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"Mathematicians Attitudes."
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Becoming a mathematician : an international perspective
Based on interviews, observations and surveys conducted in Australia, South Africa, Northern Ireland, Canada and Brunei, this book investigates the experiences and views of students and graduates in the process of seeking their identities as mathematicians.
Book
Generalized approach to Galileo's swiftest descent problem on a circle/Kiireima laskumise probleem: uldine kasitlus suvalise lahtepunkti korral
Nearly 400 years ago Galileo Galilei posed a conjecture about the fastest descent on the lower half of a circle. Namely, he assumed that the descent along the circular arc itself is faster than along any broken line of chords. Galileo studied the case when the initial speed is zero, and the paths end in the lowest point of the circle, but he made a guess that the final conclusion remains the same if the particle initially at rest falls to a starting point where the speed is not zero. Galileo was right, as is now rigorously proved. However, this intuitive idea cannot be extended to a starting point on the upper half of a circle. Indeed, as we demonstrate, the fastest descent would then correspond to a finite (not infinite!) number of connected chords. Moreover, we present an extrapolation method which can be applied to determine the optimal number and the positions of all these chords for any starting point on a circle.
Journal Article
Parental Attitudes Towards School and Everyday Mathematics and Their Engagement in Children’s Mathematical Activities
Although parental involvement plays a crucial role in children’s mathematical education, few studies examine how their attitudes towards mathematics influence their participation in specific activities. This study investigates the relationship between parental attitudes towards mathematics in school and in everyday contexts and their engagement in these activities. Results from a survey of 245 Slovenian parents reveal that most parents exhibit a neutral attitude towards school mathematics, while they generally hold a positive attitude towards the practical use of mathematics. Parents with less positive attitudes towards school mathematics tend to assist their children more often, whereas a positive attitude towards everyday mathematics is associated with greater parental involvement in daily mathematical activities at home. Vključenost staršev ima ključno vlogo pri matematičnem izobraževanju otrok, vendar obstaja le malo raziskav o vplivu odnosa staršev do matematike na njihovo vključenost v specifične aktivnosti. V raziskavi proučujemo povezavo med odnosom staršev do matematike v šolskem in vsakodnevnem kontekstu ter njihovim sodelovanjem pri teh aktivnostih. Rezultati med 245 slovenskimi starši kažejo, da večina izraža nevtralen odnos do šolske matematike, medtem ko imajo večinoma pozitiven odnos do praktične uporabe matematike. Starši z manj pozitivnim odnosom do šolske matematike pogosteje pomagajo otrokom, pozitiven odnos do vsakodnevne matematike pa je povezan z večjo vključenostjo staršev v vsakodnevne matematične aktivnosti doma.
Journal Article
Kundakunda, Cantor, and the 'Inaccessibility' of the Absolute: A Set-Theoretical Approach to Sarvajñatā
In this article, Kundakunda's theory of omniscience is defended using formal principles derived from set theory. More precisely, analogous features in the work of Jain mystic Kundakunda and the German mathematician Georg Cantor are described, demonstrating that both thinkers demanded an independently existent, transcendental Absolute to render consistent their own systems of thought. Both of their projects entailed resolving the formal quandary of inaccessibility, or the inability for any sequential, determinate objectifications to ever mereologically sum up to a genuine 'Absolute' - that is, a 'gestalt' that is 'more than the sum of its parts'.
Journal Article
Linguistic Conventions of Mathematical Proof Writing at the Undergraduate Level: Mathematicians' and Students' Perspectives
This study examined the genre of undergraduate mathematical proof writing by asking mathematicians and undergraduate students to read 7 partial proofs and identify and discuss uses of mathematical language that were out of the ordinary with respect to what they considered conventional mathematical proof writing.
Journal Article
The role of testimony in mathematics
Mathematicians appear to have quite high standards for when they will rely on testimony. Many mathematicians require that a number of experts testify that they have checked the proof of a result p before they will rely on p in their own proofs without checking the proof of p. We examine why this is. We argue that for each expert who testifies that she has checked the proof of p and found no errors, the likelihood that the proof contains no substantial errors increases because different experts will validate the proof in different ways depending on their background knowledge and individual preferences. If this is correct, there is much to be gained for a mathematician from requiring that a number of experts have checked the proof of p before she will rely on p in her own proofs without checking the proof of p. In this way a mathematician can protect her own work and the work of others from errors. Our argument thus provides an explanation for mathematicians’ attitude towards relying on testimony.
