Dual processes in mathematics: reasoning about conditionals

This thesis studies the reasoning behaviour of successful mathematicians. It is based on the philosophy that, if the goal of an advanced education in mathematics is to develop talented mathematicians, it is important to have a thorough understanding of their reasoning behaviour. In particular, one needs to know the processes which mathematicians use to accomplish mathematical tasks. However, Rav (1999) has noted that there is currently no adequate theory of the role that logic plays in informal mathematical reasoning. The goal of this thesis is to begin to answer this specific criticism of the literature by developing a model of how conditional “if…then” statements are evaluated by successful mathematics students. Two stages of empirical work are reported. In the first the various theories of reasoning are empirically evaluated to see how they account for mathematicians’ responses to the Wason Selection Task, an apparently straightforward logic problem (Wason, 1968). Mathematics undergraduates are shown to have a different range of responses to the task than the general well-educated population. This finding is followed up by an eve-tracker inspection time experiment which measured which parts of the task participants attended to. It is argued that Evans’s (1984, 1989, 1996, 2006) heuristic-analytic theory provides the best account of these data. In the second stage of empirical work an in-depth qualitative interview study is reported. Mathematics research students were asked to evaluate and prove (or disprove) a series of conjectures in a realistic mathematical context. It is argued that preconscious heuristics play an important role in determining where participants allocate their attention whilst working with mathematical conditionals. Participants’ arguments are modelled using Toulmin’s (1958) argumentation scheme, and it is suggested that to accurately account for their reasoning it is necessary to use Toulmin’s full scheme, contrary to the practice of earlier researchers. The importance of recognising that arguments may sometimes only reduce uncertainty in the conditional statement’s truth/falsity, rather than remove uncertainty, is emphasised. In the final section of the thesis, these two stages are brought together. A model is developed which attempts to account for how mathematicians evaluate conditional statements. The model proposes that when encountering a mathematical conditional “if P then Q”, the mathematician hypothetically adds P to their stock of knowledge and looks for a warrant with which to conclude Q. The level of belief that the reasoner has in the conditional statement is given by a modal qualifier which they are prepared to pair with their warrant. It is argued that this level of belief is fixed by conducting a modified version of the so-called Ramsey Test (Evans & Over, 2004). Finally the differences between the proposed model and both formal logic and everyday reasoning are discussed

Dual processes in mathematics : reasoning about conditionals

This thesis studies the reasoning behaviour of successful mathematicians. It is based on the philosophy that, if the goal of an advanced education in mathematics is to develop talented mathematicians, it is important to have a thorough understanding of their reasoning behaviour. In particular, one needs to know the processes which mathematicians use to accomplish mathematical tasks. However, Rav (1999) has noted that there is currently no adequate theory of the role that logic plays in informal mathematical reasoning. The goal of this thesis is to begin to answer this specific criticism of the literature by developing a model of how conditional “if…then” statements are evaluated by successful mathematics students. Two stages of empirical work are reported. In the first the various theories of reasoning are empirically evaluated to see how they account for mathematicians’ responses to the Wason Selection Task, an apparently straightforward logic problem (Wason, 1968). Mathematics undergraduates are shown to have a different range of responses to the task than the general well-educated population. This finding is followed up by an eve-tracker inspection time experiment which measured which parts of the task participants attended to. It is argued that Evans’s (1984, 1989, 1996, 2006) heuristic-analytic theory provides the best account of these data. In the second stage of empirical work an in-depth qualitative interview study is reported. Mathematics research students were asked to evaluate and prove (or disprove) a series of conjectures in a realistic mathematical context. It is argued that preconscious heuristics play an important role in determining where participants allocate their attention whilst working with mathematical conditionals. Participants’ arguments are modelled using Toulmin’s (1958) argumentation scheme, and it is suggested that to accurately account for their reasoning it is necessary to use Toulmin’s full scheme, contrary to the practice of earlier researchers. The importance of recognising that arguments may sometimes only reduce uncertainty in the conditional statement’s truth/falsity, rather than remove uncertainty, is emphasised. In the final section of the thesis, these two stages are brought together. A model is developed which attempts to account for how mathematicians evaluate conditional statements. The model proposes that when encountering a mathematical conditional “if P then Q”, the mathematician hypothetically adds P to their stock of knowledge and looks for a warrant with which to conclude Q. The level of belief that the reasoner has in the conditional statement is given by a modal qualifier which they are prepared to pair with their warrant. It is argued that this level of belief is fixed by conducting a modified version of the so-called Ramsey Test (Evans ;Over, 2004). Finally the differences between the proposed model and both formal logic and everyday reasoning are discussed.EThOS - Electronic Theses Online ServiceEconomic and Social Research Council (Great Britain) (ESRC)GBUnited Kingdo

