Development of Fraction Comparison Strategies: A Latent Transition Analysis

The present study investigated the development of fraction comparison strategies through a longitudinal analysis of children’s responses to a fraction comparison task in 4th through 6th grades (N = 394). Participants were asked to choose the larger value for 24 fraction pairs blocked by fraction type. Latent class analysis of performance over item blocks showed that most children initially exhibited a “whole number bias,” indicating that larger numbers in numerators and denominators produce larger fraction values. However, some children instead chose fractions with smaller numerators and denominators, demonstrating a partial understanding that smaller numbers can yield larger fractions. Latent transition analysis showed that most children eventually adopted normative comparison strategies. Children who exhibited a partial understanding by choosing fractions with smaller numbers were more likely to adopt normative comparison strategies earlier than those with larger number biases. Controlling for general math achievement and other cognitive abilities, whole number line estimation accuracy predicted the probability of transitioning to normative comparison strategies. Exploratory factor analyses showed that over time, children appeared to increasingly represent fractions as discrete magnitudes when simpler strategies were unavailable. These results support the integrated theory of numerical development, which posits that an understanding of numbers as magnitudes unifies the process of learning whole numbers and fractions. The findings contrast with conceptual change theories, which propose that children must move from a view of numbers as counting units to a new view that accommodates fractions to overcome whole number bias.

[1]  Nancy K. Mack.  Confounding whole-number and fraction concepts when building on informal knowledge. , 1995 .

[2]  D. Zabel,et al.  Diagnostic and Statistical Manual of Mental Disorders, 4th ed , 1995 .

[3]  Clarissa A. Thompson,et al.  An integrated theory of whole number and fractions development , 2011, Cognitive Psychology.

[4]  R. Siegler.  Emerging Minds: The Process of Change in Children's Thinking , 1996 .

[5]  Neil Henry.  Latent structure analysis , 1969 .

[6]  Lieven Verschaffel,et al.  Inappropriately applying natural number properties in rational number tasks: characterizing the development of the natural number bias through primary and secondary education , 2015 .

[7]  Gary J. Robertson,et al.  Wide‐Range Achievement Test , 2010 .

[8]  Marie-Pascale Noël,et al.  Rational numbers: Componential versus holistic representation of fractions in a magnitude comparison task , 2009, Quarterly journal of experimental psychology.

[9]  Erno Lehtinen,et al.  Modeling the developmental trajectories of rational number concept(s) , 2015 .

[10]  Stella Vosniadou,et al.  The representation of fraction magnitudes and the whole number bias reconsidered. , 2015 .

[11]  Young-Ok Kim.  미국 Common Core State Standards for Mathematics 소개 , 2011 .

[12]  Paul F. Lazarsfeld,et al.  Latent Structure Analysis. , 1969 .

[13]  Stella Vosniadou,et al.  The Whole Number Bias in Fraction Magnitude Comparisons with Adults , 2011, CogSci.

[14]  Linda M. Collins,et al.  Latent class and latent transition analysis , 2009 .

[15]  L. Angeles.  Latent Transition Analysis: Modeling Extensions and an Application to Peer Victimization , 2007 .

[16]  Justin Halberda,et al.  Developmental change in the acuity of the "Number Sense": The Approximate Number System in 3-, 4-, 5-, and 6-year-olds and adults. , 2008, Developmental psychology.

[17]  Andreas Obersteiner,et al.  Measuring fraction comparison strategies with eye-tracking , 2016 .

[18]  Percival G. Matthews,et al.  Neurocognitive Architectures and the Nonsymbolic Foundations of Fractions Understanding , 2016 .

[19]  Yujing Ni,et al.  Teaching and Learning Fraction and Rational Numbers: The Origins and Implications of Whole Number Bias , 2005 .

[20]  S. Dehaene,et al.  The mental representation of parity and number magnitude. , 1993 .

[21]  M. Alibali,et al.  Variability in the natural number bias: Who, when, how, and why , 2015 .

[22]  R. Spitzer.  Dsm-IV Casebook: A Learning Companion to the Diagnostic and Statistical Manual of Mental Disorders, Fourth Edition , 1994 .

[23]  Marco Zorzi,et al.  The mental representation of numerical fractions: real or integer? , 2007, Journal of experimental psychology. Human perception and performance.

