Developmental predictors of fraction concepts and procedures
TL;DR: This article investigated the developmental predictors of children's fraction concepts and procedures at the end of fourth grade in a 2-year longitudinal study and found that number line estimation, attentive behavior, calculation fluency, and working memory made unique contributions to the acquisition of fraction arithmetic procedures.
About: This article is published in Journal of Experimental Child Psychology.The article was published on 2013-09-01. It has received 224 citations till now. The article focuses on the topics: Fluency & Number line.
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TL;DR: A meta-analysis of cross-sectional and longitudinal studies revealed a moderate but statistically significant association between number acuity and math performance, and suggested that many previous studies were underpowered due to small sample sizes.
340 citations
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TL;DR: The conclusion is that there is not enough convincing evidence to support the number sense theory anymore, and researchers are encouraged not to assume that number sense is simply innate, but to put this hypothesis to the test and consider whether such an assumption is even testable in the light of the correlation of numerosity and continuous magnitudes.
Abstract: In this review, we are pitting two theories against each other: the more accepted theory, the number sense theory, suggesting that a sense of number is innate and non-symbolic numerosity is being processed independently of continuous magnitudes (e.g., size, area, and density); and the newly emerging theory suggesting that (1) both numerosities and continuous magnitudes are processed holistically when comparing numerosities and (2) a sense of number might not be innate. In the first part of this review, we discuss the number sense theory. Against this background, we demonstrate how the natural correlation between numerosities and continuous magnitudes makes it nearly impossible to study non-symbolic numerosity processing in isolation from continuous magnitudes, and therefore, the results of behavioral and imaging studies with infants, adults, and animals can be explained, at least in part, by relying on continuous magnitudes. In the second part, we explain the sense of magnitude theory and review studies that directly demonstrate that continuous magnitudes are more automatic and basic than numerosities. Finally, we present outstanding questions. Our conclusion is that there is not enough convincing evidence to support the number sense theory anymore. Therefore, we encourage researchers not to assume that number sense is simply innate, but to put this hypothesis to the test and consider whether such an assumption is even testable in the light of the correlation of numerosity and continuous magnitudes.
316 citations
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TL;DR: A meta-analysis of 110 studies with 829 effect sizes found a significant medium correlation of mathematics and WM, r =.35, 95% confidence interval [32,.37] as discussed by the authors.
Abstract: The purpose of this meta-analysis was to determine the relation between mathematics and working memory (WM) and to identify possible moderators of this relation including domains of WM, types of mathematics skills, and sample type. A meta-analysis of 110 studies with 829 effect sizes found a significant medium correlation of mathematics and WM, r = .35, 95% confidence interval [.32, .37]. Moderation analyses indicated that mathematics showed comparable association with verbal WM, numerical WM, and visuospatial WM. Word-problem solving and whole-number calculations showed the strongest relation with WM whereas geometry showed the weakest relation with WM. The relation between WM and mathematics was stronger among individuals with mathematics difficulties that are associated with other disorders or cognitive deficits compared with that among typically developing individuals and individuals with only mathematics difficulties. The implications of these findings with respect to mathematics instruction and WM training are discussed. (PsycINFO Database Record (c) 2016 APA, all rights reserved)
304 citations
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TL;DR: Analysis of longitudinal data indicated that whole number magnitude knowledge in first grade predicted knowledge of fraction magnitudes in middle school, controlling for whole number arithmetic proficiency, domain general cognitive abilities, parental income and education, race, and gender.
Abstract: Recent findings that earlier fraction knowledge predicts later mathematics achievement raise the question of what predicts later fraction knowledge. Analyses of longitudinal data indicated that whole number magnitude knowledge in first grade predicted knowledge of fraction magnitudes in middle school, controlling for whole number arithmetic proficiency, domain general cognitive abilities, parental income and education, race, and gender. Similarly, knowledge of whole number arithmetic in first grade predicted knowledge of fraction arithmetic in middle school, controlling for whole number magnitude knowledge in first grade and the other control variables. In contrast, neither type of early whole number knowledge uniquely predicted middle school reading achievement. We discuss the implications of these findings for theories of numerical development and for improving mathematics learning.
