A comparative analysis of number sense instruction in reform-based and traditional mathematics textbooks.(Report)

This study compared number sense instruction in three first-grade traditional mathematics textbooks and one reform-based textbook (Everyday Mathematics [EM]). Textbooks were evaluated with regard to their adherence to principles of effective instruction (e.g., big ideas, conspicuous instruction). The results indicated that traditional textbooks included more opportunities for number relationship tasks than did EM; in contrast, EM emphasized more real-world connections than did traditional textbooks. However, EM did better than traditional textbooks in (a) promoting relational understanding and (b) integrating spatial relationship tasks with other more complex skills. Whereas instruction was more direct and explicit and feedback was more common in traditional textbooks than it was in EM, there were differences among traditional textbooks with respect to these two criteria. Although EM excelled in scaffolding instruction by devoting more lessons to concrete and semiconcrete activities, traditional textbooks provided more opportunities for engaging in all three representations. However, EM emphasized (a) a variety of models to develop number sense concepts, (b) a concrete, or semiconcrete, to symbolic representational sequence, and (c) hands-on activities using real-world objects to enhance learner engagement. Finally, even though traditional textbooks excelled over EM in providing more opportunities to practice number sense skills, this finding may be an artifact of the worksheet format employed in traditional textbooks. At the same time, adequate distribution of review in subsequent lessons was evident in EM and in only one of the traditional textbooks. Implications for practice in accessing the general education curriculum for students with learning problems are discussed.

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Assessments of mathematics achievement of students in the United States have evoked "both a sense of despair and of hope" (National Research Council [NRC], 2001, p. 55). Although recent data from the National Assessment of Educational Progress (National Center for Education Statistics [NCES], 2006) indicate small gains in mathematics achievement among public school students, achievement gaps are wide, with low levels of achievement among minority students (i.e., African American, Native American, Latino), limited English proficient students, students with disabilities, and students from low socioeconomic status. Further, the low mathematical proficiency of U.S. students when compared to students in most developed countries is well documented (Lemke et al., 2004; Mullis, Martin, Gonzalez, & Chrostowski, 2004; NCES, 2003; Schmidt, 2002).

Several factors may explain the relatively poor performance of U.S. students. For one, increasing numbers of children with diverse learning and curricular needs receive instruction in general education classrooms (McLeskey, Henry, & Axelrod, 1999; Morocco, 2001). A potential barrier to these children's access to the general curriculum is poorly designed textbooks and educational materials that fail to provide experiences to develop critical mathematical ideas (Jones, Langrall, Thornton, & Nisbet, 2002; Suter, 2000). Evidently, mathematics content organization and instruction influence student achievement (Schmidt, Jakwerth, & McKnight, 1998). Traditional mathematics curricula have been criticized for being "relatively repetitive, unfocused, and undemanding" (Hiebert, 1999, p. 11). In addition, U.S. textbooks, when compared to textbooks from other countries, seem to lack "focus and coherence" and fail to provide "meaningful connections between the big ideas of mathematics" (Valverde & Schmidt, 1997/ 1998, p. 63). Converging evidence suggests that "differences in the quality and quantity of mathematics instruction" account for cross-national differences in mathematics achievement (e.g., Mayer, Sims, & Tajika, 1995, p. 444).

The increasing discontent with traditional mathematics textbook instruction has led to a new way of conceptualizing the teaching and learning of mathematics. With the publication of the National Council of Teachers for Mathematics's (NCTM's) Curriculum and Evaluation Standards for School Mathematics in 1989 and Principles and Standards for School Mathematics (Principles and Standards) in 2000, the emphasis has shifted from procedural knowledge and rote driven computation to conceptual knowledge (Schoenfeld, 2002). An interesting finding was that traditional textbooks emphasizing teacher-directed instruction "account for well over 80 percent of the textbooks used in schools" (Van de Walle, 2007, p. 8). However, reform-based mathematics textbooks, designed to develop conceptual understanding by engaging students in problem-solving opportunities that emphasize reasoning and thinking using open-ended approaches, are increasingly adopted in many schools across the nation (Findell, 1991; Fraivilling, Murphy, & Fuson, 1999; Greenes, 1996; Remillard, 2005).

The increased complexities of a diverse population in the U.S. educational system, along with the less than positive school outcomes in mathematics, clearly call for the need to analyze the adequacy of information presented in textbooks. Textbooks are considered a de facto national curriculum and are the primary means of imparting new information to students (Britton, Woodward, & Binkley, 1993; Chandler & Brosnan, 1994; Garner, 1992; Mayer et al., 1995; Osborne, Jones, & Stein, 1985; Porter, 1989; Valverde & Schmidt, 1997/1998). Garner (1992) noted, "Textbooks serve as critical vehicles for knowledge acquisition in school" and can "replace teacher talk as the primary source of information" (p. 53). Therefore, examining what is taught is critical in light of recent calls for challenging learning standards and school accountability (Chatterji, 2002; Nolet & McLaughlin, 2005).

Although many educators recognize that the quality and adequacy of mathematics textbooks are important factors in promoting student learning, few content analyses have focused on the implications of instructional design for learners at risk for mathematics disabilities (Camine, Jitendra, & Silbert, 1997; Jitendra, Carnine, & Silbert, 1996; Jitendra, Deatline-Buchman, & Sczesniak, 2005; Jitendra, Griffin, et al., 2005; Jitendra, Salmento, & Haydt, 1999). None of these studies, however, have compared instruction presented in reform-based and traditional textbooks or have focused on beginning mathematics skills such as number sense, which is a prerequisite for the development of higher-level mathematics skills (Isaacs & Carroll, 1999; Markovits & Sowder, 1994; McIntosh, Reys, & Reys, 1992; NCTM, 1989, 2000; NRC, 1989). The purpose of our study was to compare how number sense was taught in reform-based and traditional mathematics textbooks. In particular, we examined the quality of the instructional design features of mathematics programs with an emphasis on number sense instruction for learners at risk for mathematics disabilities.