[24]  Mary K. Hoard,et al.  Competence with fractions predicts gains in mathematics achievement. , 2012, Journal of experimental child psychology.

[25]  Megan E. Patrick,et al.  Latent Transition Analysis: Benefits of a Latent Variable Approach to Modeling Transitions in Substance Use , 2010, Journal of drug issues.

[26]  R. Siegler,et al.  Early Predictors of High School Mathematics Achievement , 2012, Psychological science.

[27]  R. Jennrich.  Admissible values of γ in direct oblimin rotation , 1979 .

[28]  Andreas Obersteiner,et al.  The natural number bias and magnitude representation in fraction comparison by expert mathematicians , 2013 .

[29]  Justin Halberda,et al.  Individual differences in non-verbal number acuity correlate with maths achievement , 2008, Nature.

[30]  Percival G. Matthews,et al.  Individual Differences in Nonsymbolic Ratio Processing Predict Symbolic Math Performance , 2016, Psychological science.

[31]  M. Mazzocco,et al.  Parts and 'holes': gaps in rational number sense among children with vs. without mathematical learning disabilities. , 2008, Developmental science.

[32]  D. Wechsler.  Wechsler Abbreviated Scale of Intelligence--Second Edition , 2018 .

[33]  ROBERT S. MOYER,et al.  Time required for Judgements of Numerical Inequality , 1967, Nature.

[34]  Ilyse Resnick,et al.  Developmental growth trajectories in understanding of fraction magnitude from fourth through sixth grade. , 2016, Developmental psychology.

[35]  Kimberley D. Lakes,et al.  Categorical and Dimensional Definitions and Evaluations of Symptoms of ADHD: History of the SNAP and the SWAN Rating Scales. , 2012, The International journal of educational and psychological assessment.

[36]  Kristopher J Preacher,et al.  On the practice of dichotomization of quantitative variables. , 2002, Psychological methods.

[37]  Robert S. Siegler,et al.  Bridging the Gap: Fraction Understanding Is Central to Mathematics Achievement in Students from Three Different Continents. , 2015 .

[38]  S. Vosniadou,et al.  The development of students’ understanding of the numerical value of fractions , 2004 .

[39]  C. A. Rashotte,et al.  Test of Word Reading Efficiency , 1999 .

[40]  Lisa K. Fazio,et al.  Strategy use and strategy choice in fraction magnitude comparison. , 2016, Journal of experimental psychology. Learning, memory, and cognition.

[41]  Aryn A Pyke,et al.  Developmental and individual differences in understanding of fractions. , 2013, Developmental psychology.

[42]  B. Muthén,et al.  Deciding on the Number of Classes in Latent Class Analysis and Growth Mixture Modeling: A Monte Carlo Simulation Study , 2007 .

[43]  Russell Gersten,et al.  Developmental predictors of fraction concepts and procedures. , 2013, Journal of experimental child psychology.

[44]  B. Thompson,et al.  Factor Analytic Evidence for the Construct Validity of Scores: A Historical Overview and Some Guidelines , 1996 .

[45]  Bengt Muthén,et al.  Auxiliary Variables in Mixture Modeling: Three-Step Approaches Using Mplus , 2014 .

[46]  R. Cattell.  The Scree Test For The Number Of Factors. , 1966, Multivariate behavioral research.

[47]  Karen Nylund-Gibson,et al.  A Latent Transition Mixture Model Using the Three-Step Specification , 2014 .

[48]  Jeroen K. Vermunt,et al.  Latent Class Modeling with Covariates: Two Improved Three-Step Approaches , 2010, Political Analysis.

[49]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[50]  Michael Schneider,et al.  Representations of the magnitudes of fractions. , 2010, Journal of experimental psychology. Human perception and performance.

[51]  Marie-Pascale Noël,et al.  Comparing 5/7 and 2/9: Adults can do it by accessing the magnitude of the whole fractions. , 2010, Acta psychologica.

[52]  L. A. Goodman.  Exploratory latent structure analysis using both identifiable and unidentifiable models , 1974 .

[53]  Linda M. Collins,et al.  Applied Latent Class Analysis: Latent Class Models for Longitudinal Data , 2002 .