Journal Article
Teachers' beliefs about school mathematics and mathematicians' mathematics and their relationship to practice
There is broad acceptance that mathematics teachers' beliefs about the nature of mathematics influence the ways in which they teach the subject. It is also recognised that mathematics as practised in typical school classrooms is different from the mathematical activity of mathematicians. This paper presents case studies of two secondary mathematics teachers, one experienced and the other relatively new to teaching, and considers their beliefs about the nature of mathematics, as a discipline and as a school subject. Possible origins and future developments of the structures of their belief systems are discussed along with implications of such structures for their practice. It is suggested that beliefs about mathematics can usefully be considered in terms of a matrix that accommodates the possibility of differing views of school mathematics and the discipline.
Journal Article
How to train your oracle
Delphi is a procedure that produces forecasts on technological and social developments. This article traces the history of Delphi’s development to the early 1950s, where a group of logicians and mathematicians working at the RAND Corporation carried out experiments to assess the predictive capacities of groups of experts. While Delphi now has a rather stable methodological shape, this was not so in its early years. The vision that Delphi’s creators had for their brainchild changed considerably. While they had initially seen it as a technique, a few years later they reconfigured it as a scientific method. After some more years, however, they conceived of Delphi as a tool. This turbulent youth of Delphi can be explained by parallel changes in the fields that were deemed relevant audiences for the technique, operations research and the policy sciences. While changing the shape of Delphi led to some success, it had severe, yet unrecognized methodological consequences. The core assumption of Delphi that the convergence of expert opinions observed over the iterative stages of the procedure can be interpreted as consensus, appears not to be justified for the third shape of Delphi as a tool that continues to be the most prominent one.
Journal Article
The role of mentorship in protégé performance
Mathematical mentors: leading by example
It is clear that mentors, in academia and elsewhere, influence the future success of their protégés, but it is unclear to what extent they influence future mentorship skills and career choices of their protégés. The records of the Mathematics Genealogy Project, which track the careers of 114,666 mathematicians since 1637, provide a data set with sufficient detail for those questions to be addressed. Malmgren
et al
. determine that career success of academic mathematicians was correlated with how many protégés they mentored, and the protégés of mentors with small trainee pools went on to have significantly larger than expected mentorship pools themselves.
Mentors influence the future success of their protégés, but to what extent do those protégés emulate their mentors? Here, one aspect of mentor emulation is studied, namely fecundity — the number of protégés a mentor trains. Analysis of data from the Mathematics Genealogy Project shows that although mentorship fecundity correlates with success, those mentors who maintain a small fecundity go on to train protégés with a larger fecundity. Moreover, the mentor's career stage influences the eventual fecundity of their protégés.
The role of mentorship in protégé performance is a matter of importance to academic, business and governmental organizations. Although the benefits of mentorship for protégés, mentors and their organizations are apparent
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, the extent to which protégés mimic their mentors’ career choices and acquire their mentorship skills is unclear
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. The importance of a science, technology, engineering and mathematics workforce to economic growth and the role of effective mentorship in maintaining a ‘healthy’ such workforce demand the study of the role of mentorship in academia. Here we investigate one aspect of mentor emulation by studying mentorship fecundity—the number of protégés a mentor trains—using data from the Mathematics Genealogy Project
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, which tracks the mentorship record of thousands of mathematicians over several centuries. We demonstrate that fecundity among academic mathematicians is correlated with other measures of academic success. We also find that the average fecundity of mentors remains stable over 60 years of recorded mentorship. We further discover three significant correlations in mentorship fecundity. First, mentors with low mentorship fecundities train protégés that go on to have mentorship fecundities 37% higher than expected. Second, in the first third of their careers, mentors with high fecundities train protégés that go on to have fecundities 29% higher than expected. Finally, in the last third of their careers, mentors with high fecundities train protégés that go on to have fecundities 31% lower than expected.
Journal Article
Discovery of Galileo’s long-lost letter shows he edited his heretical ideas to fool the Inquisition
Exclusive: Document shows that the astronomer toned down the claims that triggered science history’s most infamous battle — then lied about his edits.
Exclusive: Document shows that the astronomer toned down the claims that triggered science history’s most infamous battle — then lied about his edits.
Journal Article