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Kaye Stacey - One of the best experts on this subject based on the ideXlab platform.
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Relative Risk Analysis of Educational Data
2015Co-Authors: Kaye Stacey, Vicki SteinleAbstract:This paper demonstrates the use of relative risk, a statistic which is widely used in other areas but currently under-utilised in education. Relative risk analysis provides a language for comparing educational outcomes as well as statistical tests of significance. We illustrate this statistic with data on students ’ understanding of Decimal Notation. In particular, we determine differences in how misconceptions operate at different ages, by analysing the relative risk of primary and secondary students persisting with particular misconceptions, and becoming experts. This paper illustrates an approach to reporting, comparing and analysing educational outcomes, which is widely used in other fields but which has not been often used in mathematics education. The concept of relative risk is widely used in reporting results of medical, environmental and epidemiological research, in both scientific papers and in the popular press. Educators could capitalise on this popular familiarity in reporting their results. As well as being useful for describing results, measures of relative risk are amenable to statistical analysis. There are simple tests of statistical significance, confidence intervals and effect sizes, which are easily calculated; either manually, with
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A Longitudinal Study of Students ‘Understanding of Decimal Notation: An Overview and Refined Results.
2015Co-Authors: Vicki Steinle, Kaye StaceyAbstract:This paper provides an overview of the major results of a large-scale longitudinal study of students ’ misconceptions of Decimal Notation, drawing them together and presenting refined results. Best estimates of the prevalence of various misconceptions about Decimal numbers from both cross-sectional and longitudinal perspectives are provided, as well as some estimates of persistence. Strengths, limitations and suggestions for improvements to the Decimal Comparison Test as well as major implications for teaching are discussed. This paper aims to provide an overview and discussion of the results of a longitudinal study of students ’ understanding of Decimal Notation, and to present some refinements to previously published results. As far as we know, this is the first study to track the misconceptions of a large number of students longitudinally and hence for the first time to describe the paths that students take through misconceptions on their way to expertise. The study revealed new misconceptions of Decimals, found how common they are across the middle years of schooling, explored which misconceptions were “better ” or “worse ” to have, and showed underlying links between them. A range of associated studies not reported here and using different samples of students, investigated effective teaching
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PERSISTENCE OF Decimal MISCONCEPTIONS AND READINESS TO MOVE TO EXPERTISE
2014Co-Authors: Vicki Steinle, Kaye StaceyAbstract:This paper describes features of a group of misconceptions about Decimal Notation that lead to students selecting as larger, Decimals that look smaller. A longitudinal study identified approximately 900 students from a variety of schools who exhibited these misconceptions and whose subsequent progress could be traced. The data demonstrates that the progress that students makes depends to a certain extent both on the nature of their misconceptions and the grade at which the misconception is held. Such phenomena could be expected to hold for misconceptions in other topics. AIMS This paper aims to describe some features of several interesting misconceptions about Decimal Notation. The results are derived from careful analysis of a longitudinal study of students ’ misconceptions, with data collected in Melbourne, Australia from 199
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ANALYSING LONGITUDINAL DATA ON STUDENTS ’ Decimal UNDERSTANDING USING RELATIVE RISK AND ODDS RATIOS
2014Co-Authors: Vicki Steinle, Kaye StaceyAbstract:The purpose of this paper is to demonstrate the use of the statistics of relative risk and odds ratios in mathematics education. These statistics are widely used in other fields (especially medical research) and offer a useful but currently under-utilised alternative for education. The demonstration uses data from a longitudinal study of students ’ understanding of Decimal Notation. We investigate the statistical significance of results related to the persistence of misconceptions and the hierarchy between misconceptions. Relative risk and odds ratio techniques provide confidence intervals, which give a measure of effect size missing from simple hypothesis testing, and enable differences between phenomena to be assessed and reported with impact. This paper demonstrates some possibilities for analysing educational data, which draw upon methods that are widely used in reporting results of medical, environmental and epidemiological research. We believe that these measures provide very useful techniques for testing for statistical significance and reporting confidence intervals, which will enhance mathematics education research. Capraro (2004) draws attention to important recent policy changes within the American Psychologica
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TRAVELLING THE ROAD TO EXPERTISE: A LONGITUDINAL STUDY OF LEARNING
2014Co-Authors: Kaye StaceyAbstract:A longitudinal study of students ’ developing understanding of Decimal Notation has been conducted by testing over 3000 students in Grades 4 to 10 up to 7 times. A pencil-and-paper test based on a carefully designed set of Decimal comparison items enabled students ’ responses to be classified into 11 codes and tracked over time. The paper reports on how students ’ ideas changed across the grades, which ways of thinking were most prevalent, the most persistent and which were most likely to lead to expertise. Interestingly the answers were different for primary and secondary students. Estimates are also given of the proportion of students affected by particular ways of thinking during schooling. The conclusion shows how a careful mapping can be useful and draws out features of the learning environment that affect learning. In this presentation, we will travel on a metaphorical seven year journey with over 3000 students. As they progress from Grades 4 to 10, learning mathematics in their usual classrooms, we will think of these students as travelling along a road where the destination is to understand the meaning of Decimals. The noun “Decimal ” means a number written in base ten numeration with a visible Decimal point or decima
Ranta Aarne - One of the best experts on this subject based on the ideXlab platform.
