Effects of mathematical word problem-solving instruction on middle school students with learning problems.

This study investigated the differential effects of two problem-solving instructional approaches--schema-based instruction (SBI) and general strategy instruction (GSI)--on the mathematical word problem--solving performance of 22 middle school students who had learning disabilities or were at risk for mathematics failure. Results indicated that the SBI group significantly outperformed the GSI group on immediate and delayed posttests as well as the transfer test. Implications of the study are discussed within the context of the new IDEA amendment and access to the general education curriculum.

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Mathematics is integral to all areas of daily life; it affects successful functioning on the job, in school, at home, and in the community. The importance of mathematics literacy and problem solving is emphasized in the Goals 2000: Educate America Act of 1994 and National Council of Teachers of Mathematics' Principles and Standards for School Mathematics (NCTM, 2000; Goldman, Hasselbring, & the Cognition and Technology Group at Vanderbilt, 1997). Increasing evidence suggests that high levels of mathematical and technical skills are needed for most jobs in the 21st century. Therefore, it is important to ensure that all students, not just those planning to pursue higher education, have sufficient skills to meet the challenges of the 21st century (National Education Goals Panel, 1997). In addition, one of the provisions of the 1997 amendments to the Individuals with Disabilities Education Act (IDEA) is that students with disabilities have meaningful access to the general education curriculum. In fact, these students are held accountable to the same high academic standards required of all students (No Child Left Behind Act, 2002).

As part of the mathematics reform and standards-based reform movements, the NCTM (2000) developed the Principles and Standards for School Mathematics. The focus of the NCTM standards is on "conceptual understanding rather than procedural knowledge or rule-driven computation" (Maccini & Gagnon, 2002, p. 326). This emphasis has significant implications for classroom practice because special education typically has focused on arithmetic computation rather than higher-order skills such as reasoning and problem solving (Cawley, Parmar, Yan, & Miller, 1998). Students with learning disabilities often manifest serious deficits in mathematics, especially problem solving (Carnine, Jones, & Dixon, 1994; Cawley & Miller, 1989; Cawley, Parmar, Foley, Salmon, & Roy, 2001; Parmar, Cawley, & Frazita, 1996). Specifically, these students perform at significantly lower levels than students without disabilities on all problem types, especially problems that involve indirect language, extraneous information, and multisteps (Briars & Larkin, 1984; Cawley et al., 2001; Englert, Culatta, & Horn, 1987; Lewis & Mayer, 1987; Parmar et al., 1996). While problems in reading and basic computation skills may account for these students' poor performance, difficulties in problem representation and failure to identify relevant information and operation may exacerbate their poor performance (Hutchinson, 1993; Judd & Bilsky, 1989; Parmar, 1992).

In addition, ineffective instructional strategies may explain the poor problem-solving performance of students with learning disabilities. One commonly used instructional approach is the "key word" strategy, in which students are taught key words that cue them as to what operation to use in solving problems. For example, students learn that altogether indicates the use of the addition operation, whereas left indicates subtraction. Similarly, the word times calls for multiplication, and among indicates the need to divide. However, Parmar et al. (1996) argued that "the outcome of such training is that the student reacts to the cue word at a surface level of analysis and fails to perform a deep-structure analysis of the interrelationships among the word and the context in which it is embedded" (p. 427). That is, the focus is on whether to add, subtract, multiply, or divide rather than whether the problem makes sense. Another commonly employed problem-solving strategy is the four-step (read, plan, solve, and check) general heuristic procedure. Unfortunately, this procedure may not facilitate problem solution for students with learning disabilities, especially when the domain-specific conceptual and procedural knowledge is not adequately elaborated upon (Hutchinson, 1993; Montague, Applegate, & Marquard, 1993).

For students with learning disabilities, explicit teaching for conceptual understanding is critical to establish the necessary knowledge base for problem solution. Recent reviews provide empirical support for problem-solving instruction, such as a schema-based strategy instruction, that emphasizes conceptual understanding of the problem structure, or schemata (Xin & Jitendra, 1999). Successful problem solvers typically create a complete mental representation of the problem schema, which, in turn, facilitates the encoding and retrieval of information needed to solve problems (Didierjean & Cauzinille-Marmeche, 1998; Fuson & Willis, 1989; Marshall, 1995; Mayer, 1982). Problem schema acquisition allows the learner to use the representation to solve a range of different (i.e., containing varying surface features) but structurally similar problems (Sweller, Chandler, Tierney, & Cooper, 1990).

Schema-based strategy instruction is known to benefit both special education students (e.g., Jitendra & Hoff, 1996; Jitendra, Hoff, & Beck, 1999) and students at risk for math failure (e.g., Jitendra et al., 1998; Jitendra, DiPipi, & Grasso, 2001) in solving arithmetic word problems. However, previous research on the effects of schema-based strategy instruction is limited, for the most part, to algebra problems (Hutchinson, 1993) and addition and subtraction (e.g., change, combine, additive compare) arithmetic problems. Although the effects of semantic representation training in facilitating problem solving have been demonstrated with college students with and without disabilities, the studies are limited to a sample of comparison problems only (Lewis, 1989; Zawaiza & Gerber, 1993). Furthermore, neither the study by Lewis nor the study by Zawaiza and Gerber emphasized key components (compared, referent, and scalar function) pertinent to the compare problem schemata. In addition, the rules for figuring out the operation (e.g., if the unknown quantity is to the right of the given quantity on the number line, then addition or multiplication should be applied) cannot be directly applied to solve multiplication or division compare problems when the relational statement involves a fraction or when the unknown is the scalar function (i.e., the multiple or partial relation between two comparison quantities).