163 citations
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TL;DR: This review identifies and discusses two types of difficulties: inherent difficulties of fraction and decimal arithmetic and culturally contingent difficulties that could be reduced by improved instruction and prior knowledge of learners.
151 citations
References
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01 Jan 1983
TL;DR: In this Section: 1. Multivariate Statistics: Why? and 2. A Guide to Statistical Techniques: Using the Book Research Questions and Associated Techniques.
Abstract: In this Section: 1. Brief Table of Contents 2. Full Table of Contents 1. BRIEF TABLE OF CONTENTS Chapter 1 Introduction Chapter 2 A Guide to Statistical Techniques: Using the Book Chapter 3 Review of Univariate and Bivariate Statistics Chapter 4 Cleaning Up Your Act: Screening Data Prior to Analysis Chapter 5 Multiple Regression Chapter 6 Analysis of Covariance Chapter 7 Multivariate Analysis of Variance and Covariance Chapter 8 Profile Analysis: The Multivariate Approach to Repeated Measures Chapter 9 Discriminant Analysis Chapter 10 Logistic Regression Chapter 11 Survival/Failure Analysis Chapter 12 Canonical Correlation Chapter 13 Principal Components and Factor Analysis Chapter 14 Structural Equation Modeling Chapter 15 Multilevel Linear Modeling Chapter 16 Multiway Frequency Analysis 2. FULL TABLE OF CONTENTS Chapter 1: Introduction Multivariate Statistics: Why? Some Useful Definitions Linear Combinations of Variables Number and Nature of Variables to Include Statistical Power Data Appropriate for Multivariate Statistics Organization of the Book Chapter 2: A Guide to Statistical Techniques: Using the Book Research Questions and Associated Techniques Some Further Comparisons A Decision Tree Technique Chapters Preliminary Check of the Data Chapter 3: Review of Univariate and Bivariate Statistics Hypothesis Testing Analysis of Variance Parameter Estimation Effect Size Bivariate Statistics: Correlation and Regression. Chi-Square Analysis Chapter 4: Cleaning Up Your Act: Screening Data Prior to Analysis Important Issues in Data Screening Complete Examples of Data Screening Chapter 5: Multiple Regression General Purpose and Description Kinds of Research Questions Limitations to Regression Analyses Fundamental Equations for Multiple Regression Major Types of Multiple Regression Some Important Issues. Complete Examples of Regression Analysis Comparison of Programs Chapter 6: Analysis of Covariance General Purpose and Description Kinds of Research Questions Limitations to Analysis of Covariance Fundamental Equations for Analysis of Covariance Some Important Issues Complete Example of Analysis of Covariance Comparison of Programs Chapter 7: Multivariate Analysis of Variance and Covariance General Purpose and Description Kinds of Research Questions Limitations to Multivariate Analysis of Variance and Covariance Fundamental Equations for Multivariate Analysis of Variance and Covariance Some Important Issues Complete Examples of Multivariate Analysis of Variance and Covariance Comparison of Programs Chapter 8: Profile Analysis: The Multivariate Approach to Repeated Measures General Purpose and Description Kinds of Research Questions Limitations to Profile Analysis Fundamental Equations for Profile Analysis Some Important Issues Complete Examples of Profile Analysis Comparison of Programs Chapter 9: Discriminant Analysis General Purpose and Description Kinds of Research Questions Limitations to Discriminant Analysis Fundamental Equations for Discriminant Analysis Types of Discriminant Analysis Some Important Issues Comparison of Programs Chapter 10: Logistic Regression General Purpose and Description Kinds of Research Questions Limitations to Logistic Regression Analysis Fundamental Equations for Logistic Regression Types of Logistic Regression Some Important Issues Complete Examples of Logistic Regression Comparison of Programs Chapter 11: Survival/Failure Analysis General Purpose and Description Kinds of Research Questions Limitations to Survival Analysis Fundamental Equations for Survival Analysis Types of Survival Analysis Some Important Issues Complete Example of Survival Analysis Comparison of Programs Chapter 