The development of rational number knowledge: Old topic, new insights

Abstract The development of rational number knowledge has been studied extensively by mathematics education researchers, cognitive-developmental psychologists and, more recently, by neuroscientists as well. Building on a rich body of prior research, and some exciting new ideas, the target articles re-visit several topics, with a view to refine and deepen our understanding of how rational number understanding develops. The effect of prior natural number knowledge-either positive or adverse-on rational number learning is highlighted by all contributors. I draw on the articles to discuss five different aspects of the whole, or natural, number bias.

[1]  E. Fischbein,et al.  Intuition in Science and Mathematics: An Educational Approach , 2014 .

[2]  Leslie P. Steffe,et al.  Children's Fractional Knowledge , 2009 .

[3]  Stella Vosniadou,et al.  International handbook of research on conceptual change , 2013 .

[4]  Peter Bryant,et al.  Children Doing Mathematics , 1996 .

[5]  Precursors to number: Equivalence relations, less-than and greater-than relations, and units , 2008, Behavioral and Brain Sciences.

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

[7]  G. Brousseau.  Theory of didactical situations in mathematics , 1997 .

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

[9]  Andreas Nieder,et al.  Relating magnitudes: the brain's code for proportions , 2012, Trends in Cognitive Sciences.

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

[11]  Konstantinos P. Christou,et al.  What Kinds of Numbers Do Students Assign to Literal Symbols? Aspects of the Transition from Arithmetic to Algebra , 2012 .

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

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

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

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

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

[17]  Jeremy Kilpatrick,et al.  Adding It Up: Helping Children Learn Mathematics , 2013 .

[18]  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.

[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]  Lieven Verschaffel,et al.  In search for the natural number bias in secondary school students' interpretation of the effect of arithmetical operations , 2015 .

[21]  C. Gallistel,et al.  Non-verbal numerical cognition: from reals to integers , 2000, Trends in Cognitive Sciences.

[22]  Erno Lehtinen,et al.  Number concept and conceptual change: towards a systemic model of the processes of change , 2004 .

[23]  Susan Carey,et al.  One, two, three, four, nothing more: An investigation of the conceptual sources of the verbal counting principles , 2007, Cognition.

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

[25]  S. Vosniadou,et al.  How Many Decimals Are There Between Two Fractions? Aspects of Secondary School Students’ Understanding of Rational Numbers and Their Notation , 2010 .

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

[27]  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.

[28]  Dana Ganor-Stern,et al.  Primitives and Non-primitives of Numerical Representations , 2015 .

[29]  Kelley Durkin,et al.  Diagnosing misconceptions: Revealing changing decimal fraction knowledge. , 2015 .

[30]  Jere Confrey,et al.  The Development of multiplicative reasoning in the learning of mathematics , 1995 .

[31]  S. Vosniadou,et al.  Bridging the Gap Between the Dense and the Discrete: The Number Line and the “Rubber Line” Bridging Analogy , 2012 .

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

[33]  G. Harel,et al.  Units of quantity: A conceptual basis common to additive and multiplicative structures , 1994 .

[34]  How Valid is it to Use Number Lines to Measure Children's Conceptual Knowledge about Rational Number? , 2000 .