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Structures grammaticales dans le français mathématique : 1
'OpenEdition', 2006Co-Authors: Ranta AarneAbstract:A system of grammatical rules is presented to analyse a fragment of French that permits the expression of mathematical theorems and proofs. To this end, a version of Montague grammar is developed, with syntactic categories relativized to a context and to domains of individuals. This system can be interpreted in the constructive type theory of Martin-Löf. It is first applied to French without mathematical symbols, paying special attention to selectional restrictions and to dependencies on context. The fragment includes verbs and adjectives, plurals, relative clauses, and coordinated phrases of different categories. Second, the grammar is extended to mathematical symbolism and its embedding in French text. The fragment comprises arithmetical formulae, Decimal Notation, parenthesis conventions, explicit variables, statements of theorems, and textual structures of proofs. Finally, some applications of the grammar are studied, based on a declarative implementation in the proof editor ALF.Un système de règles grammaticales est présenté pour analyser un fragment du français permettant l'expression de théorèmes et de preuves mathématiques. Pour cet objectif, on développe une version de la grammaire de Montague, avec des catégories syntaxiques relatives au contexte et aux domaines d'individus. Ce système peut être interprété dans la théorie constructive des types de Martin-Löf. Il est appliqué, d'abord, au français sans symboles mathématiques, avec une attention spéciale aux restrictions de sélection et aux dépendances par rapport à un contexte. Le fragment comprend des verbes et des adjectifs, des formes plurielles, des propositions relatives, et des syntagmes coordonnés. Ensuite, la grammaire est étendue au symbolisme mathématique et à son usage dans le texte français. Le fragment comprend des formules arithmétiques, la Notation décimale, les conventions de parenthèses, les variables explicites, des énoncés de théorèmes et des structures textuelles de preuves. On finit par étudier quelques applications de la grammaire, basées sur l'implémentation déclarative de la grammaire dans ALF, un éditeur de preuves
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Structures grammaticales dans le français mathématique : 2 (suite et fin)
'OpenEdition', 2006Co-Authors: Ranta AarneAbstract:A system of grammatical rules is presented to analyse a fragment of French that permits the expression of mathematical theorems and proofs. To this end, a version of Montague grammar is developed, with syntactic categories relativized to a context and to domains of individuals. This system can be interpreted in the constructive type theory of Martin-Löf. It is first applied to French without mathematical symbols, paying special attention to selectional restrictions and to dependencies on context. The fragment includes verbs and adjectives, plurals, relative clauses, and coordinated phrases of different categories. Second, the grammar is extended to mathematical symbolism and its embedding in French text. The fragment comprises arithmetical formulae, Decimal Notation, parenthesis conventions, explicit variables, statements of theorems, and textual structures of proofs. Finally, some applications of the grammar are studied, based on a declarative implementation in the proof editor ALF.Un système de règles grammaticales est présenté pour analyser un ragment du français permettant l'expression de théorèmes et de preuves mathéma-tiques. Pour cet objectif, on développe une version de la grammaire de Montague, avec des catégories syntaxiques relatives au contexte et aux domaines d'individus. Ce système peut être interprété dans la théorie constructive des types de Martin-Löf. Il est appliqué, d'abord, au français sans symboles mathématiques, avec une attention spéciale aux restrictions de sélection et aux dépendances par rapport à un contexte. Le fragment comprend des verbes et des adjectifs, des formes plurielles, des propositions relatives, et des syntagmes coordonnés. Ensuite, la grammaire est étendue au symbolisme mathématique et à son usage dans le texte français. Le fragment comprend des formules arithmétiques, la Notation décimale, les conventions de paren-thèses, les variables explicites, des énoncés de théorèmes et des structures textuelles de preuves. On finit par étudier quelques applications de la gram-maire, basées sur l'implémentation déclarative de la grammaire dans ALF, un éditeur de preuves
Vicki Steinle - One of the best experts on this subject based on the ideXlab platform.
