title: On the chromatic index of Latin squares

authors: Cavenagh, Nicholas J.

abstract: A proper coloring of a Latin square of order n is an assignment of colors to its elements triples such that each row, column and symbol is assigned n distinct colors. Equivalently, a proper coloring of a Latin square is a partition into partial transversals. The chromatic index of a Latin square is the least number of colors needed for a proper coloring. We study the chromatic index of the cyclic Latin square which arises from the addition table for the integers modulo n. We obtain the best possible bounds except for the case when n=2 is odd and divisible by 3. We make some conjectures about the chromatic index, suggesting a generalization of Ryser's conjecture (that every Latin square of odd order contains a transversal).

text: On the chromatic index of Latin squares A proper coloring of a Latin square of order n is an assignment of colors to its elements triples such that each row, column and symbol is assigned n distinct colors. Equivalently, a proper coloring of a Latin square is a partition into partial transversals. The chromatic index of a Latin square is the least number of colors needed for a proper coloring. We study the chromatic index of the cyclic Latin square which arises from the addition table for the integers modulo n. We obtain the best possible bounds except for the case when n=2 is odd and divisible by 3. We make some conjectures about the chromatic index, suggesting a generalization of Ryser's conjecture (that every Latin square of odd order contains a transversal).

year: 2015

title: Learning in the 21st century: Empowering student-centred practices through curriculum integration

authors: Morgan, Jason

subjects: student-centred

abstract: The New Zealand Curriculum [NZC] provides a framework for schools to design and implement the curriculum at three levels: Nationally, school- wide, and in the classroom (Ministry of Education [MoE], 2007). Through a student-centred approach, which the NZC supports, a culturally-responsive curriculum that integrates subject matter from the learning areas with the ‘front-end’ values and key competencies is advocated, promoting the interests of students, their whānau and communities (Dowden, 2010; Fraser & Paraha, 2002). Curriculum integration [CI] is a curriculum design theory, which values students and the world they live in as the main source of curriculum, and as a result of the democratic teaching pedagogy that underpins this practice, acknowledges students as active collaborators throughout the learning process (Brough, 2008a). When students are positioned as curriculum decision-makers and have their voice heard and valued, equal power relationships can be established, enriching and positively influencing student motivation, achievement, and engagement (Brough, 2012).

As Beane (1995) explains, CI is a fundamental realignment of thinking about the purpose of schools, and the sources and purposes of curriculum and knowledge. When students are actively encouraged to engage with learning experiences relevant to their lives, knowledge is acquired in an organic way and developed through meaningful, real-life purposes. When a school’s curriculum is designed to relate to both students’ and their communities’ needs and interests, students are no longer placed as consumers of information, but rather as producers of knowledge (Boyd, 2013). By integrating powerful learning environments in their classes which values students’ voice and supports active, autonomous and collaborative learning, teachers and students are engaged in open-ended, democratic decision- making around curriculum decisions. This can enable the fostering of confident, connected, actively involved, lifelong learners (Brough, 2008b; MoE, 2007).

The aim of this thesis was to explore how teachers in the primary school, in particular the senior setting of primary schools, are integrating the curriculum to encourage student-centred practices. It is hoped that this study will illuminate potential ways classroom practice can empower student-centred pedagogy and position students at the centre of educational decision-making. It was anticipated this project would contribute to the limited research in this field in both the primary and New Zealand school contexts.

A critical theoretical framework underpinned this study as it naturally aligned with the research question and the project’s aims. A case study methodology was adopted which used a variety of methods, including semi- structured interviews, naturalistic observations, photographs and documentation.

Through this study it was found student-centred curriculum integration can provide a powerful and motivating curriculum framework and pedagogy for learning in the 21st century. Through a pedagogy that is underpinned by democratic principles and practices, students can develop critical and creative thinking skills. These skills empower students to ‘perform’ knowledge across authentic, real-life contexts. This not only prepares students to lead fulfilling and rewarding lives as active participants within society, but promotes emancipatory ideals leading to beneficial outcomes for communities. This study argues how through curriculum integration, teachers can find ways to empower student-centred practices.

text: Learning in the 21st century: Empowering student-centred practices through curriculum integration The New Zealand Curriculum [NZC] provides a framework for schools to design and implement the curriculum at three levels: Nationally, school- wide, and in the classroom (Ministry of Education [MoE], 2007). Through a student-centred approach, which the NZC supports, a culturally-responsive curriculum that integrates subject matter from the learning areas with the ‘front-end’ values and key competencies is advocated, promoting the interests of students, their whānau and communities (Dowden, 2010; Fraser & Paraha, 2002). Curriculum integration [CI] is a curriculum design theory, which values students and the world they live in as the main source of curriculum, and as a result of the democratic teaching pedagogy that underpins this practice, acknowledges students as active collaborators throughout the learning process (Brough, 2008a). When students are positioned as curriculum decision-makers and have their voice heard and valued, equal power relationships can be established, enriching and positively influencing student motivation, achievement, and engagement (Brough, 2012).

