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.     

A finite element displacement model valid for any value of the compressibility

In this paper it is shown how the displacement formulation of the theorem of minimum potential energy can be used with the finite element method to approximate both compressible and incompressible equilibria of linearly elastic, isotropic solids. The procedure is shown to be equivalent to the more complicated “mixed principle” technique, due to the use of numerical integration applied to the computation of the element stiffness matrices. Criteria for the choice of integration formulas and elements are discussed, and numerical examples are presented.     

Controlled Light‐Mediated Preparation of Gold Nanoparticles by a Norrish Type I Reaction of Photoactive Polymers

Gold nanoparticles (AuNPs) are subjects of broad interest in scientific community due to their promising physicochemical properties. Herein we report the facile and controlled light‐mediated preparation of gold nanoparticles through a Norrish type I reaction of photoactive polymers. These carefully designed polymers act as reagents for the photochemical reduction of gold ions, as well as stabilizers for the in situ generated AuNPs. Manipulating the length and composition of the photoactive polymers allows for control of AuNP size. Nanoparticle diameter can be controlled from 1.5 nm to 9.6 nm.     

Numerical breakdown at high Weissenberg number in non-Newtonian contraction flows

 Planar contraction flows of non-Newtonian fluids with integral constitutive models are studied to investigate the problem of numerical breakdown at high Weissenberg or Debrorah numbers. Spurious shear stress extrema are found on the wall downstream of the re-entrant corner for both sharp and rounded corners. Moreover, a non-monotonic relation between shear stress and strain rate is found when the Deborah number limit is approached, which correlates with these shear extrema. This strongly suggests that non-monotonicity between shear stress and strain rate may be responsible for the Deborah number limit problem in contraction flow simulations. This non-monotonicity is caused by the inaccuracy of the quadrature, using constitutive equations that do not have shear stress maxima when exactly evaluated. This conclusion agrees with recent analytical findings by others that inaccuracy of the integration along the streamlines – either by numerical integration or asymptotic approximation – makes the problem ill-conditioned, with spurious growth occurring on the wall downstream of the re-entrant corner. Received: 5 March 1999/Accepted: 1 September 1999     

RAPID STOMATAL RESPONSE TO RED LIGHT IN Zea mays

Gas exchange techniques were employed to study responses of stomatal conductance to pulses of red and blue light in the grass, Zea mays. Zea mays exhibited conductance increases following brief (< 1 min) pulses of either red or blue light, in contrast to other species (e.g. Commelina communis; Assmann, 1988, Plant Physiol. 87 , 226–231) that exhibit consistent conductance responses only to pulses of blue light. Red light pulses of 450 μmol m^?2s^?1 for x min or 225 μmol m^?2s^?1 for 2x min were used to probe the fluence dependence of the red light response. The red light-stimulated conductance increase was constant for a given fluence, and increased with increasing total fluence. The conductance response to red light was larger in field grown plants (maximum growth irradiance ? 1600 μmol m-²s^?l) than in plants raised in growth chambers (maximum growth irradiance ? 150 μmol m^?2s^?1).     

The hole-pressure effect and viscometry

This paper provides an assessment of our current understanding of the hole-pressure effect and the role it plays in the field of viscometry. The discovery in 1968 by Broadbent, Kaye, Lodge and Vale that the use of pressure taps to measure the normal stress exerted on a boundary by a non-Newtonian material can lead to significant errors has opened up interesting new avenues for making viscometric measurements. The kinematical theory developed by Higashitani and Pritchard (1972) to estimate the hole pressure, to which a deeper understanding has recently been supplied by Yao and Malkus (1990), has formed the basis of many new developments in viscometry. It is interesting that such a simplistic theory for the hole pressure apparently has a rather wide range of applicability.This paper was the basis of one of the lectures presented at the A. S. Lodge Retirement Meeting in May 1991. It was to have been published in the special issue (No. 1, Vol. 31) in honor of Professor A. S. Lodge, but was unavoidably delayed owing to the illness of one of the authors.     

Cancellation of errors in the Higashitani-Pritchard treatment of hole pressures generated by viscoelastic liquids in creeping flow past a transverse slot

Analytical studies of the hole pressure for non-Newtonian creeping flow past a transverse slot are pursued with particular interest in the formulation of Higashitani and Pritchard (HP). To correct the flaws in the treatment of HP's original work, a modified hole-pressure relation (MHPR) is employed. Some important mathematical properties of the MHPR are presented. By studying the MHPR in streamline coordinate formulation, we find a fortuitouserror cancellation phenomenon in the derivation of the HP formula: namely, the error caused by one key flaw is fortuitously cancelled out by the error introduced through another key flaw. For second-order fluids and Tanner's viscometric model (under certain assumptions) the cancellation of errors is proved to be exact. It is this cancellation of errors that provides a theoretical explanation for the paradox between an apparently flawed derivation and the fortunate success of the HP prediction.     

Boundary node correction and superconvergence in the FEM

Convergence improvement and superconvergence behaviour, obtained by the simple boundary node correction (BNC) procedure for certain stress-like variables of smoothed FEM solutions, are reported in this paper. The effectiveness of BNC is shown through three examples of steady flow problems, and a posterior error analysis based on the multiple-mesh extrapolation technique has been used for estimating the convergence rates.