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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This essay examines the impact of estimation for the training’s added value in life on level of physical training. Through a multiple-regression analysis of primary data, the study presents what impact a particular background has on the physical exercise’s position in a life puzzle of a Swedish young adult. The theoretical framework is based on the French sociologist Pierre Bourdieu’s theories of habitus, symbolic capital, symbolic value and symbolic violence. According to these concepts, all individuals operate within specific fields, within which they also hold a position. Practices and social agents in the various fields shape what Bourdieu calls habitus, an individual set of schemes that shape tastes a values. These theories help us understand why people perform different targeted actions in life based on their gender, class and grew up place. Data have been collected through a quantitative survey. The material consists of questionnaires from 179 young adults from Sweden between the ages of 19-29. The bivariate regression analysis could not prove a significant correlation between estimation of the training’s added value in life and level of physical training. When a multivariate regression design included control variables a positive correlation between the training’s added value in life and level of physical training emerged. Gender, class and age also proved important for the level of training where being a female and coming from an entrepreneurial home increased the level of training. / <p>Tillgänglig på messenger (Jonathan Westerberg), alternativt sms (0700912052).</p>