Spontaneous Focusing on Quantitative Relations: Towards a Characterization

ABSTRACT In contrast to previous studies on Spontaneous Focusing on Quantitative Relations (SFOR), the present study investigated not only the extent to which children focus on (multiplicative) quantitative relations, but also the nature of children’s quantitative focus (i.e., the types of quantitative relations that children focus on). Therefore, we offered three different SFOR tasks – a multiplicative, additive, or open SFOR task – to 315 second, fourth, and sixth graders. Results revealed, first, that most children spontaneously focused on quantitative relations. Some focused on multiplicative relations, and others on additive relations. Second, SFOR, and especially multiplicative SFOR, increased with grade, while the development of additive SFOR differed between tasks. Third, the open SFOR task seemed best suited to capture SFOR, since it evoked the largest number of each type of relational answers − while still showing substantial interindividual differences in SFOR. These results indicate that a broader conceptualization and operationalization of SFOR than the unilateral multiplicative one are warranted.

[1]  Richard Lesh,et al.  Interpreting Responses to Problems with Several Levels and Types of Correct Answers , 2013 .

[2]  Athanasios Gagatsis,et al.  Cognitive and Metacognitive Aspects of Proportional Reasoning , 2010 .

[3]  E. Lehtinen,et al.  Preschool Children’s Spontaneous Focusing on Numerosity, Subitizing, and Counting Skills as Predictors of Their Mathematical Performance Seven Years Later at School , 2015 .

[4]  L. Verschaffel,et al.  Proportional Word Problem Solving Through a Modeling Lens: A Half-Empty or Half-Full Glass? , 2016 .

[5]  Terezinha Nunes,et al.  Learning and teaching mathematics : an international perspective , 1997 .

[6]  J. R. Landis,et al.  The measurement of observer agreement for categorical data. , 1977, Biometrics.

[7]  B. Greer Modelling Reality in Mathematics Classrooms: The Case of Word Problems. , 1997 .

[8]  Lieven Verschaffel,et al.  Beyond additive and multiplicative reasoning abilities: how preference enters the picture , 2018 .

[9]  E. Lehtinen,et al.  Spontaneous Focusing on Quantitative Relations as a Predictor of the Development of Rational Number Conceptual Knowledge. , 2016 .

[10]  Lieven Verschaffel,et al.  Just Answering … or Thinking? Contrasting Pupils' Solutions and Classifications of Missing-Value Word Problems , 2010 .

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

[12]  D. Siemon,et al.  FROM ADDITIVE TO MULTIPLICATIVE THINKING – THE BIG CHALLENGE OF THE MIDDLE YEARS , 2006 .

[13]  Sue Willis,et al.  The development of multiplicative thinking in young children , 2003 .

[14]  Spontaneous Focusing on Quantitative Relations and the Development of Rational Number Conceptual Knowledge , 2014 .

[15]  Lieven Verschaffel,et al.  The development of students’ use of additive and proportional methods along primary and secondary school , 2012 .

[16]  Ty W. Boyer,et al.  Development of proportional reasoning: where young children go wrong. , 2008, Developmental psychology.

[17]  Sophie Batchelor,et al.  Dispositional factors affecting children's early numerical development , 2014 .

[18]  Peter Bryant,et al.  Assessing Quantitative Reasoning in Young Children , 2015 .

[19]  Thomas R. Post,et al.  Learning and teaching ratio and proportion: Research implications , 1993 .

[20]  Erno Lehtinen,et al.  Spontaneous Focusing on Numerosity and Mathematical Skills of Young Children. , 2005 .

[21]  Ellen F. Potter,et al.  An Exploratory Look at the Relationships Among Math Skills, Motivational Factors and Activity Choice , 2013 .

[22]  P. Bryant,et al.  Summary of paper 4 : Understanding relations and their graphical representation , 2022 .

[23]  L. Verschaffel,et al.  Kindergartners’ Spontaneous Focusing on Numerosity in Relation to Their Number-Related Utterances During Numerical Picture Book Reading , 2016 .

[24]  Lieven Verschaffel,et al.  From Addition to Multiplication … and Back: The Development of Students’ Additive and Multiplicative Reasoning Skills , 2009 .

[25]  M. Hannula-Sormunen Spontaneous Focusing On Numerosity and its Relation to Counting and Arithmetic , 2015 .

[26]  L. Verschaffel,et al.  Not Everything Is Proportional: Effects of Age and Problem Type on Propensities for Overgeneralization , 2005 .

[27]  Spontaneous Focusing on Quantitative Relations in the Development of Children's Fraction Knowledge , 2014 .

[28]  L. Verschaffel,et al.  Unraveling the gap between natural and rational numbers , 2015 .

[29]  Constance Kamii,et al.  Identification of Multiplicative Thinking in Children in Grades 1-5. , 1996 .

[30]  Erno Lehtinen,et al.  Spontaneous focusing on numerosity as a domain-specific predictor of arithmetical skills. , 2010, Journal of experimental child psychology.

[31]  Lieven Verschaffel,et al.  How do Flemish children solve ‘Greek’ word problems? On children’s quantitative analogical reasoning in mathematically neutral word problems , 2014 .

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

[33]  E. Lehtinen,et al.  Young children's recognition of quantitative relations in mathematically unspecified settings , 2013 .

[34]  Erno Lehtinen,et al.  Development of Counting Skills: Role of Spontaneous Focusing on Numerosity and Subitizing-Based Enumeration , 2007 .

[35]  L. Verschaffel,et al.  The relation between learners' spontaneous focusing on quantitative relations and their rational number knowledge , 2016 .

[36]  Ty W. Boyer,et al.  Child proportional scaling: is 1/3=2/6=3/9=4/12? , 2012, Journal of experimental child psychology.

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

[38]  T. Andreescu,et al.  Addition or Multiplication , 2004 .

[39]  E. Lehtinen,et al.  Preschool spontaneous focusing on numerosity predicts rational number conceptual knowledge 6 years later , 2015 .

[40]  Matthew Inglis,et al.  Spontaneous focusing on numerosity and the arithmetic advantage , 2015 .