The natural number bias and magnitude representation in fraction comparison by expert mathematicians

Abstract When school students compare the numerical values of fractions, they have frequently been found to be biased by the natural numbers involved (e.g., to believe that 1/4 > 1/3 because 4 > 3), thereby considering fractions componentially as two natural numbers rather than holistically as one number. Adult studies have suggested that intuitive processes could be the source of this bias, but also that adults are able to activate holistic rather than componential mental representations of fractions under some circumstances. We studied expert mathematicians on various types of fraction comparison problems to gain further evidence for the intuitive character of the bias, and to test how the mental representations depend on the type of comparison problems. We found that experts still show a tendency to be biased by natural numbers and do not activate holistic representations when fraction pairs have common numerators or denominators. With fraction pairs without common components, we found no natural number bias, and holistic representations were more likely. We discuss both findings in relation to each other, and point out implications for mathematics education.

[1]  I. Putt,et al.  Preservice Teachers Ordering of Decimal Numbers: When More Is Smaller and Less Is Larger!. , 1995 .

[2]  S. Vosniadou,et al.  Understanding the structure of the set of rational numbers: a conceptual change approach , 2004 .

[3]  Margarete Delazer,et al.  The processing and representation of fractions within the brain An fMRI investigation , 2009, NeuroImage.

[4]  Arava Y. Kallai,et al.  When meaningful components interrupt the processing of the whole: the case of fractions. , 2012, Acta psychologica.

[5]  Arava Y. Kallai,et al.  A generalized fraction: an entity smaller than one on the mental number line. , 2009, Journal of experimental psychology. Human perception and performance.

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

[7]  B. Greer.  Helping Children Develop Mathematically , 2009, Human Development.

[8]  Daniel B. Berch,et al.  Making Sense of Number Sense , 2005, Journal of learning disabilities.

[9]  Brian Butterworth,et al.  Rapid Communication: Understanding the Real Value of Fractions and Decimals , 2011, Quarterly journal of experimental psychology.

[10]  S. Dehaene,et al.  Interactions between number and space in parietal cortex , 2005, Nature Reviews Neuroscience.

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

[12]  Richard Lesh,et al.  ORDER AND EQUIVALENCE OF RATIONAL NUMBERS: A CLINICAL TEACHING EXPERIMENT , 1984 .

[13]  J. Tzelgov,et al.  Holistic representation of unit fractions. , 2011, Experimental psychology.

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

[15]  Annette R. Baturo,et al.  Preservice Teachers' Knowledge of Difficulties in Decimal Numeration , 2001 .

[16]  Robert S. Siegler,et al.  Fractions: the new frontier for theories of numerical development , 2013, Trends in Cognitive Sciences.

[17]  Thomas J. Faulkenberry,et al.  Mental representations in fraction comparison. , 2011, Experimental psychology.

[18]  Usha Goswami,et al.  Educational Neuroscience: Defining a New Discipline for the Study of Mental Representations , 2007 .

[19]  Marie-Pascale Noël,et al.  Comparing the magnitude of two fractions with common components: which representations are used by 10- and 12-year-olds? , 2010, Journal of experimental child psychology.

[20]  Robert Sekuler,et al.  Children's Judgments of Numerical Inequality. , 1977 .

[21]  Dana Ganor-Stern.  Fractions but not negative numbers are represented on the mental number line. , 2012, Acta psychologica.

[22]  M. Schocke,et al.  The processing and representation of fractions within the brain. An fMRI investigation. , 2009, NeuroImage.

[23]  Lieven Verschaffel,et al.  Are secondary school students still hampered by the natural number bias? A reaction time study on fraction comparison tasks , 2013 .

[24]  Ruth Stavy,et al.  Intuitive Interference in Probabilistic Reasoning , 2006 .

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

[26]  L. Verschaffel,et al.  Dual Processes in the Psychology of Mathematics Education and Cognitive Psychology , 2009, Human Development.

[27]  Vincent Aleven,et al.  Interleaved Practice in Multi-Dimensional Learning Tasks: Which Dimension Should We Interleave?. , 2013 .

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

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

[30]  Wim Van Dooren,et al.  What Fills the Gap between Discrete and Dense? Greek and Flemish Students' Understanding of Density. , 2011 .

[31]  Thomas P. Carpenter,et al.  Results and Implications of the Second NAEP Mathematics Assessments: Elementary School. , 1980 .

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

[33]  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.

[34]  S. Dehaene,et al.  Attention, automaticity, and levels of representation in number processing , 1995 .

[35]  Nancy C. Jordan,et al.  The Importance of Number Sense to Mathematics Achievement in First and Third Grades. , 2010, Learning and individual differences.

[36]  Lieven Verschaffel,et al.  Naturally Biased? In Search for Reaction Time Evidence for a Natural Number Bias in Adults. , 2012 .

[37]  S. Dehaene,et al.  The Number Sense: How the Mind Creates Mathematics. , 1998 .

[38]  Florence Gabriel,et al.  Developing Children’s Understanding of Fractions : An Intervention Study , 2012 .

[39]  Emmanuel Dupoux,et al.  Is numerical comparison digital? Analogical and symbolic effects in two-digit number comparison. , 1990 .

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

[41]  Anne Roche,et al.  Students’ fraction comparison strategies as a window into robust understanding and possible pointers for instruction , 2009 .