...instruction the use oral retellings as a strategy for solving word problems. Following four 30-minute instructional sessions, eight students from each class were selected to participate in one-on-one interviews. During the interview, we asked the student to retell and solve six compare word problems, three containing consistent language and three containing inconsistent language. To investigate the nature of oral retellings when used as a problem solving strategy, we analyzed responses for problem representation, strategy development, and language. Results showed that inconsistently-worded items were more difficult for students to solve than items with consistent language. Students who translated inconsistent items into consistent language more frequently selected an appropriate arithmetic operation and were more successful than those students who appeared to rely on key words rather than a meaningful representation of the problem, That is, their retelling.

Sixth Graders' Retellings of Compare Word Problems

A train leaves Chicago at 1:27 p.m. Another train leaves St. Louis at 4:30 p.m. At what time will the students reading this word problem (a) consider the exercise pointless, (b) begin doubting their ability to solve it, (c) lose interest altogether, or (d) all of the above?

Teachers struggle to motivate, empower, and enhance students' confidence as problem solvers. While the ability to solve problems is imperative to all areas of life, in mathematics, problem solving is uniquely essential; it is "a hallmark of mathematical activity and a major means of developing mathematical knowledge" (National Council of Teachers of Mathematics, 2000, p. 116). Indeed, problem solving is the substance of mathematics, with word problems "front and center" as problem solving contexts for developing mathematical power as well as analytical thinking and cognitive abilities (Latterell & Copes, 2003; Parmer, Cawley, & Frazita, 1996). Reform-minded teachers seek to create a culture of problem solving wherein students communicate mathematically about problems and their strategies for approaching them. Word problems offer a natural ingress into problem solving and a culture of mathematical power and communication.

Word problems and the strategies associated with solving them have been the subject of abundant research for many years (Lester, 1994; Trafton & Midgett, 2001). Compare word problems have been the focus of a large portion of that research. Lewis and Mayer (1987) defined compare problems as those containing a "static numerical relation between two variables" (p. 363). The static nature of the compare problem makes it the most difficult type of addition/subtraction word problem to solve (see Carpenter, Fennema, Franke, Levi, & Empson, 1999; Fuson, Carroll, & Landis, 1996; LeBlanc & Weber-Russell, 1996; Mwangi & Sweller, 1998; Okamoto, 1996; Pape, 2003; Riley & Greeno, 1988). Limited research has been conducted to assess the potential of oral retellings in measuring students' comprehension of compare word problems (Verschaffel, 1994). The purpose of this study is to tell a story of retelling that is, to examine the nature of oral retellings of compare word problems as evidenced in interviews with 16 sixth grade students.

Key Terms

Word Problems

Also called "story problems," word problems tell a mathematical story. An example of a word problem follows:

Jose has 11 butterflies in his collection. On a family vacation, he catches 4 more butterflies. How many butterflies does Jose have in his collection now?

Compare Word Problems

Compare word problems involve the static comparison of two disjoint sets and usually contain a relational statement or term (more, less or fewer). Compare problems have been labeled using a variety of classification systems (see Briars & Larkin, 1984; Carpenter et al., 1999; Fuson et al., 1996; Riley, Greeno, & Heller, 1983). This study focused on the nature of retellings of compare problems without grouping them according to the categories proposed by these researchers. Instead, problems were classified according to the manner in which the relational term was used in the context of the problem. The research of Lewis and Mayer (1987) regarding consistent and inconsistent language guided this classification.

Subject and Object of Inconsistently-worded Compare Problems

Lewis and Mayer (1987) used the terms subject and object to identify the two elements of the complex sentence usually occurring as the second sentence in an inconsistently-worded compare problem. The tradition established by Lewis and Mayer and continued by subsequent researchers allows for the use of these terms; although problematic to grammarians, the current study maintained this precedent. This specialized use is illustrated in the following word problem:

Amanda has 11 cupcakes. She has 5 fewer cupcakes [subject] than cookies [object]. How many cookies does she have? Consistent Versus Inconsistent Compare Problems Lewis and Mayer (1987) illustrated consistent and inconsistent language using the following word problems: Consistent: Joe has 3 marbles. Tom has 5 more marbles than Joe. How many marbles does Tom have? Inconsistent: Joe has 8 marbles. He has 5 more marbles than Tom. How many marbles does Tom have? (p. 354) These researchers described the language issues associated with these problems with the following explanation: In consistent language problems the unknown variable (e.g., Tom's marbles) is the subject of the second sentence, and the relational term in the second sentence (e.g., more than) is consistent with the necessary arithmetic operation (e.g., addition). On the other hand, in inconsistent language problems the unknown variable is the object of the second sentence, and the relational term (e.g., more than) conflicts with the necessary arithmetic operation (e.g., subtraction). (p. 363)

In this study, word problems in which the relational term more, less or fewer produces a conflict between the action cued by the term and an appropriate arithmetic operation for solving the problem were deemed to be inconsistently-worded. The following problem serves as an example:

Amanda has 11 cupcakes. She has 5 fewer cupcakes than cookies. How many cookies does she have?

To the problem solver, the word fewer may suggest subtraction of the...

NOTE: All illustrations and photos have been removed from this article.



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