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Sharyn Livy - One of the best experts on this subject based on the ideXlab platform.
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A 'Knowledge Quartet' Used to Identify a Second-Year Pre-Service Teacher's Primary Mathematical Content Knowledge
2020Co-Authors: Sharyn LivyAbstract:This paper draws on observation of a primary mathematics lesson prepared and taught by a second-year pre-service teacher who lacked Mathematical Content knowledge. A ‘knowledge quartet’ (Rowland, Turner, Thwaites, & Huckstep, 2009) was used to investigate when and how a pre-service teacher drew on their knowledge of mathematics during primary teaching. Data were collected from field notes, audio recording of part of a lesson, and an interview with the pre-service teacher after the lesson. Discussion focuses on the four characteristics of the ‘knowledge quartet’: foundation, connection, transformation and contingency. Conclusions suggested that pre-service teachers need to continue developing their Mathematical Content knowledge to assist with future planning and teaching of primary mathematics lessons.
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developing primary pre service teachers Mathematical Content knowledge opportunities and influences
Mathematics Education Research Journal, 2019Co-Authors: Sharyn Livy, Sandra Herbert, Colleen ValeAbstract:There is a consensus that we need to improve the quality of pre-service teacher education, and teachers’ Mathematical Content knowledge is critical for teaching. Identifying opportunities and influences that assist pre-service teachers to extend their Mathematical Content knowledge throughout their teacher education programme is important. This paper reports on qualitative data, collected over 4 years from two typical pre-service teachers whose developing Mathematical Content knowledge was investigated during their primary and secondary programme. These data were analysed and reported using the four dimensions of the Knowledge Quartet: foundation knowledge, transformation, connection and contingency. The results highlight the consequences of programme structure in order to help pre-service teachers to establish and sustain a positive mathematics learner identity, build teacher identity and develop Mathematical Content knowledge.
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Developing Primary Pre-Service Teachers' Mathematical Content Knowledge during Practicum Teaching.
Australian Journal of Teacher Education, 2016Co-Authors: Sharyn Livy, Colleen Vale, Sandra HerbertAbstract:While it is recognised that a teachers' Mathematical Content knowledge (MCK) is crucial for teaching, less is known about when different categories of MCK develop during teacher education. This paper reports on two primary pre-service teachers, whose MCK was investigated during their practicum experiences in first, second and fourth years of a four-year Bachelor of Education program. The results identify when and under what conditions pre-service teachers' developed different categories of their MCK during practicum. Factors that assisted pre-service teachers to develop their MCK included program structure providing breadth and depth of experiences; sustained engagement for learning MCK; and quality of pre-service teachers' learning experiences.
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Opportunities to promote Mathematical Content knowledge for primary teaching
Mathematics Education Research Group of Australasia, 2014Co-Authors: Sharyn Livy, Sandra HerbertAbstract:Understanding the development of pre-service teachers’ Mathematical Content knowledge (MCK) is important for improving primary mathematics’ teacher education. This paper reports on a case study, Rose , and her opportunities to develop MCK during the four years of her program. Program opportunities to promote MCK when planning and practicing primary teaching included: coursework experiences and responding to assessment requirements. Discussion includes the Knowledge Quartet: foundation knowledge, transformation, connection and contingency. By fourth-year, Rose demonstrated development of different categories of MCK when practicing her teaching because of her program experiences.
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first year pre service teachers Mathematical Content knowledge methods of solution for a ratio question
Mathematics Teacher Education and Development, 2011Co-Authors: Sharyn Livy, Colleen ValeAbstract:In this article, pre-service teachers' mathematics Content knowledge is explored through the analysis of two items about ratio from a Mathematical Competency, Skills and Knowledge Test. Pre-service teachers' thinking strategies, common errors and misconceptions in their responses are presented and discussed. Of particular interest was the range and nature of common incorrect responses for one wholewhole ratio question. Results suggested pre-service teachers had difficulty interpreting a worded multi-step, ratio (scale) question, with errors relating to ratio and/or conversion of measurement knowledge. These difficulties reveal underdevel - oped knowledge of Mathematical structure and Mathematical connections as well as an inability to deconstruct key components of a Mathematical problem. Most pre-service teachers also lacked knowledge of standard procedures and methods of solutions.
Colleen Vale - One of the best experts on this subject based on the ideXlab platform.
