The development of students’ use of additive and proportional methods along primary and secondary school

This study investigates the development of proportional and additive methods along primary and secondary school. In particular, it simultaneously investigates the use of additive methods in proportional word problems and the use of proportional methods in additive word problems. We have also studied the role played by integer and non-integer relationships between the given numbers and the nature of quantities (discrete or continuous) in the development of these phenomena. A test consisting of additive and proportional missing-value word problems was solved by 755 primary and secondary school students (from fourth to tenth grade). The findings indicate that the use of additive methods in proportional situations increased during primary school and decreased during secondary school, whereas the use of proportional methods in additive situations increased along primary and secondary school. Moreover, the presence or absence of integer ratios strongly affects this behavior, but the nature of quantities only has a small influence on the use of proportional methods.

[1]  Lieven Verschaffel,et al.  Whole number concepts and operations , 2007 .

[2]  Thomas R. Post,et al.  Learning and teaching ratio and proportion: Research implications: Middle grades mathematics , 1993 .

[3]  Robert Karplus,et al.  Early adolescents' proportional reasoning on ‘rate’ problems , 1983 .

[4]  P. Bryant,et al.  Proportional Reasoning in Young Children: Part-Part Comparisons about Continuous and Discontinuous Quantity , 1999 .

[5]  K. Hart,et al.  Children's understanding of mathematics: 11-16 , 1981 .

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

[7]  Ceneida Fernández-Verdú,et al.  Implicative analysis of strategies in solving proportional and nonproportional problems , 2008 .

[8]  P. Gerdes,et al.  Ethnomathematics and Mathematics Education , 1996 .

[9]  Douglas T. Owens.  Research Ideas for the Classroom: Middle Grades Mathematics. , 1993 .

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

[11]  Bonnie P. Schappelle,et al.  Providing a foundation for teaching mathematics in the middle grades , 1995 .

[12]  Merlyn J. Behr,et al.  Number concepts and operations in the middle grades , 1988 .

[13]  José Carrillo Yáñez,et al.  Investigación en educación matemática XIV , 2010 .

[14]  Gaea Leinhardt,et al.  Analysis of Arithmetic for Mathematics Teaching , 1992 .

[15]  Athanasios Gagatsis,et al.  Students' Improper Proportional Reasoning: A result of the epistemological obstacle of “linearity” , 2007 .

[16]  L. Verschaffel,et al.  Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education , 2009 .

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

[18]  Lieven Verschaffel,et al.  The Linear Imperative: An Inventory and Conceptual Analysis of Students' Overuse of Linearity , 2008 .

[19]  Douglas A. Grouws,et al.  Handbook of research on mathematics teaching and learning , 1992 .

[20]  Lieven Verschaffel,et al.  The Illusion of Linearity: From Analysis to Improvement , 2007 .

[21]  Lieven Verschaffel,et al.  Children's Criteria for Representational Adequacy in the Perception of Simple Sonic Stimuli , 2010 .

[22]  Lieven Verschaffel,et al.  Students' overuse of proportionality on missing-value problems: how numbers may change solutions , 2009 .

[23]  E. Corte,et al.  Making sense of word problems , 2000 .

[24]  L. Verschaffel,et al.  Students’ Overuse of Linearity: An Exploration in Physics , 2011 .

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

[26]  Steven Pulos,et al.  Proportional reasoning: A review of the literature , 1985 .

[27]  Lieven Verschaffel,et al.  Using graphical notations to assess children’s experiencing of simple and complex musical fragments , 2010 .

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

[29]  Salvador Llinares,et al.  Secondary School Students’ Construction and Use of Mathematical Models in Solving Word Problems , 2008 .

[30]  L. Verschaffel,et al.  Abstract or concrete examples in learning mathematics? A replication and elaboration of Kaminski, Sloutsky, and Heckler’s study , 2011 .

[31]  Barbara J. Dougherty,et al.  Mathematics Education Across Cultures: Proceedings of the 42nd Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education , 2020 .

[32]  Susan C. Levine,et al.  The Development of Proportional Reasoning: Effect of Continuous Versus Discrete Quantities , 2007 .

[33]  Kathleen Hart,et al.  Children's Understanding of Mathematics , 1989 .

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

[35]  L. Verschaffel,et al.  Effect of number structure and nature of quantities on secondary school students proportional reasoning , 2010 .

[36]  J. Williams,et al.  Diagnostic assessment of children’s proportional reasoning , 2003 .