title: Mixed mutually orthogonal frequency squares

authors: Poulose, Jilmy

abstract: A frequency square of type F(n; λ1, λ2, λ3, . . . , λt) is an array n × n filled with symbols ai, where each ai occurs λi times in every row and column, and n = Pt i=1 λi.
Two such frequency squares F1(n; λ1, λ2, λ3, . . . , λt) and F2(n; µ1, µ2, µ3, . . . , µs) are mutually orthogonal, when superimposed, each of the ts possible ordered pairs (i, j), where 1 ≤ i ≤ t and 1 ≤ j ≤ s, occurs exactly λiµj times.
This thesis generalizes the classical theory of mutually orthogonal Latin squares to mixed frequency settings, where symbols may appear with different frequencies. The non-uniform frequencies lead to a wider range of combinatorial structures with new methods of construction. Then, the thesis investigates maximizing the set of mixed mutually orthogonal frequency squares (MMOFS), focusing on theoretical methods rather than computational tools.
The following lemmas and theorems are new results presented in Chapter 3. In Lemma 4.1, we define mappings from two Latin squares to form two MOFS. Then in Theorem 4.7, we apply this to explore the maximum size of the sets of higher-order MMOFS by using mixed orthogonal arrays. Moreover, Section 2.2 gives an original alternative proof of an existing upper bound for sets of MOFS. Furthermore, we identify sets of MMOFS for small orders in the final chapter by using the new results and the previous theories. These new results include Corollary 4.5, Lemma 5.1, Example 22, Corollary 5.6 and Example 24.

text: Mixed mutually orthogonal frequency squares A frequency square of type F(n; λ1, λ2, λ3, . . . , λt) is an array n × n filled with symbols ai, where each ai occurs λi times in every row and column, and n = Pt i=1 λi.
Two such frequency squares F1(n; λ1, λ2, λ3, . . . , λt) and F2(n; µ1, µ2, µ3, . . . , µs) are mutually orthogonal, when superimposed, each of the ts possible ordered pairs (i, j), where 1 ≤ i ≤ t and 1 ≤ j ≤ s, occurs exactly λiµj times.
This thesis generalizes the classical theory of mutually orthogonal Latin squares to mixed frequency settings, where symbols may appear with different frequencies. The non-uniform frequencies lead to a wider range of combinatorial structures with new methods of construction. Then, the thesis investigates maximizing the set of mixed mutually orthogonal frequency squares (MMOFS), focusing on theoretical methods rather than computational tools.
The following lemmas and theorems are new results presented in Chapter 3. In Lemma 4.1, we define mappings from two Latin squares to form two MOFS. Then in Theorem 4.7, we apply this to explore the maximum size of the sets of higher-order MMOFS by using mixed orthogonal arrays. Moreover, Section 2.2 gives an original alternative proof of an existing upper bound for sets of MOFS. Furthermore, we identify sets of MMOFS for small orders in the final chapter by using the new results and the previous theories. These new results include Corollary 4.5, Lemma 5.1, Example 22, Corollary 5.6 and Example 24.

year: 2025

title: Mutualism beyond the "Mutual": The Collective Development of a New Zealand Single Industry Town Hospital

authors: Hurd, Fiona

subjects: Arts & Humanities

abstract: This paper discusses mutualism and its links to labourism. It is argued that rather than being contradictory, mutualism is incorporated into union activities in a range of ways beyond formal mutual and cooperative institutions, dependent on contextual differences in the labour movement. Using the case of mutual union and company involvement in the development of a public hospital in a single industry town in New Zealand during the 1960s and 1970s, we find evidence that the goals of management and the unions converged despite tensions at the site of production, and notions of cooperation for the benefit of workers and the wider community were brought to bear. As the workplace was an essential part of the town, the union’s interests were not limited to the workplace, but formed part of the social fabric of the town. Through this case, we see that engaging in mutualistic activities does not always demonstrate a weakening union agenda, but rather a method unions may employ towards improved worker welfare. Additionally, this example reminds us that union members are members of wider communities, families and societies, and that the boundaries between worker welfare in the workplace and those outside the workplace are not always easily drawn.

text: Mutualism beyond the "Mutual": The Collective Development of a New Zealand Single Industry Town Hospital This paper discusses mutualism and its links to labourism. It is argued that rather than being contradictory, mutualism is incorporated into union activities in a range of ways beyond formal mutual and cooperative institutions, dependent on contextual differences in the labour movement. Using the case of mutual union and company involvement in the development of a public hospital in a single industry town in New Zealand during the 1960s and 1970s, we find evidence that the goals of management and the unions converged despite tensions at the site of production, and notions of cooperation for the benefit of workers and the wider community were brought to bear. As the workplace was an essential part of the town, the union’s interests were not limited to the workplace, but formed part of the social fabric of the town. Through this case, we see that engaging in mutualistic activities does not always demonstrate a weakening union agenda, but rather a method unions may employ towards improved worker welfare. Additionally, this example reminds us that union members are members of wider communities, families and societies, and that the boundaries between worker welfare in the workplace and those outside the workplace are not always easily drawn.

year: 2017

title: “It’s becoming part of their knowing”: a study of bicultural development in an early childhood teacher education setting in Aotearoa/New Zealand

authors: Ritchie, Jenny

abstract: This study is an exploration of the implementation of bicultural development within a university-based early childhood teacher education programme in Aotearoa/New Zealand, which can be considered to be a bicultural country (Walker, 1987, p. 221). Bicultural development is a process in line with obligations contained within the Te Tiriti o Waitangi/The Treaty of Waitangi, which are reflected within the New Zealand Ministry of Education’s Revised Statement of Desirable Objectives and Practices (Ministry of Education, 1996a) and the early childhood curriculum, Te Whāriki (Ministry of Education, 1996b). The study adopts a critical, transformative theoretical paradigm, and employed an eclectic and emergent qualitative methodological approach, that reflected aspects from ethnographic, narrative and grounded theory methodologies. Interviews with university lecturers, early childhood care and education teachers, and professional development facilitators within the early childhood field; observations within early childhood care and education centres; audio taping of university class discussions; samples of university student assignments; and a written survey of graduates of the university teacher education programme provide the sources of data for the study. Central to the methodological process is a commitment to ethical care for participants, which involves attentive accountability to the interpretations that they place on the material that they contribute to the study and to the overall theorising of the significance of the findings. 

