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Article Excerpt
Abstract. High-quality core instruction in kindergarten and first and second grade is critical to prevent mathematics difficulties. Evidence-based critical features of instruction should be part of core instruction and be included in mathematics textbooks. This study examined lessons from kindergarten and first- and second-grade basal mathematics textbooks to determine the extent to which 11 critical features of instruction were present. Overall, results showed an "Approaching Acceptable" rating, meaning that the features were not fully incorporated. Implications include the need for textbook adoption committees to be mindful of the importance of including effective instructional practices when making textbook decisions and for teachers to scrutinize the components of lessons to determine if these features of effective instruction are included.
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Mathematical literacy refers to the ability to apply concepts to reason, solve problems, and communicate about mathematical situations in the classroom and everyday life (National Council of Teachers of Mathematics [NCTM], 2000). According to the NCTM's Principles and Standards for School Mathematics (2000), "those who understand and can do mathematics will have significantly enhanced opportunities and options for shaping their futures. A lack of mathematical competence keeps those doors closed" (p. 5).
Unfortunately, the mathematics performance of fourth- and eighth-grade students with disabilities who took the 2007 National Assessment of Educational Progress (NAEP; Lee, Grigg, & Dion, 2007) continues to lag behind that of their typically achieving peers even when accommodations are permitted in the testing situation. Much like current practices in early reading instruction, the achievement gap of students with mathematics disabilities compared to their typically achieving peers will remain problematic without preventive practices initiated in the primary grades (i.e., kindergarten, first, and second grade). Preventive practices should include evidence-based critical features of instruction (i.e., instructional design) that help these students access the core or primary mathematics curriculum and instruction typically found in general education classrooms. "Access to the general education curriculum" refers to students with learning difficulties receiving and benefiting from evidence-based instruction that is designed, delivered, and evaluated for effectiveness (D. Bryant, Smith, & Bryant, 2008).
While students are not usually identified as having mathematics disabilities in the primary grades, recent studies have identified procedures to determine students who are at risk for mathematics difficulties at a young age (e.g., kindergarten, first, and second grade) (B. Bryant, Bryant, Gersten, Scammacca, & Chavez, in press; B. Bryant & Bryant, 2007; L. S. Fuchs et al., 2007; Jordan, Kaplan, Olah, & Locuniak, 2006). As part of the Individuals with Education Improvement Act (IDEA, 2004), the Response to Intervention (RtI) process allows schools the opportunity to identify young children who are struggling with the core instruction and to provide secondary interventions in hopes of remediating academic weaknesses and preventing learning failure (D. Fuchs & Deshler, 2007). Core or primary instruction should include the critical features of effective instruction to enhance the ability of students at risk for mathematics difficulties to learn the core mathematics instruction.
Critical Features of Core Instruction for At-Risk Students
A key ingredient of the RtI process is the provision of high-quality core classroom instruction that is based on research (Mellard, 2004). Core mathematics instruction should be responsive to the needs of all students and include instructional design features that have been found to be critical for at-risk students. For example, a meta-analysis of academic treatment outcomes, including mathematics, for students with learning disabilities (LD) identified the positive contribution (i.e., higher effect sizes) of a combined method of instruction consisting of explicit and strategic instructional procedures compared to other instructional approaches (Swanson, Hoskyn, & Lee, 1999). Features of the combined method included sequenced subskills, instruction on prerequisite skills, multiple practice opportunities, small groupings, feedback, procedural strategies, and progress monitoring.
It stands to reason that these more general procedures could be used with younger students who are experiencing mathematics difficulties. Additionally, in the area of mathematics, the use of manipulatives to represent mathematical concepts concretely is as an effective practice for all students, and particularly so for students with mathematics difficulties (Marsh & Cook, 1996; Miller & Mercer, 1993b). Finally, Gersten, Jordan, and Flojo (2005) recommended that instruction for students with mathematics difficulties include procedures to help them learn the vocabulary of mathematics. Kindergarten teachers who participated in a focus group study on mathematics instruction supported this recommendation. Specifically, these teachers emphasized the importance of vocabulary knowledge in the early mathematics curriculum and the difficulties struggling students demonstrate with learning and applying the language (vocabulary) of mathematics instruction (D. Bryant, Bryant, Kethley, Kim, & Pool, 2004). Thus, teachers need to help students make connections among new vocabulary and prior knowledge and provide multiple opportunities to engage students in meaningful ways to apply the vocabulary across situations (D. Bryant, 2005).
