Learners’ Difficulties with Quantitative Units in Algebraic Word Problems and the Teacher’s Interpretation of those Difficulties

Abstract

This study examines 8th grade students_ coordination of quantitative units arising from word problems that can be solved via a set of equations that are reducible to a single equation with a single unknown. Along with Unit-Coordination, Quantitative Unit Conservation also emerges as a necessary construct in dealing with such problems. We base our analysis within a framework of quantitative reasoning (Thompson, 1988;1989;1993;1995) and a theory of children_s units-coordination with different levels of units (Steffe 1994) that both encompass and are extended by these two constructs. Our data consist of videotaped classroom lessons, student interviews, and teacher interviews. Ongoing analyses of these data were conducted during the teaching sequence. A retrospective analysis using constant comparison methodology was then undertaken during which the classroom video, related student interviews, and teacher interviews were revisited many times in order to generate a thematic analysis. Our results indicate that the identification and coordination of the units involved in the problem situation are critical aspects of quantitative reasoning and need to be emphasized in the teachinglearning process. We also concluded that unit coordination and unit conservation are cognitive prerequisites for constructing a meaningful algebraic equation when reasoning quantitatively about a situation.

Eighth grade students’ representations of linear equations based on a cups and tiles model

“…There is, however something more that needs to be emphasized about the cups holding tiles, i.e., that all the cups must hold the same number of tiles, as the following protocol indicates: Any representation is prone to yield some sense of dealing with different units. In fact, the cups and tiles representation necessitates a unit coordination task (Olive & Caglayan, 2008). There are two different units to be coordinated: First, each cup is to be filled with the same number of tiles; therefore, there is the same number of tiles per cup.…”

“…If it were just a scalar number, then the multiplication of 2 by c would not give a different unit; therefore, the unit for 2c would still be number of tiles per cup. However, 1, which is added to 2c, itself represents a "tile"; and has a different unit, number of tiles, therefore, we cannot add it to 2c (unit conservation, see Olive & Caglayan, 2008). We need more information about the construction of this problem, and that information will come from an appropriate definition of the number "2" in front of 2c.…”

Prospective Middle-School Mathematics Teachers’ Quantitative Reasoning and Their Support for Students’ Quantitative Reasoning

“…A solution to the candle-burning problem, which was adapted from the study of Carlson (2013), is required to build mental models of the specific quantities, relationships among these quantities and to construct a formula that represents these relationships in terms of quantitative meaning. In an attempt to solve the coin problem, which was adapted from the study conducted Olive and Çağlayan (2008), students need to determine the quantities in the context of the problem and make a quantitative-unit coordination and construct a quantitativeunit conservation to determine relationships among the quantities. In the solution process of three quantitativelyrich problems, the prospective teachers are expected to use various components of quantitative-reasoning skills such as identifying the quantities, determining relationships among the quantities, analyzing the change in quantities, creating a formula in terms of quantitative meaning, making a quantitative-unit coordination and constructing a quantitative-unit conservation.…”

“…How would you organize the solution process of this problem? (adapted from Olive and Çağlayan (2008)). …”

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).…”