12: Canonical Correlation General Purpose and Description Kinds of Research Questions Limitations Fundamental Equations for Canonical Correlation Some Important Issues Complete Example of Canonical Correlation Comparison of Programs Chapter 13: Principal Components and Factor Analysis General Purpose and Description Kinds of Research Questions Limitations Fundamental Equations for Factor Analysis Major Types of Factor Analysis Some Important Issues Complete Example of FA Comparison of Programs Chapter 14: Structural Equation Modeling General Purpose and Description Kinds of Research Questions Limitations to Structural Equation Modeling Fundamental Equations for Structural Equations Modeling Some Important Issues Complete Examples of Structural Equation Modeling Analysis. Comparison of Programs Chapter 15: Multilevel Linear Modeling General Purpose and Description Kinds of Research Questions Limitations to Multilevel Linear Modeling Fundamental Equations Types of MLM Some Important Issues Complete Example of MLM Comparison of Programs Chapter 16: Multiway Frequency Analysis General Purpose and Description Kinds of Research Questions Limitations to Multiway Frequency Analysis Fundamental Equations for Multiway Frequency Analysis Some Important Issues Complete Example of Multiway Frequency Analysis Comparison of Programs
53,113 citations
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01 Jan 1975TL;DR: In this article, the Mathematical Basis for Multiple Regression/Correlation and Identification of the Inverse Matrix Elements is presented. But it does not address the problem of missing data.
Abstract: Contents: Preface. Introduction. Bivariate Correlation and Regression. Multiple Regression/Correlation With Two or More Independent Variables. Data Visualization, Exploration, and Assumption Checking: Diagnosing and Solving Regression Problems I. Data-Analytic Strategies Using Multiple Regression/Correlation. Quantitative Scales, Curvilinear Relationships, and Transformations. Interactions Among Continuous Variables. Categorical or Nominal Independent Variables. Interactions With Categorical Variables. Outliers and Multicollinearity: Diagnosing and Solving Regression Problems II. Missing Data. Multiple Regression/Correlation and Causal Models. Alternative Regression Models: Logistic, Poisson Regression, and the Generalized Linear Model. Random Coefficient Regression and Multilevel Models. Longitudinal Regression Methods. Multiple Dependent Variables: Set Correlation. Appendices: The Mathematical Basis for Multiple Regression/Correlation and Identification of the Inverse Matrix Elements. Determination of the Inverse Matrix and Applications Thereof.
29,764 citations
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01 Jan 1986TL;DR: In this article, the authors focus on a conceptual understanding of the material rather than proving results and stress the importance of checking the data, assessing the assumptions, and ensuring adequate sample size so that the results can be generalized.
Abstract: This best-selling text is written for those who use, rather than develop, advanced statistical methods. Dr. Stevens focuses on a conceptual understanding of the material rather than proving results. Helpful narrative and numerous examples enhance understanding, and a chapter on matrix algebra serves as a review. Printouts from SPSS and SAS with annotations indicate what the numbers mean and encourage interpretation of the results. In addition to demonstrating how to use the packages effectively, the author stresses the importance of checking the data, assessing the assumptions, and ensuring adequate sample size (by providing guidelines) so that the results can be generalized. The new edition features a CD-ROM with the data sets and many new exercises. Ideal for courses on advanced or multivariate statistics found in psychology, education, and business departments, the book also appeals to practicing researchers with little or no training in multivariate methods. Prerequisites include a course on factorial analysis of variance. It does not assume a working knowledge of matrix algebra.
10,221 citations
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07 Feb 20146,430 citations