This project aims to examine the effects of incorporating collaborative learning methods extensively in a core first-year mathematics subject and to investigate students' individual learning style preferences, their attitudes towards group-work in mathematics and the objectives for setting group work, their attitudes towards using computers, in particular, Mathematica and their concerns regarding the assessment of group-based work. Following the rapid increase in the use of technology in education over the last decade, one would perhaps expect to find an overabundance of literature regarding the effects of its use. However, the number of technology related research studies has been surprisingly low, especially those pertaining to the curriculum area of Mathematics at the tertiary level. The availability of quality software, the need for curriculum redesign, and limited research on the effectiveness of computers as a teaching tool, are factors to have hindered the rate of implementation and of subsequent research. Also, despite the rapid growth in the use of collaborative methods of learning, and widespread belief in the importance of such methods, there have been calls for increased research especially at the tertiary level, and particularly in engineering education looking at students who have to study mathematics because it is a requirement and not because they are majoring in mathematics, therefore needing to determine how best to make their learning a meaningful and enjoyable experience. This project aims to investigative the effects of incorporating a rich collaborative learning based curriculum in either face-to-face or computer-supported environments in the subject Mathematical Modelling 1. The carrying out of this project is a response to the lack of research in a curriculum area of tertiary mathematics. Within the context of mathematics, issues of attitude, gender differences, motivation and achievement are considered. The chief purpose of this investigation is to explore the effectiveness of collaborative learning in mathematics at university, and to provide some insight as to what degree, if any, the use of such methods enhance mathematics learning. The research uses an experimental methodology, an attitudinal questionnaire and indepth interviews to elicit students' feelings and/or opinions toward the incorporation of collaborative learning. The questionnaire sought demographic information from the students, namely, name, age, gender, length of stay in Australia and language spoken at home, and investigates the role of these factors in the effectiveness of, and interest during the tutorial and laboratory sessions a time when students were working on collaborative-based activities. This project maintains interest in the use of collaborative problem solving, and the belief that the findings could be of international significance if the effectiveness of this style of learning can be finnly established. It is also hoped that grounding the collaborative activities in the literature, and providing statistical and theoretical support for their use might promote them more widely in mathematics in particular and more generally, across universities in Australia. The broad issue of whether the use of collaborative learning enhances mathematics learning can be broken down into a number of specific inquiries. The key research questions may thus be expressed as follows: I. What are tertiary students' preferred learning styles? 2. What are students' opinions about group work in mathematics? 3. Does collaborative group work foster a deep, meaningful understanding of mathematics? 4. What are students' attitudes about using CAS such as Mathematica? 5. What are students' attitudes about the assessment of group-based work? 6. Are there any differences in students' learning style preferences across the various demographics? 7. Are there any differences in students' attitudes towards collaborative learning methods across the various demographics? 8. Are there any differences in students' attitudes towards the use of Mathematica across the various demographics? 9. Are there any variations in students' attitudes towards the assessment of group work in mathematics across the various demographics? This study does not claim to fill the void into the effectiveness of computers or collaborative learning methods, but should provide greater insight and support to future research.