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Relative Risk Analysis of Educational Data
2015Co-Authors: Kaye Stacey, Vicki SteinleAbstract:This paper demonstrates the use of relative risk, a statistic which is widely used in other areas but currently under-utilised in education. Relative risk analysis provides a language for comparing educational outcomes as well as statistical tests of significance. We illustrate this statistic with data on students ’ understanding of Decimal Notation. In particular, we determine differences in how misconceptions operate at different ages, by analysing the relative risk of primary and secondary students persisting with particular misconceptions, and becoming experts. This paper illustrates an approach to reporting, comparing and analysing educational outcomes, which is widely used in other fields but which has not been often used in mathematics education. The concept of relative risk is widely used in reporting results of medical, environmental and epidemiological research, in both scientific papers and in the popular press. Educators could capitalise on this popular familiarity in reporting their results. As well as being useful for describing results, measures of relative risk are amenable to statistical analysis. There are simple tests of statistical significance, confidence intervals and effect sizes, which are easily calculated; either manually, with
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A Longitudinal Study of Students ‘Understanding of Decimal Notation: An Overview and Refined Results.
2015Co-Authors: Vicki Steinle, Kaye StaceyAbstract:This paper provides an overview of the major results of a large-scale longitudinal study of students ’ misconceptions of Decimal Notation, drawing them together and presenting refined results. Best estimates of the prevalence of various misconceptions about Decimal numbers from both cross-sectional and longitudinal perspectives are provided, as well as some estimates of persistence. Strengths, limitations and suggestions for improvements to the Decimal Comparison Test as well as major implications for teaching are discussed. This paper aims to provide an overview and discussion of the results of a longitudinal study of students ’ understanding of Decimal Notation, and to present some refinements to previously published results. As far as we know, this is the first study to track the misconceptions of a large number of students longitudinally and hence for the first time to describe the paths that students take through misconceptions on their way to expertise. The study revealed new misconceptions of Decimals, found how common they are across the middle years of schooling, explored which misconceptions were “better ” or “worse ” to have, and showed underlying links between them. A range of associated studies not reported here and using different samples of students, investigated effective teaching
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PERSISTENCE OF Decimal MISCONCEPTIONS AND READINESS TO MOVE TO EXPERTISE
2014Co-Authors: Vicki Steinle, Kaye StaceyAbstract:This paper describes features of a group of misconceptions about Decimal Notation that lead to students selecting as larger, Decimals that look smaller. A longitudinal study identified approximately 900 students from a variety of schools who exhibited these misconceptions and whose subsequent progress could be traced. The data demonstrates that the progress that students makes depends to a certain extent both on the nature of their misconceptions and the grade at which the misconception is held. Such phenomena could be expected to hold for misconceptions in other topics. AIMS This paper aims to describe some features of several interesting misconceptions about Decimal Notation. The results are derived from careful analysis of a longitudinal study of students ’ misconceptions, with data collected in Melbourne, Australia from 199
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ANALYSING LONGITUDINAL DATA ON STUDENTS ’ Decimal UNDERSTANDING USING RELATIVE RISK AND ODDS RATIOS
2014Co-Authors: Vicki Steinle, Kaye StaceyAbstract:The purpose of this paper is to demonstrate the use of the statistics of relative risk and odds ratios in mathematics education. These statistics are widely used in other fields (especially medical research) and offer a useful but currently under-utilised alternative for education. The demonstration uses data from a longitudinal study of students ’ understanding of Decimal Notation. We investigate the statistical significance of results related to the persistence of misconceptions and the hierarchy between misconceptions. Relative risk and odds ratio techniques provide confidence intervals, which give a measure of effect size missing from simple hypothesis testing, and enable differences between phenomena to be assessed and reported with impact. This paper demonstrates some possibilities for analysing educational data, which draw upon methods that are widely used in reporting results of medical, environmental and epidemiological research. We believe that these measures provide very useful techniques for testing for statistical significance and reporting confidence intervals, which will enhance mathematics education research. Capraro (2004) draws attention to important recent policy changes within the American Psychologica
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A case of the inapplicability of the Rasch Model to mapping conceptual learning
2006Co-Authors: Kaye Stacey, Vicki SteinleAbstract:The basic theory of Rasch measurement applies to situations where a person has a certain level of a trait being investigated, and this level of ability is what determines (to within a measurement error) how well the person does on each item in a test. This paper responds to frequent suggestions from col-leagues that the use of Rasch measurement would be profitable in analysing a set of data on students ’ understanding of Decimal Notation. We demon-strate misfit to the Rasch model by showing that item difficulty estimates show important variation by year level, that there is significant deviation from expected score curves, and that success on certain splitter items does not imply a student is more likely to score well on other items. The explana-tion given is that conceptual learning may not always be able to be measured on a scale, which is an essential feature of the Rasch approach. Instead, stu-dents move between categories of interpretations, which do not necessarily provide more correct answers even when they are based on an improved understanding of fundamental principles. In this way, the paper serves t
Ponse Alban - One of the best experts on this subject based on the ideXlab platform.
