Quantitative reasoning, complexity, and additive structures

Abstract

ABSTRACT. Six fifth-grade children participated in a four-day teaching experiment on complex additively-structured problems, which was followed by in-depth interviews of individual children. The teaching experiment was meant to investigate children's difficulties in holding in mind at once situations in which one or more items played multiple roles. Two important difficulties were identified: (1) distinguishing between "difference" as the result of subtracting and "difference" as the amount by which one quantity exceeded another; and (2) indirect evaluation of an additive comparison. Sources of these difficulties, along with pedago~cal and curricular recommendations for addressing them, are discussed.

Comparación Entre Los Promedios De La Prueba De Aptitud Académica Y La Prueba De Habilidades Cuantitativas De Los Estudiantes De La Universidad De Costa Rica

“…Los factores que cargan altamente en la dimensión razonamiento cuantitativo son medidos por tests que llevan nombres como: Razonamiento Aritmético, Aptitud Matemática, Aritmética, etc. Thompson (1993) define razonamiento cuantitativo como el análisis de una situación dentro de una estructura cuantitativa (red de relaciones entre cantidades). Lobato y Siebert (2002) mencionan que el razonamiento cuantitativo implica concebir una situación dada en términos de cantidades mediante la construcción de redes de relaciones cuantitativas.…”

Number Sense: The Underpinning Understanding for Early Quantitative Literacy

“…Children may experience difficulty with the relational dimension of number if they only experience counting in a cardinal context, without adequate consideration of its ordinal context. However, for people to develop the 'habit of mind' to identify numerical structures (Hughes-Hallett 2001) they need to construct relationships between quantities Davydov 1982;Thompson 1993), because a grasp of the part-whole relationship (which counting facilitates) is critical to the development of quantitative reasoning (Olive and Çağlayan 2008).…”

“…Numbers also express relationships between quantities as in 'there are 2 more boys than girls in my class'. The relationships between quantities are more important than the quantities themselves because relationships reflect numerical complexity (Thompson 1993).…”

“…Modelling may begin with manipulatives or pictorial referents to represent quantities (Lesh, Post, and Behr 1987) through 'seeing' quantities in one's head (Gray, Pitta, and Tall 2000) to principle-based reasoning where repeated opportunity to derive new number knowledge allows learners to construct their own rules and generalisations Baroody, Bajwa, and Eiland 2009;Palm 2008). A child at an early representational stage of modelling with manipulatives may find it difficult to simultaneously consider the relationship (Thompson 1993) indicated in the particular Story Problem's structure (Riley, Greeno, and Heller 1983).…”

Try to See It My Way: The Discursive Function of Idiosyncratic Mathematical Metaphor

“…On the other hand, it has been proposed that mathematical reasoning draws on nonquantitative resources (Arcavi, 1994;Case & Okamoto, 1996;Johnson, 1987;Pirie & Kieren, 1994;Presmeg, 1986;Radford, 2006;Resnick, 1992;Thompson, 1993) and that mathematical instruction is effective only to the extent that it engages multimodal imagery (Abrahamson, 2009a;Bartolini Bussi & Boni, 2003;Dewey, 1928Dewey, /2008Edwards, Radford, & Arzarello, 2009;Lemke, 1998;Nemirovsky & Borba, 2004;Papert, 1980;Presmeg, 2006). Indeed, dynamical images have been implicated as instrumental to scientific and mathematical creativity, discovery, and invention 56 ABRAHAMSON, GUTIÉRREZ, AND BADDORF (Benfey, 1958;Black, 1993;Hadamard, 1945;Poincaré 1897Poincaré /2003Schoenfeld, 1991;Steiner, 2001;Wilensky, 1997).…”