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developing primary pre service teachers Mathematical Content knowledge opportunities and influences
Mathematics Education Research Journal, 2019Co-Authors: Sharyn Livy, Sandra Herbert, Colleen ValeAbstract:There is a consensus that we need to improve the quality of pre-service teacher education, and teachers’ Mathematical Content knowledge is critical for teaching. Identifying opportunities and influences that assist pre-service teachers to extend their Mathematical Content knowledge throughout their teacher education programme is important. This paper reports on qualitative data, collected over 4 years from two typical pre-service teachers whose developing Mathematical Content knowledge was investigated during their primary and secondary programme. These data were analysed and reported using the four dimensions of the Knowledge Quartet: foundation knowledge, transformation, connection and contingency. The results highlight the consequences of programme structure in order to help pre-service teachers to establish and sustain a positive mathematics learner identity, build teacher identity and develop Mathematical Content knowledge.
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Developing Primary Pre-Service Teachers' Mathematical Content Knowledge during Practicum Teaching.
Australian Journal of Teacher Education, 2016Co-Authors: Sharyn Livy, Colleen Vale, Sandra HerbertAbstract:While it is recognised that a teachers' Mathematical Content knowledge (MCK) is crucial for teaching, less is known about when different categories of MCK develop during teacher education. This paper reports on two primary pre-service teachers, whose MCK was investigated during their practicum experiences in first, second and fourth years of a four-year Bachelor of Education program. The results identify when and under what conditions pre-service teachers' developed different categories of their MCK during practicum. Factors that assisted pre-service teachers to develop their MCK included program structure providing breadth and depth of experiences; sustained engagement for learning MCK; and quality of pre-service teachers' learning experiences.
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first year pre service teachers Mathematical Content knowledge methods of solution for a ratio question
Mathematics Teacher Education and Development, 2011Co-Authors: Sharyn Livy, Colleen ValeAbstract:In this article, pre-service teachers' mathematics Content knowledge is explored through the analysis of two items about ratio from a Mathematical Competency, Skills and Knowledge Test. Pre-service teachers' thinking strategies, common errors and misconceptions in their responses are presented and discussed. Of particular interest was the range and nature of common incorrect responses for one wholewhole ratio question. Results suggested pre-service teachers had difficulty interpreting a worded multi-step, ratio (scale) question, with errors relating to ratio and/or conversion of measurement knowledge. These difficulties reveal underdevel - oped knowledge of Mathematical structure and Mathematical connections as well as an inability to deconstruct key components of a Mathematical problem. Most pre-service teachers also lacked knowledge of standard procedures and methods of solutions.
K Kowalski - One of the best experts on this subject based on the ideXlab platform.
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physical and Mathematical Content of coupled cluster equations iv impact of approximations to the cluster operator on the structure of solutions
Journal of Chemical Physics, 1999Co-Authors: K Jankowski, K KowalskiAbstract:The impact of approximations to the form of the cluster operator on the structure and physical significance of the complete set of geometrically isolated solutions to the coupled-cluster (CC) equations has been studied for the first time. To systematically study the correspondence of solutions obtained at various levels of the approximation process, a continuation procedure based on a set of β-nested equations (β-NE) has been proposed and applied. Numerical studies based on a homotopy method for obtaining full solutions to sets of polynomial equations have been performed for the H4 and P4 models which belong to the simplest realistic many-electron model systems. Two examples of approximation procedures have been considered. The first one involved, for the P4 model, the approximation leading from the full CC (FCC) method to the CC method based on double excitations (CCD). As a result of this approximations the number of solutions has increased from 8 to 20. In the second example, for H4, we have studied th...
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physical and Mathematical Content of coupled cluster equations iii model studies of dissociation processes for various reference states
Journal of Chemical Physics, 1999Co-Authors: K Jankowski, K KowalskiAbstract:The structure and physical significance of the full set of solutions to coupled-cluster (CC) equations at various stages of the dissociation process and the impact of the choice of reference functions on these solutions have been studied for the first time. The equations for the CC method involving double excitations (CCD) are obtained for the P4 model consisting of two H2 molecules in a rectangular nuclear configuration determined by a geometry parameter α. We consider equations for the reference states |ΦA〉, |ΦQ〉, and |ΦB〉 corresponding to the lowest, highest, and intermediate Hartree–Fock (HF) energies, respectively. The first two states provide a size-consistent description of the dissociation process. For the compact-molecule geometries (α<10.0) the sets of complete solutions to the standard CCD equations [based on molecular orbitals (MOs) of D2h symmetry] in the spin–orbital and spin–symmetry-adapted versions always consist of 20 and 12 entries, respectively. For |ΦA〉 and |ΦB〉 in the dissociation li...