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Datatype defining rewrite systems for naturals and integers
2020Co-Authors: Bergstra, Jan A., Ponse AlbanAbstract:A datatype defining rewrite system (DDRS) is an algebraic (equational) specification intended to specify a concrete datatype. When interpreting the equations from left-to-right, a DDRS defines a term rewriting system that must be ground-complete. First we define two DDRSs for the ring of integers, each comprising twelve rewrite rules, and prove their ground-completeness. Then we introduce natural number and integer arithmetic specified according to unary view, that is, arithmetic based on a postfix unary append constructor (a form of tallying). Next we specify arithmetic based on two other views: binary and Decimal Notation. The binary and Decimal view have as their characteristic that each normal form resembles common number Notation, that is, either a digit, or a string of digits without leading zero, or the negated versions of the latter. Integer arithmetic in binary and Decimal Notation is based on (postfix) digit append functions. For each view we define a DDRS, and in each case the resulting datatype is a canonical term algebra that extends a corresponding canonical term algebra for natural numbers. Then, for each view, we consider an alternative DDRS based on tree constructors that yields comparable normal forms, which for that view admits expressions that are algorithmically more involved. For all DDRSs considered, ground-completeness is proven.Comment: 31 pages; 14 tables. Changes with respect to arXiv:1608.06212v2: we left out Section 3.3; the DDRSs in Tables 8, 9, 11, 12 and 13 are simplified; all termination proofs were found by the tool Aprove, and some (gc-)confluence proofs were found by the tools CSI and AGCP. arXiv admin note: text overlap with arXiv:1406.328
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Three Datatype Defining Rewrite Systems for Datatypes of Integers each extending a Datatype of Naturals
2016Co-Authors: Bergstra, Jan A., Ponse AlbanAbstract:Integer arithmetic is specified according to three views: unary, binary, and Decimal Notation. The binary and Decimal view have as their characteristic that each normal form resembles common number Notation, that is, either a digit, or a string of digits without leading zero, or the negated versions of the latter. The unary view comprises a specification of integer arithmetic based on 0, successor function $S$, and predecessor function, with negative normal forms $-S^i(0)$. Integer arithmetic in binary and Decimal Notation is based on (postfix) digit append functions. For each view we define a ground-confluent and terminating datatype defining rewrite system (DDRS), and in each case the resulting datatype is a canonical term algebra that extends a corresponding canonical term algebra for natural numbers. Then, for each view, we consider an alternative DDRS based on tree constructors that yield comparable normal forms, which for that binary and Decimal view admits expressions that are algorithmically more involved. These DDRSes are incorporated because they are closer to existing literature. For these DDRSes we also provide ground-completeness results. Finally, we define a DDRS for the ring of Integers (comprising fifteen rewrite rules) and prove its ground-completeness.Comment: 33 pages; 19 tables. All DDRSes in S.2 are proven ground-complete (gc). In S.3, the DDRS for Z_{ut} contains 16 equations and is proven gc; the DDRS for Z_{bt} has one more equation ([bt22]) and is proven gc; the DDRSes for N_{dt} (Table 14) and Z_{dt} (Table 16) are proven gc in [13]. In Appendix C, corrected versions of the DDRSes for N_{u'} and Z_{u'} are proven g
Green Rebecca - One of the best experts on this subject based on the ideXlab platform.
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Melvil Deweyâ s Ingenious Notational System Item type Conference Paper
2016Co-Authors: Green RebeccaAbstract:Historically, the Notational system of the Dewey Decimal Classification provided for non-institution-specific, relative location shelf arrangements, thus substantially reducing bibliographic classification effort. Today its Decimal Notation continues to provide the classification scheme with flexible granularity, is hospitable to expansion, expresses relationships, interfaces well with modern retrieval systems, and is internationally understood.
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Melvil Dewey’s Ingenious Notational System
'University of Washington Libraries', 2011Co-Authors: Green RebeccaAbstract:Historically, the Notational system of the Dewey Decimal Classification provided for non-institution-specific, relative location shelf arrangements, thus substantially reducing bibliographic classification effort. Today its Decimal Notation continues to provide the classification scheme with flexible granularity, is hospitable to expansion, expresses relationships, interfaces well with modern retrieval systems, and is internationally understood
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Melvil Deweyâ s Ingenious Notational System
2009Co-Authors: Green RebeccaAbstract:Historically, the Notational system of the Dewey Decimal Classification provided for non-institution-specific, relative location shelf arrangements, thus substantially reducing bibliographic classification effort. Today its Decimal Notation continues to provide the classification scheme with flexible granularity, is hospitable to expansion, expresses relationships, interfaces well with modern retrieval systems, and is internationally understood
