Effects of semantic cues on mathematical modeling: Evidence from word-problem solving and equation construction tasks

Abstract

Mathematical solutions to textbook word problems are correlated with semantic relations between the objects described in the problem texts. In particular, division problems usually involve functionally related objects (e.g., tulips-vases) and rarely involve categorically related objects (e.g., tulips-daisies). We examined whether middle school, high school, and college students use object relations when they solve division word problems (WP) or perform the less familiar task of representing verbal statements with algebraic equations (EQ). Both tasks involved multiplicative comparison statements with either categorically or functionally related objects (e.g., "four times as many cupcakes [commuters] as brownies [automobiles]"). Object relations affected the frequency of correct solutions in the WP task but not in the EQ task. In the latter task, object relations did affect the structure of nonalgebraic equation errors. We argue that students use object relations as "semantic cues" when they engage in the sense-making activity of mathematical modeling.

Following the standard form: Effects of equation format on algebraic modeling

“…Many of the participants in that study (37%) failed to construct the correct algebraic equation (6P = S), and their most common error was a reversal of the two variables (6S = P). Over the years, these and other researchers have reported similar findings, with error rates ranging between 40 and 60% (Bernardo & Okagaki, 1994;Clement, 1982;Clement, Lochhead, & Monk, 1981;Fisher, 1988;Graf, Bassok, Hunt, & Minstrell, 2004;Kaput & Sims-Knight, 1983;Martin & Bassok, 2005;Rosnick, 1981;Rosnick & Clement, 1980;Wollman, 1983).…”

“…In the context of numerical problem solving, mathematical modeling entails choosing an arithmetic operation that corresponds to the situation described in the problem text. College students have extensive experience with this modeling task, and are highly fluent in selecting arithmetic operations that are structurally analogous to the quantitative relations in word problems (e.g., Bassok, Chase, & Martin, 1998;Bassok, Wu, and Olseth, 1995;Dixon, 2005;Dixon, Deets & Bangert, 2001;Martin & Bassok, 2005). When students transition to algebra and learn that algebraic models have a standard (multiplication) form, some of them may be unsure whether and, if so, how algebraic modeling involves selecting an appropriate arithmetic operation.…”

“…Irrespective of the specific reason that leads students to commit reversal errors, such modeling errors are much more common in algebra than in numerical problem solving (Bernardo & Okagaki, 1994, Martin & Bassok, 2005Wollman, 1983), a domain for which students have developed strong modeling skills. For example, Martin and Bassok (2005) asked middle school, high school, and college students to either construct algebraic equations that represent relational statements, such as "There are 6 times as many nails as screws," or to solve numerical word problems with the same relational statements, such as "A certain factory produces 6 times as many nails as screws.…”

“…For example, Martin and Bassok (2005) asked middle school, high school, and college students to either construct algebraic equations that represent relational statements, such as "There are 6 times as many nails as screws," or to solve numerical word problems with the same relational statements, such as "A certain factory produces 6 times as many nails as screws. Yesterday, it produced 3,450 nails.…”

Two Approaches to the Concept of Knowledge Application: Transfer and Modeling Overview and Criticism

“…We demonstrate that semantics has an unconscious influence on mathematical model construction. For example, the use of functionally related items (boxes -oranges) activates the division model, while categorically related items (oranges -lemons) activate the addition model [Martin, Bassok, 2005].…”