Winona Daily News

https://openriver.winona.edu/winonadailynews/1666/thumbnail.jp

Journal of Mormon History Vol. 31, No. 3, 2005

CONTENTS ARTICLES --The Case for Sidney Rigdon as Author of the Lectures on Faith Noel B. Reynolds, 1 --Reconstructing the Y-Chromosome of Joseph Smith: Genealogical Applications Ugo A. Perego, Natalie M. Myres, and Scott R. Woodward, 42 --Lucy\u27s Image: A Recently Discovered Photograph of Lucy Mack Smith Ronald E. Romig and Lachlan Mackay, 61 --Eyes on the Whole European World : Mormon Observers of the 1848 Revolutions Craig Livingston, 78 --Missouri\u27s Failed Compromise: The Creation of Caldwell County for the Mormons Stephen C. LeSueur, 113 --Artois Hamilton: A Good Man in Carthage? Susan Easton Black, 145 --One Masterpiece, Four Masters: Reconsidering the Authorship of the Salt Lake Tabernacle Nathan D. Grow, 170 --The Salt Lake Tabernacle in the Nineteenth Century: A Glimpse of Early Mormonism Ronald W. Walker, 198 --Kerstina Nilsdotter: A Story of the Swedish Saints Leslie Albrecht Huber, 241 REVIEWS --John Sillito, ed., History\u27s Apprentice: The Diaries of B. H. Roberts, 1880-1898 Davis Bitton, 264 --Martha Beck, Leaving the Saints: How I Lost the Mormons and Found My Faith Boyd Jay Petersen, 267 --Donald Q. Cannon, Richard O. Cowan, et al., Unto Every Nation: Gospel Light Reaches Every Land Kahlile B. Mehr, 271 --Scott H. Faulring, Kent P.Jackson, and Robert J. Matthews, eds., Joseph Smith\u27s New Translation of the Bible: Original Manuscripts H. Michael Marquardt, 27

Kunapipi 18 (2 & 3) 1996 Full Version

Kunapipi 18 (2 & 3) 1996 Full Version

Evaluating economic policies for promoting rainforest conservation in developing countries

Economic policies are often suggested as mechanisms for promoting rainforest conservation in developing countries. To help decide whether international resources should be used to protect specific rainforcsts, the calculation of a "rainforest supply price" (RSP) is proposed. If protection is warranted, then empirical analyses explore the conditions under which selected policies within developing countries might be effective in protecting rainforests. Korup National Park in Cameroon contains the oldest rainforest in Africa and - as a haven for important endangered species- it is the subject of active international conservation efforts. A cost-bencfit analysis of a conservation project to protect Korup from increased land-use pressures suggests that it is not in Cameroon's interest unless a 5.4 million ECU inducement is transferred to Cameroon. Given the protection afforded, the transfer is equivalent to a RSP of 1060 ECU per km2 per year. Evaluations of six other tropical rainforest projects suggests that international donors made transfers having values ranging from 15 to 1575ECU per km2 per year. It is thus concluded that the inducements required are within a range which conservation interests are apparently willing to mobilise. To target inducements the provision of incentives in a "buffer zone" around a park is often believed to promote conservation. This is based on the hypothesis that increased incomes will draw individuals out of the park and will give them something better to do than exploit the park. A survey of 341 households around Korup was analysed in detail to test this hypothesis. Evidence suggests that economic development in the buffer zone would increase pressures on the park because: a) higher incomes would reduce emigration from the region and would thus cause greater population pressure on the Park; and, b) hunting effort increases as non-hunting income increases