This project aims to examine the effects of incorporating collaborative learning methods extensively in a core first-year mathematics subject and to investigate students' individual learning style preferences, their attitudes towards group-work in mathematics and the objectives for setting group work, their attitudes towards using computers, in particular, Mathematica and their concerns regarding the assessment of group-based work. Following the rapid increase in the use of technology in education over the last decade, one would perhaps expect to find an overabundance of literature regarding the effects of its use. However, the number of technology related research studies has been surprisingly low, especially those pertaining to the curriculum area of Mathematics at the tertiary level. The availability of quality software, the need for curriculum redesign, and limited research on the effectiveness of computers as a teaching tool, are factors to have hindered the rate of implementation and of subsequent research. Also, despite the rapid growth in the use of collaborative methods of learning, and widespread belief in the importance of such methods, there have been calls for increased research especially at the tertiary level, and particularly in engineering education looking at students who have to study mathematics because it is a requirement and not because they are majoring in mathematics, therefore needing to determine how best to make their learning a meaningful and enjoyable experience. This project aims to investigative the effects of incorporating a rich collaborative learning based curriculum in either face-to-face or computer-supported environments in the subject Mathematical Modelling 1. The carrying out of this project is a response to the lack of research in a curriculum area of tertiary mathematics. Within the context of mathematics, issues of attitude, gender differences, motivation and achievement are considered. The chief purpose of this investigation is to explore the effectiveness of collaborative learning in mathematics at university, and to provide some insight as to what degree, if any, the use of such methods enhance mathematics learning. The research uses an experimental methodology, an attitudinal questionnaire and indepth interviews to elicit students' feelings and/or opinions toward the incorporation of collaborative learning. The questionnaire sought demographic information from the students, namely, name, age, gender, length of stay in Australia and language spoken at home, and investigates the role of these factors in the effectiveness of, and interest during the tutorial and laboratory sessions a time when students were working on collaborative-based activities. This project maintains interest in the use of collaborative problem solving, and the belief that the findings could be of international significance if the effectiveness of this style of learning can be finnly established. It is also hoped that grounding the collaborative activities in the literature, and providing statistical and theoretical support for their use might promote them more widely in mathematics in particular and more generally, across universities in Australia. The broad issue of whether the use of collaborative learning enhances mathematics learning can be broken down into a number of specific inquiries. The key research questions may thus be expressed as follows: I. What are tertiary students' preferred learning styles? 2. What are students' opinions about group work in mathematics? 3. Does collaborative group work foster a deep, meaningful understanding of mathematics? 4. What are students' attitudes about using CAS such as Mathematica? 5. What are students' attitudes about the assessment of group-based work? 6. Are there any differences in students' learning style preferences across the various demographics? 7. Are there any differences in students' attitudes towards collaborative learning methods across the various demographics? 8. Are there any differences in students' attitudes towards the use of Mathematica across the various demographics? 9. Are there any variations in students' attitudes towards the assessment of group work in mathematics across the various demographics? This study does not claim to fill the void into the effectiveness of computers or collaborative learning methods, but should provide greater insight and support to future research.

In this thesis a number of mathematical models are investigated with the aid of the modelling package Mathematica. Some of the models are of a mechanical nature and some of the models are laboratories that have been constructed for the purpose of assisting researchers in a particular field. In the early sections of the thesis mechanical models are investigated. After the equations of motion for the model have been presented, Mathematica is employed to generate solutions which are then used to drive animations of the model. The frames of the animations are graphical snapshots of the model in motion. Mathematica proves to be an ideal tool for this type of modelling since it combines algebraic, numeric and graphics capabilities on one platform. In the later sections of this thesis, Mathematica laboratories are created for investigating models in two different fields. The first laboratory is a collection of routines for performing Phase-Plane analysis of planar autonomous systems of ordinary differential equations. A model of a mathematical concept called a bifurcation is investigated and an animation of this mathematical event is produced. The second laboratory is intended to help researchers in the tomography field. A standard filtered back-projection algorithm for reconstructing images from their projections is implemented. In the final section of the thesis an indication of how the tomography laboratory could be used is presented. Wavelet theory is used to construct a new filter that could be used in filtered back-projection tomography. / Thesis (Ph.D.)-University of Natal, Durban, 1994.

In this thesis we analyse particular differential equations that arise in physical situations. This is achieved with the aid of the computer software package called Mathematica. We first describe the basic features of Mathematica highlighting its capabilities in performing calculations in mathematics. Then we consider a first order Newtonian equation representing the trajectory of a particle around a spherical object. Mathematica is used to solve the Newtonian equation both analytically and numerically. Graphical plots of the trajectories of the planetary bodies Mercury, Earth and Jupiter are presented. We attempt a similar analysis for the corresponding relativistic equation governing the orbits of gravitational objects. Only numerical results are possible in this case. We also perform a perturbative analysis of the relativistic equation and determine the amount of perihelion shift. The second equation considered is the Emden-Fowler equation of order two which arises in many physical problems, including certain inhomogeneous cosmological applications. The analytical features of this equation are investigated using Mathematica and the Lie analysis of differential equations. Different cases of the related autonomous form of the Emden-Fowler equation are investigated and graphically represented. Thereafter, we generate a number of profiles of the energy density and the pressure for a particular solution which demonstrates that a numerical approach for studying inhomogeneity, in cosmological models in general relativity, is feasible. / Thesis (M.Sc.)-University of Natal, Durban, 2001.