title: Extended Kepler–Coulomb quantum superintegrable systems in three dimensions

authors: Kalnins, Ernie G.

subjects: quadratic algebras

abstract: The quantum Kepler-Coulomb system in three dimensions is well known to be second order superintegrable, with a symmetry algebra that closes polynomially under commutators. This polynomial closure is also typical for second order superintegrable systems in 2D and for second order systems in 3D with nondegenerate (four-parameter) potentials. However, the degenerate three-parameter potential for the 3D Kepler-Coulomb system (also second order superintegrable) is an exception, as its symmetry algebra does not close polynomially. The 3D four-parameter potential for the extended Kepler-Coulomb system is not even second order superintegrable, but Verrier and Evans (2008 J. Math. Phys. 49 022902) showed it was fourth order superintegrable, and Tanoudis and Daskaloyannis (2011 arXiv:11020397v1) showed that, if a second fourth order symmetry is added to the generators, the symmetry algebra closes polynomially. Here, based on the Tremblay, Turbiner and Winternitz construction, we consider an infinite class of quantum extended Kepler-Coulomb three-and four-parameter systems indexed by a pair of rational numbers (k(1), k(2)) and reducing to the usual systems when k(1) = k(2) = 1. We show these systems to be superintegrable of arbitrarily high order and determine the structure of their symmetry algebras. We demonstrate that the symmetry algebras close algebraically; only for systems admitting extra discrete symmetries is polynomial closure achieved. Underlying the structure theory is the existence of raising and lowering operators, not themselves symmetry operators or even defined independent of basis, that can be employed to construct the symmetry operators and their structure relations.

text: Extended Kepler–Coulomb quantum superintegrable systems in three dimensions The quantum Kepler-Coulomb system in three dimensions is well known to be second order superintegrable, with a symmetry algebra that closes polynomially under commutators. This polynomial closure is also typical for second order superintegrable systems in 2D and for second order systems in 3D with nondegenerate (four-parameter) potentials. However, the degenerate three-parameter potential for the 3D Kepler-Coulomb system (also second order superintegrable) is an exception, as its symmetry algebra does not close polynomially. The 3D four-parameter potential for the extended Kepler-Coulomb system is not even second order superintegrable, but Verrier and Evans (2008 J. Math. Phys. 49 022902) showed it was fourth order superintegrable, and Tanoudis and Daskaloyannis (2011 arXiv:11020397v1) showed that, if a second fourth order symmetry is added to the generators, the symmetry algebra closes polynomially. Here, based on the Tremblay, Turbiner and Winternitz construction, we consider an infinite class of quantum extended Kepler-Coulomb three-and four-parameter systems indexed by a pair of rational numbers (k(1), k(2)) and reducing to the usual systems when k(1) = k(2) = 1. We show these systems to be superintegrable of arbitrarily high order and determine the structure of their symmetry algebras. We demonstrate that the symmetry algebras close algebraically; only for systems admitting extra discrete symmetries is polynomial closure achieved. Underlying the structure theory is the existence of raising and lowering operators, not themselves symmetry operators or even defined independent of basis, that can be employed to construct the symmetry operators and their structure relations.

year: 2013

title: Relational pedagogy and the arts.

authors: Fraser, Deborah

abstract: This article explores the notion of “relational pedagogy” in the Arts (drama, dance, music, and visual art) starting with the teacher–child relationship, peer relationships, and building communities of enquiry.

text: Relational pedagogy and the arts. This article explores the notion of “relational pedagogy” in the Arts (drama, dance, music, and visual art) starting with the teacher–child relationship, peer relationships, and building communities of enquiry.

year: 2007

title: “I learned quite a lot of the Maths stuff now that I think of it”: Māori Medium Students reflecting on their Initial Teacher Education

authors: Hawera, Ngārewa

subjects: initial teacher education

abstract: Research involving preservice or initial teacher education (ITE) indicates that mathematics education is a vital component of study. Little is known however, of indigenous student views of their compulsory mathematics education courses for a teaching degree. This research contributes to that knowledge space as it explores Māori medium ITE students’ perceptions of mathematics education in Aotearoa New Zealand. A thematic and qualitative analysis of a focused group discussion provides insights into key factors that students reported as significant links between their university and practicum experiences (teaching practice in schools). Some suggestions for strengthening that programme were also expressed. Findings indicate that factors linked to teacher ‘cultural competencies’, including an ethic of care, respectful relationships, revision of content knowledge, language learning, assessment practices, exposure to different ideas and planning and pedagogy were important. Navigating student-mentor teacher tensions as well as clarifying the significance of ideas in texts were highlighted as areas for strengthening. Mathematics educators preparing students for indigenous primary school settings may find this study useful.

text: “I learned quite a lot of the Maths stuff now that I think of it”: Māori Medium Students reflecting on their Initial Teacher Education Research involving preservice or initial teacher education (ITE) indicates that mathematics education is a vital component of study. Little is known however, of indigenous student views of their compulsory mathematics education courses for a teaching degree. This research contributes to that knowledge space as it explores Māori medium ITE students’ perceptions of mathematics education in Aotearoa New Zealand. A thematic and qualitative analysis of a focused group discussion provides insights into key factors that students reported as significant links between their university and practicum experiences (teaching practice in schools). Some suggestions for strengthening that programme were also expressed. Findings indicate that factors linked to teacher ‘cultural competencies’, including an ethic of care, respectful relationships, revision of content knowledge, language learning, assessment practices, exposure to different ideas and planning and pedagogy were important. Navigating student-mentor teacher tensions as well as clarifying the significance of ideas in texts were highlighted as areas for strengthening. Mathematics educators preparing students for indigenous primary school settings may find this study useful.

year: 2017