A Systematic Analysis of Errors in the Simplification of a Rational Expression

Exploring the errors that mathematics students frequently make is a means by which teachers can gain a better understanding of students’ difficulties. Reported here are the process by which the algebraic working of 95 undergraduate students who incorrectly simplified a rational expression was analysed and the results of the analysis. Initially, a deductive approach to analysing the errors was planned, categorising students’ mistakes using the error types identified, named and described in the literature. In reviewing the literature, however, it became clear that this would be no simple task. The large body of literature, while rich in examples of “typical errors” that could be expected in students’ working, had two limitations. Firstly, the error types lacked precise descriptions and were mainly described by example only. Secondly, insufficient details of the procedures used to categorise the errors prevented replication of the categorising process. Consequently, a mainly inductive approach, that categorised the errors by their location and inferred student operation was devised. This systematic approach resulted in generating descriptions of three error categories

Atmospheric interfacial waves

Waves at the interface of a two-layer fluid are considered. The fluid in the lower layer is incompressible with constant density and is flowing irrotationally. In the upper layer, the fluid is stationary but compressible, and corresponds to an isothermal atmosphere with a density profile that decreases exponentially with height. The interface between the two fluids is assumed sharp. The formation of waves at the interface would come about typically as a result of the interaction of the moving lower layer of fluid with local topographical features, as with the classical problem of the generation of waves on the lee side of a mountain range. It is shown that the present model is capable of supporting the formation of interfacial waves that are similar in many respects to the classical gravity wave of Stokes, and that are ultimately limited in every case by the formation of a 120° angle at the wave crest. The highly nonlinear wave profiles are computed numerically and compared with the predictions of linearized theory. An extended perturbation analysis is given near the point at which the interfacial waves break down as a result of the Kelvin-Helmholtz instability

Interfacial waves and hydraulic falls: Some applications to atmospheric flows in the lee of mountains

Two dimensional flow of a layer of constant density fluid over arbitrary topography, beneath a compressible, isothermal and stationary fluid is considered. Both downstream wave and critical flow solutions are obtained using a boundary integral formulation which is solved numerically by Newton's method. The resulting solutions are compared against waves produced behind similar obstacles in which the compressible upper layer is absent (single layer flow) and against the predictions of a linearised theory. The limiting waves predicted by the full non-linear equations are contrasted with those predicted by the forced Korteweg-de Vries theory. In particular, it is shown that at some parameter values a multiplicity of solutions exists in the full nonlinear theory

Fully non-linear two-layer flow over arbitrary topography

Steady, two-dimensional, two-layer flow over an arbitrary topography is considered. The fluid in each layer is assumed to be inviscid and incompressible and flows irrotationally. The interfacial surface is found using a boundary integral formulation, and the resulting integrodifferential equations are solved iteratively using Newton's method. A linear theory is presented for a given topography and the non-linear theory is compared against this to show how the non-linearity affects the problem

A simple model for oil spill containment using a boom - exact solutions

This paper revisits the mathematical model of an oilslick presented in [1]. Through various simplifying assumptions the model is one for which the equations may be solved exactly\ud using a series solution approach. Steady solutions are found. The model may be used as a predictive tool to help determine when deploying a boom may be successful in limiting the\ud spread of oil after a spill

Efficient series solutions for non-linear flow over topography

Fluid flowing over topography occurs in many physical situations. As a consequence, study of flow over topography has been a research topic of prime interest for many decades. Formally, the problem can be modelled as a nonlinear free boundary problem. Although methods such as boundary integrals are typically used, analytic series methods have also been developed to solve some of these problems. Arguably the hardest problem to solve is the lee wave problem: when the flow conditions are suitable, waves form downstream of the obstacle. Wave solutions pose several problems for the analytic series methods. The solution method is iterative, and at each step the existing solution must be updated. For the iterative scheme to converge, very accurate series solutions must be obtained at each step. The convergence rate of the series solution itself is critical in this process, and depends to a large extent on the free boundary representation. In this paper, we compare and discuss the convergence rates for a variety of free surface representations. We show that spectral convergence is possible if the correct representation is used. \ud \u

Bridging the gap: teaching university mathematics to high school students

Over recent years there has been a lot of emphasis placed on the drop in standards of students entering first year university mathematics in Australia. The tertiary sector struggles to handle this increasing gap and, with pressure to maintain student numbers, a common response has been to reduce the difficulty level of the first year mathematics courses. This approach has had limited success, with students passing first year mathematics but lacking preparation for the higher years. If realistic change is to be made in bridging this gap, then the problem needs to be addressed at both the tertiary and secondary level. We investigate the successes and potential failures of running a tertiary level mathematics course over five years at four high schools in North Queensland. This has been a genuine team approach by both university academics and high school mathematics teachers and forged solid links between the sectors. The presence of academics in the high school classrooms as well as students and teachers attending university activities led to a greater understanding of perceived difficulties on both sides. The `Mathematics into high schools' program proves to be a small but significant start in helping to bridge the secondary-tertiary gap

Wave patterns and sediment mixing near coral reefs

Most models of the wave patterns surrounding coral reefs are for incident waves and are based on the assumption that the waves are infinitesimally small. The research effort thus far has focused on the attenuation and/or refraction of these waves as they pass over the reef. However, it is now possible to accurately model finite amplitude waves and flow patterns that do not depend on the amplitude of the wave. These solutions are obtained from the fully non‐linear equations for fluid flow over topography. In contrast to previous research, we will consider the wave pattern generated by the flow of water over the reef. Solutions are presented for the wave patterns and velocity profiles for a reef located off the North Queensland coast near Townsville. We present results for fully non‐linear flow and compare them with the linear solutions. We investigate the parameter values that change the wave profile, in particular the size and shape of the reef. As these solutions are analytical, the velocity field is immediately available. We provide velocity fields for the reef and show that there is a significant change to the vertical component of the velocity for the finite amplitude waves compared with linear wave theory. We briefly discuss the effect this has on sediment mixing and deposition around the reef

Optimising series solution methods for flow over topography - Part 1

Series solution methods have recently been used to solve fully nonlinear flow over topography problems. These methods are iterative schemes that update an initial estimate of the fluid surface (a free boundary) using a cost function. Series solutions are obtained efficiently and accurately with exact error bounds immediately available. Critical to the speed of the procedure is the implementation of efficient computer code and numerical techniques. In this paper we discuss methods that improve the computational time of the original implementation by several orders of magnitude, without any loss of accuracy. The efficiency of the improved method is demonstrated by generating two dimensional solutions to subcritical flow over an isolated cosine shaped obstacle.\ud \u

A potential update method for series methods in steady seepage

Computationally, steady seepage in arbitrary shaped aquifers reduces to a Laplacian free boundary problem. Analytic series solutions can be obtained by iteratively improving an initial estimate. Although the free water table is slowly varying in shape, accurate solutions can\ud be extremely difficult to obtain for the long aspect ratios encountered in typical problems. In previous research, the Neumann condition along the free boundary was used as a cost function. In this paper, we present an update method that uses the Dirichlet condition as the cost function. We find that this approach is significantly more efficient and\ud stable than the previous approach. This enables the determination of very accurate solutions for the water table at low computational cost