It is essential that general education teachers who are teaching young students with risk status for mathematics difficulties employ evidence-based instructional practices found to improve mathematical performance and preventing learning problems (D. Bryant et al., 2008). In the general education classroom, the mathematics textbook or basal is an important component of early mathematics education. Textbooks play a crucial role in what teachers do during instruction (Nathan, Long, & Alibali, 2002). Further, how the lessons are implemented and supplemented by the classroom teacher has a major influence on student learning (Sood & Jitendra, 2007).
In recent years, core reading programs have undergone considerable scrutiny to determine the presence of evidence-based practices (Simmons & Kame'enui, 2000). Carnine (1991) called for a similar critical review of mathematics textbooks. Unfortunately, such a review showed that "A close look at traditional basals suggests that publishers are not meeting their responsibilities to assist teachers in providing suitable development for students" (Carnine, 1991, p. 55). In the years since, several mathematics textbooks examinations have been conducted, as outlined below.
Mathematics Textbook Evaluation
A number of mathematics textbook evaluations have been conducted to examine the extent to which components of effective instructional design (i.e., critical features of effective instruction) are present as well as the extent to which textbooks include instructional content that reflects trends (e.g., reform-based mathematics instruction, "number sense") in mathematics education. Additionally, evaluations have focused on a particular grade (e.g., fourth grade, fifth grade) and content area (e.g., division, addition, subtraction).
Findings from these evaluations have been interpreted in terms of the instructional implications for students with mild disabilities. For example, Jitendra, Carnine, and Silbert (1996) examined fifth-grade basal instruction for teaching division. They looked at two basal textbooks published before and after the NCTM Standards (1989) in light of nine components of effective instruction (prior knowledge, introducing new concepts, coherence, clarity of teacher communication, manipulative activities, guided practice, initial practice, later practice, and review) to determine the extent to which the elements of effective instruction were built into division lessons. These researchers concluded that "the inadequacy of traditional basals to meet the needs of most students will continue to widen the gap between students with mild disabilities and the non-disabled population" (Jitendra et al., 1996, p. 401).
In another study, Carnine, Jitendra, and Silbert (1997) used what they termed "pedagogical criteria" to analyze three fifth-grade basal programs for teaching adding and subtracting fractions. Pedagogical criteria were described as fundamental concepts and principles, big ideas, pacing of instructional content introduction, teaching demonstrations, manipulative activities, and review. The researchers viewed these components as particularly relevant when dealing with students who have mild disabilities. Carnine et al. concluded that their analysis of traditional basals resulted in disturbing findings and that the lack of pedagogical criteria should be of concern to teachers of diverse learners.
In yet another textbook evaluation, Jitendra, Salmento, and Haydt (1999) examined fourth-grade subtraction instruction. Using nine components of instructional design (clarity of objective, additional concepts and skills taught, prerequisite skills taught, explicit teaching explanations, efficient use of instructional time, sufficient and appropriate teaching examples, adequate practice, appropriate review, and effective feedback), they evaluated seven math basals to determine the extent to which the components were included in the subtraction lessons. Only two of the basals incorporated most (seven or eight) of the instructional components examined. Further, only two components (clarity of objective and number of additional concepts) were present across all of the basals. Based on these findings, Jitendra et al. concluded that students with learning disabilities will need instructional adaptations if they are to benefit sufficiently from typical textbook-based, fourth-grade subtraction instruction.
The adoption of standards-based mathematics instruction (i.e., NCTM, 2000), inspired by lackluster student performance on national assessments, initiated efforts to focus instruction on higher order thinking and problem solving. An inquiry-based approach to instruction (e.g., discovery approach) was embraced as an effective way to help students construct their understanding of mathematical relationships and share their mathematical reasoning and solutions (Baxter, Woodward, & Olson, 2001). In this type of learning environment, students assume responsibility for organizing and integrating their learning experiences. Jitendra et al. (2005) examined five third-grade mathematics textbooks to determine the extent to which the textbook publishers adhered to the emphasis on problem solving espoused in the Standards (NCTM, 2000) and addressed instructional design features that are critical for students with learning disabilities.
The authors found that while problem-solving opportunities were typically present, textbooks were less likely to include activities for students to generate representations or identify problem-solution representations. In the area of instructional design, the authors noted improvements in the extent to which the design features occurred compared to their previous studies. However, such improvement was found in only three out of the five textbooks reviewed.
In a more recent analysis, Sood and Jitendra (2007) reviewed...
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