title: Teachers’ reflecting on professional knowledge in the numeracy (mathematics) classroom

authors: Mills, Judith Patricia

abstract: New Zealand primary school teachers are expected to regularly reflect on their teaching practice in order to consider the implications of past teaching on future planning. Aligned to teachers’ ongoing reflection, the New Zealand Curriculum (Ministry of Education, 2007) contains a section on effective pedagogy—teacher actions promoting student learning, which includes a Teaching as Inquiry Cycle (pp. 34–35). Embedded within their inquiry, teachers consider the teaching-learning relationship and often turn to frameworks of knowledge for guidance. This article shares the implications of using a framework of teacher knowledge in research. While the framework used contained much detail for the researcher, it overlapped categories and at the same time lacked acknowledgement of some important concepts for teachers in classroom practice. Findings from using a framework in this research were combined with findings from previous research to formulate the Wheel of Professional Knowledge, which was developed for mathematics teachers to use when reflecting on their practice.

text: Teachers’ reflecting on professional knowledge in the numeracy (mathematics) classroom New Zealand primary school teachers are expected to regularly reflect on their teaching practice in order to consider the implications of past teaching on future planning. Aligned to teachers’ ongoing reflection, the New Zealand Curriculum (Ministry of Education, 2007) contains a section on effective pedagogy—teacher actions promoting student learning, which includes a Teaching as Inquiry Cycle (pp. 34–35). Embedded within their inquiry, teachers consider the teaching-learning relationship and often turn to frameworks of knowledge for guidance. This article shares the implications of using a framework of teacher knowledge in research. While the framework used contained much detail for the researcher, it overlapped categories and at the same time lacked acknowledgement of some important concepts for teachers in classroom practice. Findings from using a framework in this research were combined with findings from previous research to formulate the Wheel of Professional Knowledge, which was developed for mathematics teachers to use when reflecting on their practice.

year: 2020

title: Semigroups with if-then-else and halting programs

authors: Jackson, Marcel

subjects: transformation

abstract: The "if–then–else" construction is one of the most elementary programming commands, and its abstract laws have been widely studied, starting with McCarthy. Possibly, the most obvious extension of this is to include the operation of composition of programs, which gives a semigroup of functions (total, partial, or possibly general binary relations) that can be recombined using if–then–else. We show that this particular extension admits no finite complete axiomatization and instead focus on the case where composition of functions with predicates is also allowed (and we argue there is good reason to take this approach). In the case of total functions — modeling halting programs — we give a complete axiomatization for the theory in terms of a finite system of equations. We obtain a similar result when an operation of equality test and/or fixed point test is included.

text: Semigroups with if-then-else and halting programs The "if–then–else" construction is one of the most elementary programming commands, and its abstract laws have been widely studied, starting with McCarthy. Possibly, the most obvious extension of this is to include the operation of composition of programs, which gives a semigroup of functions (total, partial, or possibly general binary relations) that can be recombined using if–then–else. We show that this particular extension admits no finite complete axiomatization and instead focus on the case where composition of functions with predicates is also allowed (and we argue there is good reason to take this approach). In the case of total functions — modeling halting programs — we give a complete axiomatization for the theory in terms of a finite system of equations. We obtain a similar result when an operation of equality test and/or fixed point test is included.

year: 2009

title: Spot sampling of nutrient concentrations in the Puarenga catchment, Rotorua

authors: Abell, Jonathan Michael

subjects: New Zealand

abstract: The Centre for Biodiversity and Ecology Research was approached by Tūhourangi Tribal Authority for assistance with measuring water quality in streams in the Puarenga Stream catchment. Water sampling was subsequently undertaken on 18 July 2011 and samples were analysed to determine concentrations of total and dissolved fractions of nitrogen and phosphorus. 
Nitrogen and phosphorus are both essential plant nutrients which, when present in excess, can cause eutrophication and associated water quality decline of freshwaters. High concentrations of dissolved forms of nitrogen can also be toxic to aquatic organisms. Excessive nitrogen and phosphorus concentrations are typically the result of pollution due to human activities, although groundwater in the Central Volcanic Plateau region can have elevated concentrations of phosphorus arising from natural geological sources.
This report summarises the methods used, presents the results and places measured concentrations in context by drawing comparisons with both guideline and regional mean values.

text: Spot sampling of nutrient concentrations in the Puarenga catchment, Rotorua The Centre for Biodiversity and Ecology Research was approached by Tūhourangi Tribal Authority for assistance with measuring water quality in streams in the Puarenga Stream catchment. Water sampling was subsequently undertaken on 18 July 2011 and samples were analysed to determine concentrations of total and dissolved fractions of nitrogen and phosphorus. 
Nitrogen and phosphorus are both essential plant nutrients which, when present in excess, can cause eutrophication and associated water quality decline of freshwaters. High concentrations of dissolved forms of nitrogen can also be toxic to aquatic organisms. Excessive nitrogen and phosphorus concentrations are typically the result of pollution due to human activities, although groundwater in the Central Volcanic Plateau region can have elevated concentrations of phosphorus arising from natural geological sources.
This report summarises the methods used, presents the results and places measured concentrations in context by drawing comparisons with both guideline and regional mean values.

year: 2011-09