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Article Excerpt
Abstract. Responsiveness to Intervention (RtI) is recommended both as an essential step before identifying learning disabilities (LD) and as a mechanism for preventing learning difficulties. The use of evidence-based multi-tiered interventions is of critical importance when implementing RtI. This article presents the results of a study that examined the effects of Tier 2 intervention on the performance of first-grade students who were identified as at risk for mathematics difficulties. Participants included 161 (Tier 2, N = 42) first graders. Tier 2 students received 20-minute intervention booster lessons in number and operation skills and concepts for 23 weeks. Results showed a significant intervention effect on the Texas Early Mathematics Inventories-Progress Monitoring (TEMI-PM, University of Texas System/Texas Education Agency) total standard score.
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There is a growing interest in early mathematics difficulties, stemming in part from prevalence figures indicating that 5% to 10% of school-age children exhibit mathematics disabilities (L. Fuchs, Fuchs, & Hollenbeck, 2007; Gross-Tsur, Manor, & Shalev, 1996; Ostad, 1998). The reauthorization of the Individuals with Disabilities Education Improvement Act (2004) supports the use of Response to Intervention (RtI) as a way of identifying students with learning disabilities (LD), including students who may have LD in mathematics. Initially conceptualized by Heller, Holtzman and Messick (1982), and further developed by Fuchs and Fuchs (1998), Fuchs, Fuchs, and Speece (2002), and Vaughn and Fuchs (2003), RtI holds promise as an alternative to more traditional approaches to LD identification and as a means to improve procedures associated with prevention and remediation (e.g., implementation of validated practices and assessment of student response to treatment).
Briefly, the RtI approach is characterized by (a) a high-quality general education program that includes universal screening procedures to identify students at risk for academic difficulties, (b) secondary intervention consisting of a standard, evidence-based treatment protocol with progress monitoring for a specified duration, and (c) tertiary intervention that is more intensive and tailored to individual student needs (Fuchs, Mock, Morgan, & Young, 2003; Vaughn & Fuchs, 2003).
Tier 1 is characterized by implementation of evidence-based core instruction for all students (Chard et al., 2008; L. Fuchs, Fuchs, Yazdian, & Powell, 2002). Tier 2 includes intervention to prevent further mathematics difficulties with ongoing progress monitoring to assess response to treatment for students who are identified with risk status in early mathematics skills and concepts. In mathematics, Tier 2 intervention consists of small-group, explicit and systematic instructional procedures incorporating concrete-representation-abstract sequences (Miller & Hudson, 2007) with a fixed duration of instruction. Tier 3, or tertiary instruction, is reserved for students who are struggling to the extent that they require more intensive intervention than a small-group session conducted in their classroom 3-5 days a week.
To date, a multi-tiered prevention and intervention model for operationalizing RtI has been applied in early reading (e.g., Vaughn, Linan-Thompson, & Hickman-Davis, 2003) and, to some extent, in early (primary level) mathematics instruction (D. Bryant, Bryant, Gersten, Scammacca, & Chavez, 2008; L. Fuchs et al., 2007). More research is needed in early mathematics (Chard et al., 2005; L. Fuchs et al., 2005; Gersten, Jordan, & Flojo, 2005).
Measures for screening and progress monitoring are increasingly available for schools (e.g., B. Bryant, Bryant, Gersten, Wagner, Roberts, Kim et al., 2008; Chard et al., 2005; L. Fuchs et al., 2007; VanDerHeyden, Witt, Naquin, & Noell, 2001) to identify students at risk. An emerging body of research on young children's mathematics cognition and the way they learn early mathematics concepts is contributing to our understanding of the early numeracy skills that prove problematic for students at risk for mathematics disabilities and should serve as the core of screening measures (Fuchs & Fuchs, 2001; L. Fuchs et al., 2005; L. Fuchs et al., 2007; Geary, Hamson, & Hoard, 2000; Jordan, Kaplan, & Hanich, 2002; Jordan, Kaplan, Olah, & Locuniak, 2006).
Research results have indicated that students with early mathematics problems exhibit difficulties understanding number sense as demonstrated in number knowledge and relationships activities (e.g., magnitude, sequencing, base ten) (Jordan et al., 2006); solving word problems (L. Fuchs et al., 2007); and using efficient counting and calculation strategies (e.g., counting on, doubles + 1) to solve arithmetic combinations (i.e., number facts) (D. Bryant et al., in press; L. Fuchs, Fuchs, Hamlet, Powell, Capizzi, & Seethaler, 2006). Findings from studies in these areas informed the design of the preventive intervention practices described in this article, specifically in the area of number sense (number knowledge and relationships, base ten) and arithmetic combinations.
