M.Ed. / Recently there has been renewed interest in proof and proving in schools worldwide. However, many school students and even teachers of mathematics have only superficial ideas on the nature of proof. Proof is considered the heart of mathematics as individuals explore, make conjectures and try to convince themselves and others about the truth or falsity of their conjectures. There are basically two categories of deductive proof, namely proof by direct argument and indirect proofs. The aim of this study was to examine the structural features common to most of the mathematical proofs for formalised mathematical systems, with the emphasis on indirect proof techniques. The main question was to investigate which mathematical activities and logical principles at secondary school level are necessary for students to become proficient with proof writing. A great deal of specialised language is associated with reasoning. Such words as axiom, theorem, proof, and conjecture are just some of the terms that students must understand as they engage in the proof-making task. The formal aspect of mathematics at secondary school is extremely important. It is inevitable that students become involved with hypothetical arguments. They use among others, proofs by contradiction. Furthermore, necessary and sufficient conditions are related to theorems and their converses. It is therefore apparent that the study of logic is necessary already at secondary school level in order to practise mathematics satisfactorily. An analysis of the mathematics syllabus of the Department of Education has indicated that students should use indirect techniques of proof. According to this syllabus students should be familiar with logical arguments. The conclusion which is reached, gives evidence that students’ background in logic is completely lacking and inadequate. As a result they cannot cope adequately with argumentation and this causes a poor perception of what mathematics entails. Although proof writing can never be reduced to a mechanical process, considerable anxiety and uncertainty can be eliminated from the process if students are exposed to the principles of elementary logic and techniques. Mathematics educators and education researchers have reported students’ difficulties with mathematical proof and point out the conflict between the nature of this essential mathematical activity and current approaches to teaching it. This recent interest has led to an increased effort to teach proof in innovative ways.

Diagrams have been used for over a millennium to communicate rich meaning for diverse purposes. Three major and persistent problems concerning our understanding of diagrams have been identified and must be addressed: 1. The variety of diagram types 2. Handling changes while retaining well formedness 3. Semantically mixed diagrams. A variety of both scientific and philosophical approaches to understanding diagrams is examined, and all are found unable to meet these challenges in full.

The thesis presents a survey of formalisms for non-monotonic reasoning, providing a sketch of the "state of the art" in the field. Reiter's logic for default reasoning is discussed in detail. Following this, a procedure which can determine the extensions of general finite default theories is demonstrated. The potential impact of this procedure on some of the other research in the field is explored, and some promising areas for future research are indicated. Grounds for cautious optimism about the tractability of default theories capable of representing a wide variety of common situations are presented. / Science, Faculty of / Computer Science, Department of / Graduate

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The trumpet is among the most ancient of all musical instruments, and an examination of its history reveals that it has consistently maintained important and specific symbolic roles in society. Although from its origins this symbolic identity was linked to the instrument’s limited ceremonial and signaling function, the seventeenth century represents a period in which a variety of new roles and identities emerged. Bologna and Venice represent the two most important centers for trumpet writing in Italy during the seventeenth century. Because of the differing ideologies at work in these cities, two distinctive symbolic images of the instrument and two different ways of writing for it emerged. The trumpet’s ecclesiastic role in Bologna and its participation in Venetian opera put the instrument at the service of two societies, one centered around the Church, and another around a more permissive state. Against the backdrop of the social and political structures in Venice and Bologna, and through an examination of its newly-emerging musical roles in each city, the trumpet’s changing identities during a most important point in the history of the instrument will be examined.