Reviving Pólya's “Look Back” in a Singapore school

This study is based on the stance that Pólya's “Look Back,” though understudied, remains relevant to Mathematics curricula that place emphasis on problem solving. Although the Singapore Mathematics curriculum adopts the goal of teaching Look Back, research about how it is carried out in actual classroom practice is rare. In our project, we focus on a redesign of a teacher development programme that is targeted to help teachers realize Pólya's original vision of Look Back in the classroom. We report the cases of two teachers who have participated in the teacher development programme: their thinking about Look Back (through interview records) and their interpretation of Look Back in their classroom practice (through video records). By bringing these two spheres of data together, we discuss their degree of ‘buy-in’ to Look Back in the overall problem solving enterprise.     

Diagonal quasi-Newton methods via least change updating principle with weighted Frobenius norm

Numerical Algorithms - This paper presents a class of low memory quasi-Newton methods with standard backtracking line search for large-scale unconstrained minimization. The methods are derived by...     

PIV measurements of the flow field inside an enclosed cubical cavity in natural convection

The natural convective velocity field in an enclosed air-filled cubical cavity with two opposing isothermal faces and the remaining four sides having a well-defined linear temperature rise from the cold to the hot face has been measured at different physical orientations and Rayleigh (Ra) numbers. In particular, two components of the velocity at the mid-plane have been measured by using particle image velocimetry (PIV) at Ra = 10⁶ and 6 × 10⁶ at each of two different physical orientations: heating-from-the-side (HFS), and heating-from-below (HFB). The 95% confidence limit uncertainties in the measured velocity vectors are about 2% for laminar flow. The accuracy and integrity of the experiments were validated by the comparison to some well-established CFD results at the HFS orientation at Ra = 10⁶. It was concluded that the experimental method is sound and so findings at other orientations and at other values of Ra should have an accuracy consistent with the findings of the uncertainty analysis. Therefore, the other results can be confidently used as benchmark data for testing CFD codes. The turbulence intensities at the mid-plane are also presented.     

Interparticle forces arising from an adsorbed strong polyelectrolyte in colloidal dispersions: charged patch attraction

Adsorbed polystyrene sulphonate (PSS) shifts the pH of the zero zeta potential, pH_ς = 0, of ZrO₂ to a lower pH. The positive charge density of ZrO₂ at pH = pH_ς=0 determined from the amount of PSS adsorbed was in excellent agreement with that obtained from charge titration. Polystyrene sulphonic acid shifts pH_ς=0 to a greater degree compared with polyacrylic acid because it is a much stronger acid. A patch is likely to just consist of one adsorbed molecule. The patch is negative when the charges of the molecule exceed the underlying positive surface charge. Attraction between the negative patch and the bare positive surface of a second particle is responsible for increasing the yield stress of concentrated ZrO₂ dispersions at pH_ς=0. Its magnitude is only of the order of the van der Waals attraction. Increasing ionic strength and patch misalignment diminish the attraction. The upper limit of the patch area was estimated from the radius of gyration of the molecule in solution. With a known patch area, the patch charged density can be calculated. With the selection of an appropriate patch area, the yield stress due to charged patch attraction increases linearly with the product of the negative and positive patch densities. Received: 30 March 1998 Accepted: 18 September 1998     

Influence of hue origin on the hue-temperature calibration of thermochromic liquid crystals

Liquid crystal thermography incorporates a process by which the liquid crystal color, bearing temperature information, is analyzed using image processing techniques to quantify complex temperature distributions on heat transfer surfaces. Previous researches have found that hue can be used to represent the liquid crystal color since there exists a monotonic relation between the hue and temperature of liquid crystals. It is found in this study that the white-light point, which is usually adopted as the hue reference point, does not generally lie at the center of the liquid crystal colors defined on a chromaticity diagram. This results that hue does not have uniform resolution to all liquid crystal colors and a biased error is produced subsequently in the temperature measurement. The non-uniform hue can be corrected by shifting the hue origin to the center of the colors or by a color calibration scheme so that the liquid crystal colors distribute evenly around the white-light point. Received on 15 November 1996     

Development of an integrated optical system for warpage and hermeticity test of microdisplay

We have developed an integrated optical system to measure the warpage of a microdisplay product at post-packaging die-attach temperature and to test its hermeticity after manufacturing. This system is constructed based on the principle of Twyman/Green (T/G) interferometry and Newton interferometry. T/G interferometry is used to characterize warpage of the microdisplay's base-plate, while Newton interferometry is used to characterize the gap variance between face-plate and base-plate of the microdisplay. With these measurements, deformation and shape of the microdisplay can be comprehensively determined during its manufacturing process, which is critical for its quality test and reliability evaluation. Delicate mechanical system was designed to facilitate the optics integration and alignment. To evaluate the performance of the optical system itself, warpage of a sample was re-measured with Tailor–Hobson profilometer whose resolution is 10 nm, the results from these two testing systems agree well with each other. Some application examples of the integrated testing system were presented to verify its work efficiency.     

Thermal Conductivity of Soils at Very Low Moisture Content and Moderate Temperatures

An analysis of soil thermal conductivity data shows that, at very low moisture content, this property first varies insignificantly and then begins to increase from a certain critical moisture content, whose value tends to depend on clay mass fraction. Two simple models evaluating the critical moisture content were developed; the first one is a fraction of the permanent wilting point; the second one is a simple linear function dependent on clay mass fraction. An insignificant variation of soil thermal conductivity is observed at 20°C, within a water pressure head ranging from 1×10³ to 1×10³m, while for higher temperatures (45–50°C) from 5000 to 100000m. Three extensions of the enhanced thermal conductivity model by Sundberg, namely SUN-1, SUN-2 and SUN-3, were proposed and tested. They produce an average root mean square error of 27%, 24% and 30%, respectively, with respect to experimental data. SUN-1and SUN-2 predictions could be further improved if better estimates of thermal conductivity at the dry state and the permanent wilting point were provided. SUN-3 is a simple model which does not require information about the PWP and critical moisture content. All SUN models have a potential for implementing the latent heat transfer component.     

Enantioselective palladium-catalyzed carbozincation of cyclopropenes

A highly enantioselective palladium-catalyzed carbozincation of cyclopropenes has been developed. The intermediate cyclopropylzinc species, after transmetalation with copper, were trapped with various electrophiles. This one-pot procedure furnished functionalizied cyclopropenes with excellent diastereo- and enantioselectivity.     

Antibacterial Thiopeptide GE2270-Congeners from Nonomuraea jiangxiensis

Thiopeptides are macrocyclic natural products with potent bioactivity. Nine new natural thiopeptides (1–9) were obtained from a Nonomuraea jiangxiensis isolated from a terrestrial soil sample collected in Singapore. Even though some of these compounds were previously synthesized or isolated from engineered strains, herein we report the unprecedented isolation of these thiopeptides from a native Nonomuraea jiangxiensis. A comparison with the literature and a detailed analysis of the NMR and HRMS of compounds 1–9 was conducted to assign their chemical structures. The structures of all new compounds were highly related to the thiopeptide antibiotics GE2270, with variations in the substituents on the thiazole and amino acid moieties. Thiopeptides 1–9 exhibited a potent antimicrobial activity against the Gram-positive bacteria, Staphylococcus aureus with MIC₉₀ values ranging from 2 µM to 11 µM. In addition, all compounds were investigated for their cytotoxicity against the human cancer cell line A549, none of the compounds were cytotoxic.