Number Sense
For young students, developing number sense of mathematical concepts and mastery and fluency with arithmetic combinations is critical. Number sense is defined as "moving from the initial development of basic counting techniques to more sophisticated understandings of the size of numbers, number relationships, patterns, operations, and place value" (National Council of Teachers of Mathematics [NCTM], 2000, p. 79). Gersten and Chard (1999), Gersten et al. (2005), and Okamoto and Case (1996) further operationalized number sense as the ability to understand the magnitude of numbers, the ability to use representations, and ease of use with mental computation.
Number sense components. Jordan et al. (2006) identified a broader array of number sense components in their kindergarten assessment battery, including counting (e.g., counting sequence, counting principles); number knowledge (e.g., quantity discrimination); number transformation (e.g., addition and subtraction verbal and nonverbal calculations); estimation (e.g., of group size using reference points); and number patterns (e.g., extending number patterns, discerning numerical relationships).
According to Jordan et al., these skills relate to the primary-level mathematics curriculum and have been validated as important for developing early mathematics concepts in young children (e.g., Griffin, 2004; Griffin & Case, 1997). For example, studies have shown that many children enter kindergarten understanding counting principles, such as one-to-one correspondence and the cardinality principle, and acquire more advanced counting skills (e.g., counting backwards, counting objects in groups, counting by 10s) in the primary grades (Gelman & Gallistel, 1978; Jordan et al., 2006).
However, young students with mathematics problems have difficulty with the conceptual understanding of some counting principles (e.g., order irrelevance), and counting difficulties affect the use of more advanced counting abilities (e.g., counting on: 8 + 2 = 11) to solve arithmetic combinations (Case & Okamoto, 1996; Geary, 2004; Griffin, 2004). Number knowledge represents the ability to understand the concept of quantity; that numbers have magnitude and that this magnitude relates to a counting sequence. Importantly, number knowledge has been linked to arithmetic achievement in first grade (Baker et al., 2002). Students use their understanding of number knowledge to develop a "mental number line" (i.e., linear increases of magnitude) to solve calculations "in their heads" and to comprehend place value (Jordan et al., 2006; Siegler & Booth, 2004). Thus, students begin to integrate their conceptual understanding of counting with quantity (Griffin, 2004).
Importance of place value. Conceptual understanding and conceptual proficiency for whole numbers--the base-ten system (i.e., place value, computation)--is an important component of mathematics instruction that students must fully grasp (Van de Walle, 2004). Place value understanding can be developed by building connections between important features of instruction, such as grouping objects by 10 and units and using written notations (e.g., numerals) to convey information about the groupings (e.g., 3 groups of 10 and 4 units = 34) (Hiebert & Wearne, 1992).
According to Ross (1989), there are five levels of place value understanding, as follows.
* Single numeral: Individual digits in numerals such as 52 are not understood as representing specific values in the number. Instead, 52 is merely a single numeral.
* Position names: The student can name the position of the digits, for example, in 52, 5 is in the tens place and 2 is in the ones place, but does not associate value with the position.
* Face value: Each digit is taken at face value. In 52, the student selects 5 blocks to make up the 5 and 2 blocks make up the 2. The value of the position is not understood.
* Transition to place value: In 52, 2 blocks are selected for the ones place and the remaining 50 blocks are selected for the 5; no grouping of tens is demonstrated.
* Full understanding: In 52, 5 groups of 10 are selected, and 2 remaining blocks are chosen for the 2.
Unfortunately, evidence suggests that students do not learn place value concepts sufficiently to understand procedures for multi-digit calculations. Consequently, some students solve computational problems correctly but lack the conceptual understanding of what they are doing (Fuson, 1990).
Jordan, Hanich, and Kaplan (2003) conducted a longitudinal study of 180 students in second grade and followed them to third grade. Students were administered a battery of tests designed to assess performance on a variety of early mathematics tasks that included place value. Place value tasks involved problems with standard (e.g., 43 = 4 tens and 3 ones) and nonstandard (e.g., 43 = 3 tens and 13 ones) place value and digit representations (e.g., 43: show with concrete models what 4 stands for; count out 40 chips). Jordan et al. found that over time students with mathematics difficulties scored lower on place value tasks than average students.
These findings suggest that students with mathematics difficulties require sustained instructional time in place value concepts beginning in the early grades and continuing...
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