMRES) method for solving nonsingular systems of linear equationsAx =b in finite and infinite dimensional spaces. One application of our methods is the solution of discretised infinite dimensional problems, such as integral equations, where the constants in the asymptotic bounds are independent of the mesh size.Our model provides simple, general bounds that explain the convergence of GMRES as follows: If the eigenvalues ofA consist of a single cluster plus outliers then the convergence factor is bounded by the cluster radius, while the asymptotic error constant reflects the non-normality ofA and the distance of the outliers from the cluster. If the eigenvalues ofA consist of several close clusters, then GMRES treats the clusters as a single big cluster, and the convergence factor is the radius of this big cluster. We exhibit matrices for which these bounds are tight.Our bounds also lead to a simpler proof of existing r-superlinear convergence results in Hilbert space.This research was partially supported by National Science Foundation grants DMS-9122745, DMS-9423705, CCR-9102853, CCR-9400921, DMS-9321938, DMS-9020915, and DMS-9403224.     

PHELIX – Status and First Experiments

The high-energy high-power laser system PHELIX (Petawatt High Energy Laser for heavy Ion eXperiments) [1] is currently under construction at the Gesellschaft fuer Schwerionenforschung mbH (GSI) Darmstadt. With PHELIX GSI will offer the unique combination of a high-current, high-energy (GeV/u) heavy-ion beam with an intense laser beam. This will open the door to a variety of fundamental science issues in the field of atomic physics, plasma physics and nuclear physics. The project will gain further interest in the near future by the dramatic increase of the accelerator performance with the starting FAIR project at GSI [2]. This paper reports the current status of the project as well as the laser architecture. The proposed physics program and a first experiment carried out with PHELIX, the realization of a transient collisionally excited x-ray laser [3], will also be reviewed briefly.