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physical and Mathematical Content of coupled cluster equations ii on the origin of irregular solutions and their elimination via symmetry adaptation
Journal of Chemical Physics, 1999Co-Authors: K Jankowski, K KowalskiAbstract:To establish the existence and origin of the nonalgebraic irregularities of solutions to coupled-cluster (CC) equations and to indicate ways of their elimination, we have revisited the two analytically solvable characteristic equations (CE) studied by Živkovic and Monkhorst [J. Math. Phys. 19, 1007 (1978)]. The results of these studies have strongly influenced the general conclusions concerning the possible types of singularities. We present some arguments that the most serious irregularities—the nonnormal and resonance ones—are a result of the special structures of the CEs considered. The CE employed for the demonstration of resonance solutions is not physically representable, which raises the hope that such solutions will not appear in quantum-chemical applications of the coupled-cluster method. It is proved that the presence of nonnormal solutions is a consequence of the existence of such passive diagonal blocks of the Hamiltonian matrix which share a common eigenvalue. Such blocks can be eliminated by taking into account the symmetry species of the basis functions involved, which is most effectively done by proceeding to a symmetry-adapted formulations. Therefore, one may eliminate or at least reduce the number of nonnormal solutions to the CC equations by proceeding to their symmetry-adapted versions.
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physical and Mathematical Content of coupled cluster equations correspondence between coupled cluster and configuration interaction solutions
Journal of Chemical Physics, 1999Co-Authors: K Jankowski, K KowalskiAbstract:To gain more insight into the physical and Mathematical Content of the equations of the coupled–cluster (CC) method, comprehensive numerical studies have been performed for various geometries of the H4 model which belongs to the simplest and best understood among the realistic many–electron model systems. These studies are for the first time based on the knowledge of the complete sets of geometrically isolated solutions of the relevant equations that are obtained when using a special version of the homotopy methods. The equations of the CC method including two–electron excitations (CCD) both in the spin–orbital and spin–symmetry–adapted versions are considered. To establish the correspondence of the solutions attained with those of the configuration interaction (CID) method, we have for the first time solved the unabridged characteristic equations (CE) of T. P. Živkovic and H. J. Monkhorst [J. Math. Phys. 19, 1007 (1978)]. The complete sets of solutions to the spin–orbital and spin–symmetry–adapted versio...
K Jankowski - One of the best experts on this subject based on the ideXlab platform.
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physical and Mathematical Content of coupled cluster equations iv impact of approximations to the cluster operator on the structure of solutions
Journal of Chemical Physics, 1999Co-Authors: K Jankowski, K KowalskiAbstract:The impact of approximations to the form of the cluster operator on the structure and physical significance of the complete set of geometrically isolated solutions to the coupled-cluster (CC) equations has been studied for the first time. To systematically study the correspondence of solutions obtained at various levels of the approximation process, a continuation procedure based on a set of β-nested equations (β-NE) has been proposed and applied. Numerical studies based on a homotopy method for obtaining full solutions to sets of polynomial equations have been performed for the H4 and P4 models which belong to the simplest realistic many-electron model systems. Two examples of approximation procedures have been considered. The first one involved, for the P4 model, the approximation leading from the full CC (FCC) method to the CC method based on double excitations (CCD). As a result of this approximations the number of solutions has increased from 8 to 20. In the second example, for H4, we have studied th...
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physical and Mathematical Content of coupled cluster equations iii model studies of dissociation processes for various reference states
Journal of Chemical Physics, 1999Co-Authors: K Jankowski, K KowalskiAbstract:The structure and physical significance of the full set of solutions to coupled-cluster (CC) equations at various stages of the dissociation process and the impact of the choice of reference functions on these solutions have been studied for the first time. The equations for the CC method involving double excitations (CCD) are obtained for the P4 model consisting of two H2 molecules in a rectangular nuclear configuration determined by a geometry parameter α. We consider equations for the reference states |ΦA〉, |ΦQ〉, and |ΦB〉 corresponding to the lowest, highest, and intermediate Hartree–Fock (HF) energies, respectively. The first two states provide a size-consistent description of the dissociation process. For the compact-molecule geometries (α<10.0) the sets of complete solutions to the standard CCD equations [based on molecular orbitals (MOs) of D2h symmetry] in the spin–orbital and spin–symmetry-adapted versions always consist of 20 and 12 entries, respectively. For |ΦA〉 and |ΦB〉 in the dissociation li...
