Developing teacher competencies through videos for facilitation of mathematical modelling in Singapore primary schools

Mathematical modelling tasks which are situated in real-world contexts encourage students to draw connections between school-based mathematics and the real-world, enhancing their engagement in learning. Such tasks often require varied interpretations of the real-world problem context resulting in multiple pathways of solutions. Although mathematical modelling has been introduced in the Singapore mathematics curriculum since 2007, its incorporation in schools has been limited. One reason for this could be that teachers are challenged by how best to facilitate for rich student mathematisation processes during such tasks. This chapter reports how a multi-tiered teaching experiment using design research methodology was conducted to build teachers’ capacity in designing, facilitating, and evaluating student mathematisation during mathematical modelling tasks with an intact class of Primary 5 students (aged 10-11). The use of videos was critical because grounded images helped capture the dynamics and complexity of authentic classroom interactions. This chapter highlights how video recordings of teacher-student interactions during a modelling task were harnessed during design methodology cycles, particularly during the Retrospective Analysis phase, to activate critical moments of learning for the teacher towards developing her competencies in facilitating students’ mathematisation processes.

City-level auditor industry specialization, economies of scale, and audit pricing

We examine the effects of city-level auditor industry specialization and scale economies on audit pricing in the United States. Using a sample of Big N clients for the 2000-2007 period, and a scale measure based on percentile rankings of the number of audit clients at the city-industry level, we document significant specialization premiums and scale discounts in both the pre- and post-Sarbanes-Oxley Act (SOX) periods. However, the effects of industry specialization and scale economies on audit pricing are highly interactive. The negative effect of city-industry scale on audit fees obtains only for clients of specialist auditors. By contrast, clients of non-specialist auditors obtain scale discounts only when they enjoy strong bargaining power, suggesting that auditors are "forced" to pass on scale economies to clients with greater bargaining power.

Machining of titanium alloy (Ti-6Al-4V)-theory to application

This article correlates laboratory-based understanding in machining of titanium alloys with the industry based outputs and finds possible solutions to improve machining efficiency of titanium alloy Ti-6Al-4V. The machining outputs are explained based on different aspects of chip formation mechanism and practical issues faced by industries during titanium machining. This study also analyzed and linked the methods that effectively improve the machinability of titanium alloys. It is found that the deformation mechanism during machining of titanium alloys is complex and causes basic challenges, such as sawtooth chips, high temperature, high stress on cutting tool, high tool wear and undercut parts. These challenges are correlated and affected by each other. Sawtooth chips cause variation in cutting forces which results in high cyclic stress on cutting tools. On the other hand, low thermal conductivity of titanium alloy causes high temperature. These cause a favorable environment for high tool wear. Thus, improvements in machining titanium alloy depend mainly on overcoming the complexities associated with the inherent properties of this alloy. Vibration analysis kit, high pressure coolant, cryogenic cooling, thermally enhanced machining, hybrid machining and, use of high conductive cutting tool and tool holders improve the machinability of titanium alloy.