title: Making multiplication meaningful: Teaching for conceptual understanding

authors: Mills, Judith Patricia

subjects: Conceptual understanding

abstract: The term numeracy is used widely in schools today and brings with it the expectation that students will be taught both how to do the mathematics, alongside an understanding of the concepts associated with the procedural application. One issue, which has arisen with the terminology ‘numeracy classroom’, is how to best support teachers to enhance their teaching of mathematics to allow this understanding to occur. This article stems from a larger research study that analysed the professional knowledge of teachers when teaching numeracy, and the impact their mathematics knowledge and procedural application had on children’s learning. This article presents observations of three teachers teaching a multiplication lesson (the first in a series of lessons over a six-week period) as they developed their students’ understanding of the mathematical concepts associated with the interpretation of the multiplication symbol. An analysis of the findings shows when the teachers used manipulatives, related word problems to the children’s lives, and promoted discussion in groups, a greater understanding of multiplication was apparent.

text: Making multiplication meaningful: Teaching for conceptual understanding The term numeracy is used widely in schools today and brings with it the expectation that students will be taught both how to do the mathematics, alongside an understanding of the concepts associated with the procedural application. One issue, which has arisen with the terminology ‘numeracy classroom’, is how to best support teachers to enhance their teaching of mathematics to allow this understanding to occur. This article stems from a larger research study that analysed the professional knowledge of teachers when teaching numeracy, and the impact their mathematics knowledge and procedural application had on children’s learning. This article presents observations of three teachers teaching a multiplication lesson (the first in a series of lessons over a six-week period) as they developed their students’ understanding of the mathematical concepts associated with the interpretation of the multiplication symbol. An analysis of the findings shows when the teachers used manipulatives, related word problems to the children’s lives, and promoted discussion in groups, a greater understanding of multiplication was apparent.

year: 2019

title: Non-orthogonally transitive G₂ spike solution

authors: Lim, Woei Chet

subjects: Science & Technology

abstract: We generalize the orthogonally transitive (OT) G₂ spike solution to the non-OT G₂ case. This is achieved by applying Geroch's transformation on a Kasner seed. The new solution contains two more parameters than the OT G₂ spike solution. Unlike the OT G₂ spike solution, the new solution always resolves its spike.

text: Non-orthogonally transitive G₂ spike solution We generalize the orthogonally transitive (OT) G₂ spike solution to the non-OT G₂ case. This is achieved by applying Geroch's transformation on a Kasner seed. The new solution contains two more parameters than the OT G₂ spike solution. Unlike the OT G₂ spike solution, the new solution always resolves its spike.

year: 2015

title: Tephras and New Zealand archaeology

authors: Lowe, David J.

subjects: tephrochronology

abstract: Establishing an accurate date for earliest Polynesian settlement in New Zealand is essential for understanding patterns of settlement and associated environmental impacts, and the processes and rates of cultural change in Eastern Polynesia. Tephra deposits from five volcanic centres, together with exotic sea-rafted pumice, provide isochronous constraints on the timing of earliest settlement and human impacts in northern New Zealand. A local basaltic tephra from Rangitoto Island (Auckland) and locally distributed andesitic tephras from Egmont volcano directly date human occupation to c. AD 1400–1450. Distal andesitic tephras (Tufa Trig Formation) from Mt Ruapehu, Tongariro volcanic centre, help constrain the timing of earliest anthropogenic deforestation signals in Hawke's Bay. Sea-rafted Loisels Pumice(s), although of uncertain stratigraphic reliability in places, overlies cultural remains that can be no younger than c. AD 1350 along the east coast, North Island. The regionally extensive rhyolitic Kaharoa Tephra, which erupted from Okataina volcano between c. AD  1300–1390, is the critical “settlement layer” datum for dating prehistory in the North Island: no human artefacts are recorded beneath it and the earliest inferred environmental impacts by humans are dated to c. AD  1280, just prior to its deposition. This maximum date matches the earliest radiocarbon dates derived for both settlement and human impacts from archaeological and natural sites (c. AD 1250), and implies that the onset of deforestation was essentially contemporaneous with initial settlement.
The widespread rhyolitic Taupo Tephra, which erupted from Taupo volcano c. AD 200, provides an isochronous benchmark well before earliest settlement. The tephra may coincide approximately with a putative earlier transient contact in New Zealand based on Pacific rat-bone (Rattus exulans) dates. More precise calendrical dates on the tephras—via dendrochronology or ice-core records or other dating methods—would help refine assessment of the timing of earliest settlement, while extending the distributional range of critical tephra layers, through application of crypto-tephra analysis, could lead to a greater understanding of settlement patterns.

text: Tephras and New Zealand archaeology Establishing an accurate date for earliest Polynesian settlement in New Zealand is essential for understanding patterns of settlement and associated environmental impacts, and the processes and rates of cultural change in Eastern Polynesia. Tephra deposits from five volcanic centres, together with exotic sea-rafted pumice, provide isochronous constraints on the timing of earliest settlement and human impacts in northern New Zealand. A local basaltic tephra from Rangitoto Island (Auckland) and locally distributed andesitic tephras from Egmont volcano directly date human occupation to c. AD 1400–1450. Distal andesitic tephras (Tufa Trig Formation) from Mt Ruapehu, Tongariro volcanic centre, help constrain the timing of earliest anthropogenic deforestation signals in Hawke's Bay. Sea-rafted Loisels Pumice(s), although of uncertain stratigraphic reliability in places, overlies cultural remains that can be no younger than c. AD 1350 along the east coast, North Island. The regionally extensive rhyolitic Kaharoa Tephra, which erupted from Okataina volcano between c. AD  1300–1390, is the critical “settlement layer” datum for dating prehistory in the North Island: no human artefacts are recorded beneath it and the earliest inferred environmental impacts by humans are dated to c. AD  1280, just prior to its deposition. This maximum date matches the earliest radiocarbon dates derived for both settlement and human impacts from archaeological and natural sites (c. AD 1250), and implies that the onset of deforestation was essentially contemporaneous with initial settlement.
The widespread rhyolitic Taupo Tephra, which erupted from Taupo volcano c. AD 200, provides an isochronous benchmark well before earliest settlement. The tephra may coincide approximately with a putative earlier transient contact in New Zealand based on Pacific rat-bone (Rattus exulans) dates. More precise calendrical dates on the tephras—via dendrochronology or ice-core records or other dating methods—would help refine assessment of the timing of earliest settlement, while extending the distributional range of critical tephra layers, through application of crypto-tephra analysis, could lead to a greater understanding of settlement patterns.

year: 2000