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Article Excerpt
Abstract. Competence in algebra is linked to access to higher education, employment in better-paying jobs, and, increasingly, the ability to earn a high school diploma. For many students with learning disabilities, developing proficiency in algebra represents a challenging, but necessary goal. Teachers of students with learning disabilities need access to assessment tools and instructional strategies that support algebra learning. This article reports research on a group of measures designed to monitor student progress in algebra and highlights findings specific to students with disabilities. In addition, evidence-based instructional strategies for algebra are summarized. Implications for practitioners and future research are discussed for both progress monitoring assessment tools and algebra instructional practices.
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Proficiency in mathematics is strongly associated with students' access to higher education and quality employment (U.S. Department of Education, 1997). That is, students who complete advanced mathematics courses, such as algebra, are more likely to succeed in college and obtain better-paying jobs (Cavanagh, 2007) than those who don't. The importance of higher standards for mathematics, and proficiency in algebra in particular, is evident in changing graduation requirements. Currently, 24 states require Algebra I or will have such a requirement in place in the next three years (Dounay, 2007).
National and international assessments across multiple years have highlighted the desperate need for more effective teaching and learning of mathematics in general, and algebra in particular (Carpenter et al., 1981; Silver & Kenney, 2000; U.S. Department of Education, 1997). For students with disabilities, reports of mathematics achievement are particularly discouraging. The National Longitudinal Transition Study-2 (Wagner, Newman, Cameto, & Levine, 2006) found that more than half of high school students with disabilities demonstrated mathematics computation and problem-solving levels below the 25th percentile on an individually administered achievement test. Results of the 2005 National Assessment of Education Progress (NAEP) Mathematics assessment revealed that 69% of eighth-grade students with disabilities in the sample performed at the "below basic" level, while only 28% of nondisabled students performed at this level (Perle, Grigg, & Dion, 2005). Similar results are found when looking specifically at achievement levels in algebra. More than 75% of eighth-grade students with disabilities earned a scale score on the Algebra and Functions strand of NAEP Mathematics that was below the mean score for the full sample (National Center for Education Statistics, n.d.).
The underlying causes for these difficulties are not clear. Gersten, Clarke, and Mazzocco (2007) observed that no consensus exists on the components that contribute to mathematics difficulties. In an effort to identify possible causes, investigations of children's mathematics difficulties have spanned a diverse range of fields and theoretical perspectives (Berch & Mazzocco, 2007), including behavioral genetics (Petrill & Plomin, 2007), neuropsychology (Zamarian, Lopez-Rolon, & Delazer, 2007), and cognitive science (Butterworth & Reigosa, 2007). Proposed mechanisms include deficits in general cognitive abilities (e.g., working memory capacity, strategy selection) (Geary, Hoard, Nugent, & Byrd-Craven, 2007), and domain-specific cognitive abilities such as recognizing "numerosities" and comparing quantities (Butterworth & Reigosa, 2007).
However, the vast majority of this work has focused on students' initial development of mathematical thinking and has been conducted almost exclusively with mathematics topics and content typical of elementary school classrooms. Geary et al. (2007) noted that the bulk of the work to date has focused on basic number concepts and simple arithmetic, with little attention to conceptual understanding and even less research in other mathematical domains. Bull (2007) commented that "researchers still shy away from trying to pinpoint the cognitive skills supporting complex tasks like geometry and algebra" (p. 270).
Despite the absence of specific theories about the sources of students' difficulties with algebra, the existing literature does suggest potential avenues for future investigation. For example, Hecht, Vagi, and Torgesen (2007) described a line of research investigating students' understanding and computational skill with fractions. They suggested that difficulties with problems involving rational (e.g., fraction) numbers are associated with a separation between conceptual understanding and fraction problem-solving procedures. This proposition is consistent with research by Siegler (1996) illustrating the interrelationships between conceptual knowledge and procedural knowledge. Rittle-Johnson, Siegler, and Alibali (2001) asserted that conceptual knowledge facilitates effective selection and execution of procedures, while the use of the procedures affords students an opportunity to refine their knowledge of mathematical concepts. It is likely that the mechanisms that underpin competence in algebra show a similar interrelationship between conceptual understanding and the efficient and accurate selection and execution of problem solving procedures.