As Beane (1995) explains, CI is a fundamental realignment of thinking about the purpose of schools, and the sources and purposes of curriculum and knowledge. When students are actively encouraged to engage with learning experiences relevant to their lives, knowledge is acquired in an organic way and developed through meaningful, real-life purposes. When a school’s curriculum is designed to relate to both students’ and their communities’ needs and interests, students are no longer placed as consumers of information, but rather as producers of knowledge (Boyd, 2013). By integrating powerful learning environments in their classes which values students’ voice and supports active, autonomous and collaborative learning, teachers and students are engaged in open-ended, democratic decision- making around curriculum decisions. This can enable the fostering of confident, connected, actively involved, lifelong learners (Brough, 2008b; MoE, 2007).

The aim of this thesis was to explore how teachers in the primary school, in particular the senior setting of primary schools, are integrating the curriculum to encourage student-centred practices. It is hoped that this study will illuminate potential ways classroom practice can empower student-centred pedagogy and position students at the centre of educational decision-making. It was anticipated this project would contribute to the limited research in this field in both the primary and New Zealand school contexts.

A critical theoretical framework underpinned this study as it naturally aligned with the research question and the project’s aims. A case study methodology was adopted which used a variety of methods, including semi- structured interviews, naturalistic observations, photographs and documentation.

Through this study it was found student-centred curriculum integration can provide a powerful and motivating curriculum framework and pedagogy for learning in the 21st century. Through a pedagogy that is underpinned by democratic principles and practices, students can develop critical and creative thinking skills. These skills empower students to ‘perform’ knowledge across authentic, real-life contexts. This not only prepares students to lead fulfilling and rewarding lives as active participants within society, but promotes emancipatory ideals leading to beneficial outcomes for communities. This study argues how through curriculum integration, teachers can find ways to empower student-centred practices.

year: 2014

title: Geographical and ethnic differences of osteoarthritis-associated hip and knee replacement surgeries in New Zealand: a population-based cross-sectional study

authors: Lao, Chunhuan

abstract: Objectives To (1) explore the regional and ethnic differences in rates of publicly funded osteoarthritis-associated hip and knee replacement surgeries and (2) investigate the mortality after surgery.

Design Population-based, retrospective, cross-sectional study.

Setting General population in New Zealand.

Participants Patients with osteoarthritis who underwent publicly funded primary hip and knee replacement surgeries in 2005–2017. Patients aged 14–99 years were included.

Primary and secondary outcome measures Age-standardised rate, standardised mortality ratio (SMR) and 30 days, 90 days and 1 year mortality.

Results We identified 53 439 primary hip replacements and 50 072 primary knee replacements with a diagnosis of osteoarthritis. The number and age-standardised rates of hip and knee replacements increased over time. Māori had the highest age-standardised rate of hip replacements, followed by European/others and Pacific, and Asian had the lowest rate. Pacific had the highest age-standardised rate of knee replacements, followed by Māori and European/others, and Asian had the lowest rate. The Northern Health Network had the lowest rate of hip surgeries, and the Southern Health Network had the lowest rate of knee surgeries. The SMRs of patients undergoing hip and knee replacements were lower than the general population: 0.92 (95% CI 0.89 to 0.95) for hip and 0.79 (95% CI 0.76 to 0.82) for knee. The SMRs were decreasing over time. The patterns of 30 days, 90 days and 1 year mortality were similar to the SMR.

Conclusions The numbers of publicly funded osteoarthritis-associated primary hip and knee replacements are steadily increasing. Māori people had the highest age-standardised rate of hip replacements and Pacific people had the highest rate of knee replacements. The Northern Health Network had the lowest rate of hip surgeries, and the Southern Health Network had the lowest rate of knee surgeries. Compared with the general population, patients who had hip and knee replacements have a better life expectancy.

text: Geographical and ethnic differences of osteoarthritis-associated hip and knee replacement surgeries in New Zealand: a population-based cross-sectional study Objectives To (1) explore the regional and ethnic differences in rates of publicly funded osteoarthritis-associated hip and knee replacement surgeries and (2) investigate the mortality after surgery.

Design Population-based, retrospective, cross-sectional study.

Setting General population in New Zealand.

Participants Patients with osteoarthritis who underwent publicly funded primary hip and knee replacement surgeries in 2005–2017. Patients aged 14–99 years were included.

Primary and secondary outcome measures Age-standardised rate, standardised mortality ratio (SMR) and 30 days, 90 days and 1 year mortality.

Results We identified 53 439 primary hip replacements and 50 072 primary knee replacements with a diagnosis of osteoarthritis. The number and age-standardised rates of hip and knee replacements increased over time. Māori had the highest age-standardised rate of hip replacements, followed by European/others and Pacific, and Asian had the lowest rate. Pacific had the highest age-standardised rate of knee replacements, followed by Māori and European/others, and Asian had the lowest rate. The Northern Health Network had the lowest rate of hip surgeries, and the Southern Health Network had the lowest rate of knee surgeries. The SMRs of patients undergoing hip and knee replacements were lower than the general population: 0.92 (95% CI 0.89 to 0.95) for hip and 0.79 (95% CI 0.76 to 0.82) for knee. The SMRs were decreasing over time. The patterns of 30 days, 90 days and 1 year mortality were similar to the SMR.

Conclusions The numbers of publicly funded osteoarthritis-associated primary hip and knee replacements are steadily increasing. Māori people had the highest age-standardised rate of hip replacements and Pacific people had the highest rate of knee replacements. The Northern Health Network had the lowest rate of hip surgeries, and the Southern Health Network had the lowest rate of knee surgeries. Compared with the general population, patients who had hip and knee replacements have a better life expectancy.

year: 2019