A teaching experiment to foster the conceptual understanding of multiplication based on children's literature to facilitate dialogic learning

The importance of conceptual understanding as opposed to low-level procedural knowledge in mathematics has been well documented (Hiebert & Carpenter, 1992). Development of conceptual understanding of multiplication is fostered when students recognise the equal group structure that is common in all multiplicative problems (Mulligan & Mitchelmore, 1996). This paper reports on the theoretical development of a transformative teaching experiment based on conjecture-driven research design (Confrey & Lachance, 1999) that aims to enhance Year 3 students’ conceptual understanding of multiplication. The teaching experiment employs children’s literature as a motivational catalyst and mediational tool for students to explore and engage in multiplication activities and dialogue. The SOLO taxonomy (Biggs & Collis, 1989) is used to both frame the novel teaching and learning activities, as well as assess the level of students’ conceptual understanding of multiplication as displayed in the products derived from the experiment. Further, student’s group interactions will be analysed in order to investigate the social processes that may contribute positively to learning

Mean flow instabilities of two-dimensional convection in strong magnetic fields

The interaction of magnetic fields with convection is of great importance in astrophysics. Two well-known aspects of the interaction are the tendency of convection cells to become narrow in the perpendicular direction when the imposed field is strong, and the occurrence of streaming instabilities involving horizontal shears. Previous studies have found that the latter instability mechanism operates only when the cells are narrow, and so we investigate the occurrence of the streaming instability for large imposed fields, when the cells are naturally narrow near onset. The basic cellular solution can be treated in the asymptotic limit as a nonlinear eigenvalue problem. In the limit of large imposed field, the instability occurs for asymptotically small Prandtl number. The determination of the stability boundary turns out to be surprisingly complicated. At leading order, the linear stability problem is the linearisation of the same nonlinear eigenvalue problem, and as a result, it is necessary to go to higher order to obtain a stability criterion. We establish that the flow can only be unstable to a horizontal mean flow if the Prandtl number is smaller than order , where B0 is the imposed magnetic field, and that the mean flow is concentrated in a horizontal jet of width in the middle of the layer. The result applies to stress-free or no-slip boundary conditions at the top and bottom of the layer

Combustor flame flashback

A stainless steel, two-dimensional (rectangular), center-dump, premixed-prevaporized combustor with quartz window sidewalls for visual access was designed, built, and used to study flashback. A parametric study revealed that the flashback equivalence ratio decreased slightly as the inlet air temperature increased. It also indicated that the average premixer velocity and premixer wall temperature were not governing parameters of flashback. The steady-state velocity balance concept as the flashback mechanism was not supported. From visual observation several stages of burning were identified. High speed photography verified upstream flame propagation with the leading edge of the flame front near the premixer wall. Combustion instabilities (spontaneous pressure oscillations) were discovered during combustion at the dump plane and during flashback. The pressure oscillation frequency ranged from 40 to 80 Hz. The peak-to-peak amplitude (up to 1.4 psi) increased as the fuel/air equivalence ratio was increased attaining a maximum value just before flashback. The amplitude suddenly decreased when the flame stabilized in the premixer. The pressure oscillations were large enough to cause a local flow reversal. A simple test using ceramic fiber tufts indicated flow reversals existed at the premixer exit during flickering. It is suspected that flashback occurs through the premixer wall boundary layer flow reversal caused by combustion instability. A theoretical analysis of periodic flow in the premixing channel has been made. The theory supports the flow reversal mechanism

Transient technique for measuring heat transfer coefficients on stator airfoils in a jet engine environment

A transient technique was used to measure heat transfer coefficients on stator airfoils in a high-temperature annular cascade at real engine conditions. The transient response of thin film thermocouples on the airfoil surface to step changes in the gas stream temperature was used to determine these coefficients. In addition, gardon gages and paired thermocouples were also utilized to measure heat flux on the airfoil pressure surface at steady state conditions. The tests were conducted at exit gas stream Reynolds numbers of one-half to 1.9 million based on true chord. The results from the transient technique show good comparison with the steady-state results in both trend and magnitude. In addition, comparison is made with the STAN5 boundary layer code and shows good comparison with the trends. However, the magnitude of the experimental data is consistently higher than the analysis

Random Field XY Model in Three Dimensions: The Role of Vortices

We study vortex states in a 3d random-field XY model of up to one billion lattice spins. Starting with random spin orientations, the sample freezes into the vortex-glass state with a stretched-exponential decay of spin correlations, having short correlation length and a low susceptibility, compared to vortex-free states. In a field opposite to the initial magnetization, peculiar topological objects -- walls of spins still opposite to the field -- emerge along the hysteresis curve. On increasing the field strength, the walls develop cracks bounded by vortex loops. The loops then grow in size and eat the walls away. Applications to magnets and superconductors are discussed.Comment: 5 pages, 8 figure