Objectifying early-number : a visual nomenclature to express mathematical domain knowledge

To address issues of divisive ideologies in the Mathematics Education community and to subsequently advance educational practice, an alternative theoretical framework and operational model is proposed which represents a consilience of constructivist learning theories whilst acknowledging the objective but improvable nature of domain knowledge. Based upon Popper’s three-world model of knowledge, the proposed theory supports the differentiation and explicit modelling of both shared domain knowledge and idiosyncratic personal understanding using a visual nomenclature. The visual nomenclature embodies Piaget’s notion of reflective abstraction and so may support an individual’s experience-based transformation of personal understanding with regards to shared domain knowledge. Using the operational model and visual nomenclature, seminal literature regarding early-number counting and addition was analysed and described. Exemplars of the resultant visual artefacts demonstrate the proposed theory’s viability as a tool with which to characterise the reflective abstraction-based organisation of a domain’s shared knowledge. Utilising such a description of knowledge, future research needs to consider the refinement of the operational model and visual nomenclature to include the analysis, description and scaffolded transformation of personal understanding. A detailed model of knowledge and understanding may then underpin the future development of educational software tools such as computer-mediated teaching and learning environments.

A visual description of early-number counting, addition and subtraction knowledge

Early-number is a rich fabric of interconnected ideas that is often misunderstood and thus taught in ways that do not lead to rich understanding. In this presentation, a visual language is used to describe the organisation of this domain of knowledge. This visual language is based upon Piaget’s notion of reflective abstraction (Dubinsky, 1991; Piaget, 1977/2001), and thus captures the epistemological associations that link the problems, concepts and representations of the domain. The constructs of this visual language are introduced and then applied to the early-number domain. The introduction to this visual language may prompt reflection upon its suitability and significance to the description of other domains of knowledge. Through such a process of analysis and description, the visual language may serve as a scaffold for enhancing pedagogical content knowledge and thus ultimately improve learning outcomes.

Teaching and learning early number.

Libro orientado a profesores en formación, en prácticas y ayudantes de aula en educación infantil y primaria para la enseñanza de los primeros conceptos sobre números y matemáticas. Incluye nuevos estudios sobre el proceso de cálculo y sobre las notas de los niños en matemáticas, actividades de cálculo en casa y como juego en el entorno escolar, la importancia de las representaciones matemáticas y de las tecnologías de la información y la comunicación (TIC) en la compresión de los números por los niños, errores y falsas ideas y la evaluación del conocimiento numérico de los niños.

The identification of teaching interactions used in one-to-one teaching of number in the early years of schooling

This research paper reports on phase one of an investigation of video recorded intensive one-to-one teaching interactions with 6–7-year-old students who were in their second year of schooling in Australia and identified by the their teacher as low attaining in early number. The two-phased study from which this paper emerges was originally conducted in 1998 as part of my Bachelor of Teaching Honours (Research) program at Southern Cross University Lismore, New South Wales. That study identified teaching interactions particularly suited to one-to-one teaching in the Maths Recovery Program, a program designed for these students who were at risk of failure in early number. Since that time a great deal has not changed with limited literature available that comprehensively reports on teaching interactions in intensive one-to-one settings. Revisiting the original study is considered timely given the increasing number of withdrawal and intensive programs now funded and adopted by schools and yet, rarely reported on in terms of the effectiveness of the teaching interactions that occur in such settings. This paper then presents a discussion of a preliminary series of teaching interactions that either positively and or negatively influence an intensive one-to-one teaching and learning setting

Towards a model for the analysis and description of mathematical knowledge and understanding

