title: The development of pre-service teachers’ competence to teach mental calculation strategies

authors: Vale, Pamela

subjects: pre-service teachers

abstract: Background: There is a concern in South Africa that pre-service teachers do not have the required knowledge to teach mathematics in primary school and to develop learners’ number sense. In this study, pre-service teachers taught the mental strategy of bridging through ten through a structured teaching sequence from the Mental Starters Assessment Project (MSAP) materials as a work-integrated learning opportunity.

Aim: We ask the question: How do the MSAP materials support pre-service teachers in competently teaching mental mathematics?

Setting: Thirty-eight Bachelor of Education (Foundation Phase) third-year preservice teachers from an Eastern Cape university participated in this study.

Methods: Participants taught the strategy during their Teaching Practice, quantitatively analysed the results of their classes and reflected on the experience in a questionnaire and focus group interviews.

Results: Results indicate that the teachers were relatively successful in their teaching of the strategy; however, all indicated that they taught the sequence for a more extended period than recommended. Qualitative responses provide evidence of the teachers’ development in their knowledge of learners and their characteristics, general pedagogical knowledge, pedagogical content knowledge and knowledge of educational contexts.

Conclusion: This study offers evidence of the professional learning of pre-service teachers that resulted from taking an integrated approach to facilitating a mathematics teaching methodology course through requiring a work-integrated learning component.

Contribution: We argue that such an approach is necessary for pre-service teachers to be adequately prepared for the challenges of teaching mathematics in the South African classroom.

text: The development of pre-service teachers’ competence to teach mental calculation strategies Background: There is a concern in South Africa that pre-service teachers do not have the required knowledge to teach mathematics in primary school and to develop learners’ number sense. In this study, pre-service teachers taught the mental strategy of bridging through ten through a structured teaching sequence from the Mental Starters Assessment Project (MSAP) materials as a work-integrated learning opportunity.

Aim: We ask the question: How do the MSAP materials support pre-service teachers in competently teaching mental mathematics?

Setting: Thirty-eight Bachelor of Education (Foundation Phase) third-year preservice teachers from an Eastern Cape university participated in this study.

Methods: Participants taught the strategy during their Teaching Practice, quantitatively analysed the results of their classes and reflected on the experience in a questionnaire and focus group interviews.

Results: Results indicate that the teachers were relatively successful in their teaching of the strategy; however, all indicated that they taught the sequence for a more extended period than recommended. Qualitative responses provide evidence of the teachers’ development in their knowledge of learners and their characteristics, general pedagogical knowledge, pedagogical content knowledge and knowledge of educational contexts.

Conclusion: This study offers evidence of the professional learning of pre-service teachers that resulted from taking an integrated approach to facilitating a mathematics teaching methodology course through requiring a work-integrated learning component.

Contribution: We argue that such an approach is necessary for pre-service teachers to be adequately prepared for the challenges of teaching mathematics in the South African classroom.

year: 2024

title: Critical sets of full Latin squares

authors: Raass, Petelo Vaipuna

abstract: This thesis explores the properties of critical sets of the full n-Latin square and related combinatorial structures including full designs, (m,n,2)-balanced Latin rectangles and n-Latin cubes.

In Chapter 3 we study known results on designs and the analogies between critical sets of the full n-Latin square and minimal defining sets of the full designs.

Next in Chapter 4 we fully classify the critical sets of the full (m,n,2)-balanced Latin square, by describing the precise structures of these critical sets from the smallest to the largest.

Properties of different types of critical sets of the full n-Latin square are investigated in Chapter 5. We fully classify the structure of any saturated critical set of the full n-Latin square. We show in Theorem 5.8 that such a critical set has size exactly equal to n³ - 2n² - n. In Section 5.2 we give a construction which provides an upper bound for the size of the smallest critical set of the full n-Latin square. Similarly in Section 5.4, another construction gives a lower bound for the size of the largest non-saturated critical set. We conjecture that these bounds are best possible.

Using the results from Chapter 5, we obtain spectrum results on critical sets of the full n-Latin square in Chapter 6. In particular, we show that a critical set of each size between (n - 1)³ + 1 and n(n - 1)² + n - 2 exists.

In Chapter 7, we turn our focus to the completability of partial k-Latin squares. The relationship between partial k-Latin squares and semi-k-Latin squares is used to show that any partial k-Latin square of order n with at most (n - 1) non-empty cells is completable.

As Latin cubes generalize Latin squares, we attempt to generalize some of the results we have established on k-Latin squares so that they apply to k-Latin cubes. These results are presented in Chapter 8.

text: Critical sets of full Latin squares This thesis explores the properties of critical sets of the full n-Latin square and related combinatorial structures including full designs, (m,n,2)-balanced Latin rectangles and n-Latin cubes.

In Chapter 3 we study known results on designs and the analogies between critical sets of the full n-Latin square and minimal defining sets of the full designs.

Next in Chapter 4 we fully classify the critical sets of the full (m,n,2)-balanced Latin square, by describing the precise structures of these critical sets from the smallest to the largest.

Properties of different types of critical sets of the full n-Latin square are investigated in Chapter 5. We fully classify the structure of any saturated critical set of the full n-Latin square. We show in Theorem 5.8 that such a critical set has size exactly equal to n³ - 2n² - n. In Section 5.2 we give a construction which provides an upper bound for the size of the smallest critical set of the full n-Latin square. Similarly in Section 5.4, another construction gives a lower bound for the size of the largest non-saturated critical set. We conjecture that these bounds are best possible.

Using the results from Chapter 5, we obtain spectrum results on critical sets of the full n-Latin square in Chapter 6. In particular, we show that a critical set of each size between (n - 1)³ + 1 and n(n - 1)² + n - 2 exists.

In Chapter 7, we turn our focus to the completability of partial k-Latin squares. The relationship between partial k-Latin squares and semi-k-Latin squares is used to show that any partial k-Latin square of order n with at most (n - 1) non-empty cells is completable.

As Latin cubes generalize Latin squares, we attempt to generalize some of the results we have established on k-Latin squares so that they apply to k-Latin cubes. These results are presented in Chapter 8.

year: 2016

title: La gouvernance des rivières en Nouvelle-Zélande : une solution élégante ?

authors: Te Aho, Linda

subjects: Water governance,

abstract: Governance of Water in New Zealand: An Elegant Solution? This chapter is grounded in the context of Aotearoa New Zealand while drawing on larger references, such as the United Nations Declaration on the Rights of Indigenous peoples, the Rio+20 Declaration and constitutional provisions for Rights of Nature. The chapter analyses a proposed new model for the contentious issue of freshwater governance in New Zealand based on responsible use which also addresses the complex world of state-Māori relations.

text: La gouvernance des rivières en Nouvelle-Zélande : une solution élégante ? Governance of Water in New Zealand: An Elegant Solution? This chapter is grounded in the context of Aotearoa New Zealand while drawing on larger references, such as the United Nations Declaration on the Rights of Indigenous peoples, the Rio+20 Declaration and constitutional provisions for Rights of Nature. The chapter analyses a proposed new model for the contentious issue of freshwater governance in New Zealand based on responsible use which also addresses the complex world of state-Māori relations.

year: 2018