The Real Story Behind Story Problems: Effects of Representations on Quantitative Reasoning

Abstract

This article explores how differences in problem representations change both the performance and underlying cognitive processes of beginning algebra students engaged in quantitative reasoning. Contrary to beliefs held by practitioners and researchers in mathematics education, students were more successful solving simple algebra story problems than solving mathematically equivalent equations. Contrary to some views of situated cognition, this result is not simply a consequence of situated world knowledge facilitating problem-solving performance, but rather a consequence of student difficulties with comprehending the formal symbolic representation of quantitative relations. We draw on analyses of students' strategies and errors as the basis for a cognitive process explanation of when, why, and how differences in problem representation affect problem solving. We conclude that differences in external representations can affect performance and learning when one representation is easier to comprehend than another or when one representation elicits more reliable and meaningful solution strategies than another.A commonly held belief about story problems at both the arithmetic and algebra levels is that they are notoriously difficult for students. Support for this belief can be seen among a variety of populations including the general public, textbook au-

Performance and Difficulties of Students in Formulating and Solving Quadratic Equations with One Unknown

“…Studies have revealed that solving algebra word problems is challenging for the majority of students because the formal algebraic system creates a serious barrier to generating equations that represent the relationships within the problem (Kieran, 1992). However, students' challenges depend, not only on a formal algebraic system in the solution phase, but also on the linguistic form of the word problems in the comprehension phase (Koedinger & Nathan, 2004). Various researchers have proposed that solving algebra word problems consists of a "comprehension phase and [a] solution phase" (e.g., Cummins, Kintsch, Reusser, & Weimer, 1988;Mayer, 1982;Nathan, Kintsch, & Young, 1992).…”

“…In the comprehension phase, a problem-solver comprehends and then forms the text base of the problem, utilizing words as an internal representation in his or her memory. In the solution phase, she or he expresses this internal representation externally and applies the rules of algebra to reach a conclusion (Koedinger & Nathan, 2004;Mayer, 1982).…”

“…As a result of coding students' incorrect answers, we decided to categorize student performance in solving quadratic equations with respect to their use of methods (i.e., quadratic formula, factoring method, completing square method, and square root property) to solve quadratic equations. In terms of word problems, the codes and categories came from conceptual frameworks and prior research relating to solving arithmetic and algebraic word problems (Cummins et al, 1988;Koedinger & Nathan, 2004). Students' incorrect and incomplete answers were coded and categorized with respect to their comprehension of the problem statement, setting up the correct relationship and formulating quadratic equations in one unknown, and solving the formulated quadratic equation.…”

Learning to Think: Cognitive Mechanisms of Knowledge Transfer

“…When asked to predict student performance, teachers and educators indicate that algebra story problems are harder for students to solve than matched symbolic equations, since students need to fi rst translate these word problems into symbolic notation (Nathan & Koedinger, 2000). Koedinger and Nathan (2004) compared students' performance on story problems and matched equations, and discovered that the assumed knowledge component analysis (e.g., that equations are needed to solve story problems) was incorrect. ey found that beginning algebra students are actually better able to solve introductory story and word problems than matched equations.…”

Longer bars for bigger numbers? children’s usage and understanding of graphical representations of algebraic problems

“…One pedagogical approach that may further benefit student is to emphasize the different situations under which the two approaches are best suited. Although not specifically focused on the model method, previous research have shown that students are more successful when they use more concrete or grounded representations to solve simple algebra questions, but are more successful with more abstract, symbolic representations with more complex problems (Koedinger, Alibali, & Nathan, 2008;Koedinger & Nathan, 2004).…”