Young Children's Intuitive Models of Multiplication and Division

Abstract

In this study, an intuitive model was defined as an internal mental structure corresponding to a class of calculation strategies. A sample of female students was observed 4 times during Grades 2 and 3 as they solved the same set of 24 word problems. From the correct responses, 12 distinct calculation strategies were identified and grouped into categories from which the children's intuitive models of multiplication and division were inferred. It was found that the students used 3 main intuitive models: direct counting, repeated addition, and multiplicative operation. A fourth model, repeated subtraction, only occurred in division problems. All the intuitive models were used with all semantic structures, their frequency varying as a complex interaction of age, size of numbers, language, and semantic structure. The results are interpreted as showing that children acquire an expanding repertoire of intuitive models and that the model they employ to solve any particular problem reflects the mathematical structure they impose on it. Several recent studies have shown that students can solve a variety of multiplicative problems long before formal instruction on the operations of multiplication and division. For example, Kouba (1989) found that 30% of Grade 1 and 70% of Grade 2 students could solve simple equivalent group problems and Mulligan (1992) found a steady increase in success rate on similar problems from over 50% at the beginning of Grade 2 to nearly 95% at the end of Grade 3. More recently Carpenter, Ansell, Franke, Fennema, and Weisbeck (1993) found that even kindergarten students could learn to solve multiplicative problems. Students use a range of solution strategies to solve multiplication and division word problems, and from this it has been inferred that they acquire various intuitive models of multiplication and division (Fischbein, Deri, Nello, & Merino, 1985; Kouba, 1989; Greer, 1992). The interest in intuitive models lies in the proposition that they are formed early on in elementary contexts and can strongly influence students' understanding of more complex multiplicative situations in secondary school and adulthood, often negatively (Fischbein et al. 1985; Graeber, Tirosh, & Glover 1989; Simon, 1993). However, it is not clear what intuitive models young children form, how they are related to the semantic structure of the problems to be solved, and how the models develop over time. The present paper attempts to throw light on these questions using data from a longitudinal study of students in Grades 2 and 3.

The Effects of Presenting Multidigit Mathematics Problems in a Realistic Context on Sixth Graders' Problem Solving

“…The current study focuses on the first three of these aspects, on the domain of multidigit arithmetic. In the domain of elementary or simple arithmetic, strategy use has been studied extensively: in elementary addition and subtraction (e.g., Carr & Jessup, 1997;Carr & Davis, 2001;Torbeyns, Verschaffel, & Ghesquière, 2004, 2005, in elementary multiplication (e.g., Anghileri, 1989;Imbo & Vandierendonck, 2007;Lemaire & Siegler, 1995;Mabbott & Bisanz, 2003;Mulligan & Mitchelmore, 1997;Sherin & Fuson, 2005;Siegler, 1988b), and in elementary division (e.g., Robinson et al, 2006). By contrast, research on solution strategies in complex or multidigit arithmetic problems is less extensive, but there is a growing body of studies in multidigit addition and subtraction (e.g., Beishuizen, 1993;Beishuizen, Van Putten, & Van Mulken, 1997;Blöte al., 2001;Torbeyns, Verschaffel, & Ghesquière, 2006) and in multidigit multiplication and division (e.g., Ambrose, Baek, & Carpenter, 2003;Buijs, 2008;Hickendorff, Heiser, Van Putten, & Verhelst, 2009;Hickendorff & Van Putten, 2012;Hickendorff, Van Putten, Verhelst, & Heiser, 2010;Van Putten, Van den Brom-Snijders, & Beishuizen, 2005).…”

Children’s Conceptions of Area Measurement and Their Strategies for Solving Area Measurement Problems

“…As children develop more sophisticated procedures for the numeric calculation of grid squares, they can realize that the formula "Area = Length x Width (A=LxW)" for measuring rectangular figures is a faster alternative than employing the square counting method (Outhred & Mitchelmore, 2000). Thus, it is argued that seeing the structure of rectangular arrays should be more sophisticated than seeing the repeated addition model and set model of multiplication (i.e., 2x3 can be modeled as two groups of 3) (Mulligan & Mitchelmore, 1997).…”

Problemas de estructura multiplicativa: Evolución de niveles de éxito y estrategias en estudiantes de 6 a 12 años

“…Las estrategias que los niños emplean en la resolución de estos problemas y los diferentes niveles de éxito alcanzados han puesto de manifiesto la dificultad que tienen para comprender las diferentes situaciones multiplicativas (Clark y Pere Ivars y Ceneida Fernández Kamii, 1996;García, Vázquez y Zarzosa, 2013;Mulligan y Mitchelmore, 1997;Peled y Nesher, 1988). Esta investigación tiene como objetivo caracterizar la evolución del nivel de éxito y de las estrategias en los problemas de estructura multiplicativa a lo largo de la Educación Primaria (6-12 años).…”