Mathematics education literature has called for an abandonment of ontological and epistemological ideologies that have often divided theory-based practice. Instead, a consilience of theories has been sought which would leverage the strengths of each learning theory and so positively impact upon contemporary educational practice. This research activity is based upon Popper’s notion of three knowledge worlds which differentiates the knowledge shared in a community from the personal knowledge of the individual, and Bereiter’s characterisation of understanding as the individual’s relationship to tool-like knowledge. Using these notions, a re-conceptualisation of knowledge and understanding and a subsequent re-consideration of learning theories are proposed as a way to address the challenge set by literature. Referred to as the alternative theoretical framework, the proposed theory accounts for the scaffolded transformation of each individual’s unique understanding, whilst acknowledging the existence of a body of domain knowledge shared amongst participants in a scientific community of practice. The alternative theoretical framework is embodied within an operational model that is accompanied by a visual nomenclature with which to describe consensually developed shared knowledge and personal understanding. This research activity has sought to iteratively evaluate this proposed theory through the practical application of the operational model and visual nomenclature to the domain of early-number counting, addition and subtraction. This domain of mathematical knowledge has been comprehensively analysed and described. Through this process, the viability of the proposed theory as a tool with which to discuss and thus improve the knowledge and understanding with the domain of mathematics has been validated. Putting of the proposed theory into practice has lead to the theory’s refinement and the subsequent achievement of a solid theoretical base for the future development of educational tools to support teaching and learning practice, including computer-mediated learning environments. Such future activity, using the proposed theory, will advance contemporary mathematics educational practice by bringing together the strengths of cognitivist, constructivist and post-constructivist learning theories.

A Popperian consilience : modelling mathematical knowledge and understanding

Goldin (2003) and McDonald, Yanchar, and Osguthorpe (2005) have called for mathematics learning theory that reconciles the chasm between ideologies, and which may advance mathematics teaching and learning practice. This paper discusses the theoretical underpinnings of a recently completed PhD study that draws upon Popper’s (1978) three-world model of knowledge as a lens through which to reconsider a variety of learning theories, including Piaget’s reflective abstraction. Based upon this consideration of theories, an alternative theoretical framework and complementary operational model was synthesised, the viability of which was demonstrated by its use to analyse the domain of early-number counting, addition and subtraction.

Early mathematical learning: Number processing skills and executive function at 5 and 8 years of age

This research investigated differences and associations in performance in number processing and executive function for children attending primary school in a large Australian metropolitan city. In a cross-sectional study, performance of 25 children in the first full-time year of school, (Prep; mean age = 5.5 years) and 21 children in Year 3 (mean age = 8.5 years) completed three number processing tasks and three executive function tasks. Year 3 children consistently outperformed the Prep year children on measures of accuracy and reaction time, on the tasks of number comparison, calculation, shifting, and inhibition but not on number line estimation. The components of executive function (shifting, inhibition, and working memory) showed different patterns of correlation to performance on number processing tasks across the early years of school. Findings could be used to enhance teachers’ understanding about the role of the cognitive processes employed by children in numeracy learning, and so inform teachers’ classroom practices.

Genome-wide association of early-onset myocardial infarction with single nucleotide polymorphisms and copy number variants

We conducted a genome-wide association study testing single nucleotide polymorphisms (SNPs) and copy number variants (CNVs) for association with early-onset myocardial infarction in 2,967 cases and 3,075 controls. We carried out replication in an independent sample with an effective sample size of up to 19,492. SNPs at nine loci reached genome-wide significance: three are newly identified (21q22 near MRPS6-SLC5A3-KCNE2, 6p24 in PHACTR1 and 2q33 in WDR12) and six replicated prior observations1-4 (9p21, 1p13 near CELSR2-PSRC1-SORT1, 10q11 near CXCL12, 1q41 in MIA3, 19p13 near LDLR and 1p32 near PCSK9). We tested 554 common copy number polymorphisms (>1% allele frequency) and none met the pre-specified threshold for replication (P < 10-3). We identified 8,065 rare CNVs but did not detect a greater CNV burden in cases compared to controls, in genes compared to the genome as a whole, or at any individual locus. SNPs at nine loci were reproducibly associated with myocardial infarction, but tests of common and rare CNVs failed to identify additional associations with myocardial infarction risk.

The acquisition of person and number morphology within the verbal domain in Early Greek

The aim of this paper is to examine the acquisition pattern of person and number verb morphology within the generative framework and to compare the results of the analyses with previous research in Greek and other European languages. The study considers previous data on the acquisition of subject-verb agreement, and thereafter, examines the acquisition of person and number morphology in a new dataset of two monolingual Greek-speaking children. The analyses present quantitative data of accuracy of person and number marking, error data, and qualitative analyses addressing the productivity of person and number marking. The results suggest that person and number morphology is used correctly and productively from a very early age in Greek speaking children. The findings provide new insight into early Greek language acquisition and are also relevant for research in early development of languages with rich inflectional morphology.