The challenge of learning algebra is obvious to students with and without disabilities. Thus, when surveyed about their perceptions, students with learning disabilities were more likely than their peers (55% vs. 32%) to identify mathematics as their least favorite high school class (Kotering, deBettencourt, & Braziel, 2005). In the same study, students with learning disabilities identified providing more assistance, altering typical teaching styles, incorporating group work, and increasing the interest level of the instruction as teacher strategies that would assist them in improving their performance.
As schools respond to federal and state mandates for more challenging instructional curricula and more highly qualified teachers, increasing numbers of students with learning disabilities are receiving their mathematics instruction in general education classrooms from general education teachers or from a co-teaching pair of teachers consisting of a general education teacher and a special education teacher.
Maccini and Gagnon (2006) conducted a national survey of secondary general and special education teachers who taught mathematics to students with disabilities. They found that special education teachers often lacked sufficient content preparation relative to the demands of the high school curriculum. At the same time, general education teachers were less likely than their special education colleagues to implement recommended instructional practices or assessment accommodations for students with disabilities. These findings are consistent with earlier work by Schumaker et al. (2002), who conducted extensive descriptive studies in nine high schools across four states using classroom observations, as well as staff, student, and parent interviews/questionnaires. Schumaker et al. found that only one of the nine high schools that they studied was using evidence-based methods to instruct students with disabilities in general education classrooms. Not surprisingly, this school obtained the highest levels of staff and student satisfaction ratings.
If students with learning disabilities are to succeed in algebra, the use of evidence-based practices for assessment and instruction must become standard practice. Educators need effective tools for tracking student learning and determining when instructional changes are needed. They also need proven strategies for providing supplemental instruction in algebra when students experience difficulty. This article reports on an emerging approach to monitoring student progress in algebra and presents evidence-based strategies for enhancing the algebra learning of students with disabilities.
MONITORING STUDENT PROGRESS IN ALGEBRA
Progress monitoring (also termed curriculum-based measurement or general outcome measurement) is an empirically developed approach to formative evaluation that relies on frequent assessment using brief measures that serve as indicators of general proficiency in a content area. Originally developed by Stan Deno and his colleagues at the University of Minnesota (Deno, 1985), curriculum-based measurement strategies for basic skills at the elementary level have expanded to explore progress monitoring tools for early literacy (Kaminski & Good, 1996; Lembke, Deno, & Hall, 2003; McConnell, McEvoy, & Priest, 2002) and secondary content-area instruction (Busch & Espin, 2003; Espin, Shin, & Busch, 2005). Critical features of all of these tools include the use of frequent assessment with brief (often 1- to 5-minute) measures that have empirically documented levels of reliability and criterion validity. Most important, the measures are intended to provide teachers with objective data on student performance that can be used to track progress and indicate the need for instructional changes when students are not progressing at acceptable levels. An extensive research base documents the technical features of the measures and their use for improving student performance (see McMaster & Espin, 2007; Stecker, Fuchs, & Fuchs, 2005; Wayman, Wallace, Wiley, Ticha, & Espin, 2007).
In the area of mathematics, the bulk of the research in progress monitoring has been conducted in the elementary grades (Foegen, Jiban, & Deno, 2007). Although some extensions of this work exist at the preschool and middle school levels, Foegen et al. (2007) were unable to identify any published studies of progress monitoring tools designed for high school or advanced mathematics content.
To address this gap, my colleagues and I have been engaged in a three-year project to develop and validate progress monitoring tools for Pre-Algebra and initial...
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