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physical and Mathematical Content of coupled cluster equations ii on the origin of irregular solutions and their elimination via symmetry adaptation
Journal of Chemical Physics, 1999Co-Authors: K Jankowski, K KowalskiAbstract:To establish the existence and origin of the nonalgebraic irregularities of solutions to coupled-cluster (CC) equations and to indicate ways of their elimination, we have revisited the two analytically solvable characteristic equations (CE) studied by Živkovic and Monkhorst [J. Math. Phys. 19, 1007 (1978)]. The results of these studies have strongly influenced the general conclusions concerning the possible types of singularities. We present some arguments that the most serious irregularities—the nonnormal and resonance ones—are a result of the special structures of the CEs considered. The CE employed for the demonstration of resonance solutions is not physically representable, which raises the hope that such solutions will not appear in quantum-chemical applications of the coupled-cluster method. It is proved that the presence of nonnormal solutions is a consequence of the existence of such passive diagonal blocks of the Hamiltonian matrix which share a common eigenvalue. Such blocks can be eliminated by taking into account the symmetry species of the basis functions involved, which is most effectively done by proceeding to a symmetry-adapted formulations. Therefore, one may eliminate or at least reduce the number of nonnormal solutions to the CC equations by proceeding to their symmetry-adapted versions.
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physical and Mathematical Content of coupled cluster equations correspondence between coupled cluster and configuration interaction solutions
Journal of Chemical Physics, 1999Co-Authors: K Jankowski, K KowalskiAbstract:To gain more insight into the physical and Mathematical Content of the equations of the coupled–cluster (CC) method, comprehensive numerical studies have been performed for various geometries of the H4 model which belongs to the simplest and best understood among the realistic many–electron model systems. These studies are for the first time based on the knowledge of the complete sets of geometrically isolated solutions of the relevant equations that are obtained when using a special version of the homotopy methods. The equations of the CC method including two–electron excitations (CCD) both in the spin–orbital and spin–symmetry–adapted versions are considered. To establish the correspondence of the solutions attained with those of the configuration interaction (CID) method, we have for the first time solved the unabridged characteristic equations (CE) of T. P. Živkovic and H. J. Monkhorst [J. Math. Phys. 19, 1007 (1978)]. The complete sets of solutions to the spin–orbital and spin–symmetry–adapted versio...
Sandra Herbert - One of the best experts on this subject based on the ideXlab platform.
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developing primary pre service teachers Mathematical Content knowledge opportunities and influences
Mathematics Education Research Journal, 2019Co-Authors: Sharyn Livy, Sandra Herbert, Colleen ValeAbstract:There is a consensus that we need to improve the quality of pre-service teacher education, and teachers’ Mathematical Content knowledge is critical for teaching. Identifying opportunities and influences that assist pre-service teachers to extend their Mathematical Content knowledge throughout their teacher education programme is important. This paper reports on qualitative data, collected over 4 years from two typical pre-service teachers whose developing Mathematical Content knowledge was investigated during their primary and secondary programme. These data were analysed and reported using the four dimensions of the Knowledge Quartet: foundation knowledge, transformation, connection and contingency. The results highlight the consequences of programme structure in order to help pre-service teachers to establish and sustain a positive mathematics learner identity, build teacher identity and develop Mathematical Content knowledge.
-
Developing Primary Pre-Service Teachers' Mathematical Content Knowledge during Practicum Teaching.
Australian Journal of Teacher Education, 2016Co-Authors: Sharyn Livy, Colleen Vale, Sandra HerbertAbstract:While it is recognised that a teachers' Mathematical Content knowledge (MCK) is crucial for teaching, less is known about when different categories of MCK develop during teacher education. This paper reports on two primary pre-service teachers, whose MCK was investigated during their practicum experiences in first, second and fourth years of a four-year Bachelor of Education program. The results identify when and under what conditions pre-service teachers' developed different categories of their MCK during practicum. Factors that assisted pre-service teachers to develop their MCK included program structure providing breadth and depth of experiences; sustained engagement for learning MCK; and quality of pre-service teachers' learning experiences.
-
Opportunities to promote Mathematical Content knowledge for primary teaching
Mathematics Education Research Group of Australasia, 2014Co-Authors: Sharyn Livy, Sandra HerbertAbstract:Understanding the development of pre-service teachers’ Mathematical Content knowledge (MCK) is important for improving primary mathematics’ teacher education. This paper reports on a case study, Rose , and her opportunities to develop MCK during the four years of her program. Program opportunities to promote MCK when planning and practicing primary teaching included: coursework experiences and responding to assessment requirements. Discussion includes the Knowledge Quartet: foundation knowledge, transformation, connection and contingency. By fourth-year, Rose demonstrated development of different categories of MCK when practicing her teaching because of her program experiences.