Germline DNA copy number variation in familial and early-onset breast cancer

Introduction: Genetic factors predisposing individuals to cancer remain elusive in the majority of patients with a familial or clinical history suggestive of hereditary breast cancer. Germline DNA copy number variation (CNV) has recently been implicated in predisposition to cancers such as neuroblastomas as well as prostate and colorectal cancer. We evaluated the role of germline CNVs in breast cancer susceptibility, in particular those with low population frequencies (rare CNVs), which are more likely to cause disease." Methods: Using whole-genome comparative genomic hybridization on microarrays, we screened a cohort of women fulfilling criteria for hereditary breast cancer who did not carry BRCA1/BRCA2 mutations. Results: The median numbers of total and rare CNVs per genome were not different between controls and patients. A total of 26 rare germline CNVs were identified in 68 cancer patients, however, a proportion that was significantly different (P = 0.0311) from the control group (23 rare CNVs in 100 individuals). Several of the genes affected by CNV in patients and controls had already been implicated in cancer. Conclusions: This study is the first to explore the contribution of germline CNVs to BRCA1/2-negative familial and early-onset breast cancer. The data suggest that rare CNVs may contribute to cancer predisposition in this small cohort of patients, and this trend needs to be confirmed in larger population samples.

Time courses and quantitative analysis of atrial fibrillation episode number and duration after circular plus linear left atrial lesions: trigger elimination or substrate modification: early or delayed cure?

OBJECTIVES We sought to analyze the time course of atrial fibrillation (AF) episodes before and after circular plus linear left atrial ablation and the percentage of patients with complete freedom from AF after ablation by using serial seven-day electrocardiograms (ECGs). BACKGROUND The curative treatment of AF targets the pathophysiological corner stones of AF (i.e., the initiating triggers and/or the perpetuation of AF). The pathophysiological complexity of both may not result in an "all-or-nothing" response but may modify number and duration of AF episodes. METHODS In patients with highly symptomatic AF, circular plus linear ablation lesions were placed around the left and right pulmonary veins, between the two circles, and from the left circle to the mitral annulus using the electroanatomic mapping system. Repetitive continuous 7-day ECGs administered before and after catheter ablation were used for rhythm follow-up. RESULTS In 100 patients with paroxysmal (n = 80) and persistent (n = 20) AF, relative duration of time spent in AF significantly decreased over time (35 +/- 37% before ablation, 26 +/- 41% directly after ablation, and 10 +/- 22% after 12 months). Freedom from AF stepwise increased in patients with paroxysmal AF and after 12 months measured at 88% or 74% depending on whether 24-h ECG or 7-day ECG was used. Complete pulmonary vein isolation was demonstrated in <20% of the circular lesions. CONCLUSIONS The results obtained in patients with AF treated with circular plus linear left atrial lesions strongly indicate that substrate modification is the main underlying pathophysiologic mechanism and that it results in a delayed cure instead of an immediate cure.

Second history of Charlestown, N.H., the Old Number Four; embracing a summary of the early history of the town from its settlement to 1876 and more particularly developments and events in the town from 1876 to 1954.

Bibliography: p. 430-433.

History of Kanawha County : from its organization in 1789 until the present time : embracing accounts of early settlements, and thrilling adventures with the Indians derived from history and aged citizens ; also, biographical sketches of a large number of the early settlers of the Great Kanawha Valley /

Illustrated by numerous engravings.

Annals of Annapolis : comprising sundry notices of that old city from the period of the first settlements in its vicinity in the year 1649, until the war of 1812 : together with various incidents in the history of Maryland, derived from early records, public documents, and other sources ; with an appendix, containing a number of letters from General Washington, and other distinguished persons, which letters have never been published before /

Sabin

A collection of old paintings by English and continental masters from London and other sources with an unusual number of early American portraits.

Mode of access: Internet.