Key research questions of the study were to define ‘bicultural development’ in the context of early childhood care and education in this country; to identify key components of the bicultural development implementation process employed within the university teacher early childhood education programme; to distinguish some useful ‘indicators’ of bicultural development within early childhood centres; and to consider the implications for the teacher education programme in terms of preparing graduates to facilitate bicultural development within their future work in early childhood education. 

A significant finding of the study was the identification of a range of strategies for implementing bicultural development in early childhood education teacher education and care and education settings. The key strategies include Tiriti-based partnership models of teaching whereby Māori and Pākehā work alongside each other, enabling the paralleling of Māori and Western perspectives within both university teacher education and early childhood care and education programmes. It was considered essential that universities employ Māori lecturers to enable Māori content within the teacher education programme to be adequately provisioned. Since there are few Māori educators employed in mainstream early childhood services, it was suggested that teachers might seek to employ a “whanaungatanga” approach. This involves the early childhood centre staff creating a climate and environment in which Māori families/whānau are comfortable to participate fully in programme planning, implementation, and evaluation, which will enable all children attending to access Māori perspectives alongside the Western curriculum content. This approach is consistent with Article Two of Te Tiriti o Waitangi, which guaranteed to Māori their tino rangatiratanga, or self-determination. 

An implication for management and staff of early childhood teacher education programmes in Aotearoa/New Zealand is the need to equip graduates with the knowledge bases and commitment that will enable them to implement bicultural development, ideally through adopting, a whanaungatanga approach. This involves facilitating teacher education students to move beyond the dysconscious racism (King, 1994) that is deeply embedded in mainstream Pākehā culture in this country, to the position where they are able to adopt a critical, transformative stance that is humble, respectful and responsive to cultural difference. This stance includes valuing the inclusion of te reo me ōna tikanga throughout early childhood programmes, in a partnership initiated by teachers, but one in which whānau Māori are then able to take the lead in determining the nature and application of Māori content.

text: “It’s becoming part of their knowing”: a study of bicultural development in an early childhood teacher education setting in Aotearoa/New Zealand This study is an exploration of the implementation of bicultural development within a university-based early childhood teacher education programme in Aotearoa/New Zealand, which can be considered to be a bicultural country (Walker, 1987, p. 221). Bicultural development is a process in line with obligations contained within the Te Tiriti o Waitangi/The Treaty of Waitangi, which are reflected within the New Zealand Ministry of Education’s Revised Statement of Desirable Objectives and Practices (Ministry of Education, 1996a) and the early childhood curriculum, Te Whāriki (Ministry of Education, 1996b). The study adopts a critical, transformative theoretical paradigm, and employed an eclectic and emergent qualitative methodological approach, that reflected aspects from ethnographic, narrative and grounded theory methodologies. Interviews with university lecturers, early childhood care and education teachers, and professional development facilitators within the early childhood field; observations within early childhood care and education centres; audio taping of university class discussions; samples of university student assignments; and a written survey of graduates of the university teacher education programme provide the sources of data for the study. Central to the methodological process is a commitment to ethical care for participants, which involves attentive accountability to the interpretations that they place on the material that they contribute to the study and to the overall theorising of the significance of the findings. 

Key research questions of the study were to define ‘bicultural development’ in the context of early childhood care and education in this country; to identify key components of the bicultural development implementation process employed within the university teacher early childhood education programme; to distinguish some useful ‘indicators’ of bicultural development within early childhood centres; and to consider the implications for the teacher education programme in terms of preparing graduates to facilitate bicultural development within their future work in early childhood education. 

A significant finding of the study was the identification of a range of strategies for implementing bicultural development in early childhood education teacher education and care and education settings. The key strategies include Tiriti-based partnership models of teaching whereby Māori and Pākehā work alongside each other, enabling the paralleling of Māori and Western perspectives within both university teacher education and early childhood care and education programmes. It was considered essential that universities employ Māori lecturers to enable Māori content within the teacher education programme to be adequately provisioned. Since there are few Māori educators employed in mainstream early childhood services, it was suggested that teachers might seek to employ a “whanaungatanga” approach. This involves the early childhood centre staff creating a climate and environment in which Māori families/whānau are comfortable to participate fully in programme planning, implementation, and evaluation, which will enable all children attending to access Māori perspectives alongside the Western curriculum content. This approach is consistent with Article Two of Te Tiriti o Waitangi, which guaranteed to Māori their tino rangatiratanga, or self-determination. 

An implication for management and staff of early childhood teacher education programmes in Aotearoa/New Zealand is the need to equip graduates with the knowledge bases and commitment that will enable them to implement bicultural development, ideally through adopting, a whanaungatanga approach. This involves facilitating teacher education students to move beyond the dysconscious racism (King, 1994) that is deeply embedded in mainstream Pākehā culture in this country, to the position where they are able to adopt a critical, transformative stance that is humble, respectful and responsive to cultural difference. This stance includes valuing the inclusion of te reo me ōna tikanga throughout early childhood programmes, in a partnership initiated by teachers, but one in which whānau Māori are then able to take the lead in determining the nature and application of Māori content.

year: 2002