Algebra progress monitoring and interventions for students with learning disabilities.

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 Algebra 1 courses (Project AAIMS; Foegen, 2003). This project has investigated multiple types of algebra progress monitoring tools using an iterative research process to refine the measures (Foegen, Olson, & Perkmen, 2005a, 2005b; Perkmen, Foegen, & Olson, 2006a, 2006b).

Algebra Progress Monitoring Measures

Four algebra progress monitoring measures were found to have sufficient levels of technical adequacy to serve as static indicators of student proficiency. Described in greater detail by Foegen, Olson, and Impecoven-Lind (in press), a brief summary of these measures is presented in Table 1. (1) The measures were created to reflect two different approaches to the design of progress monitoring tools (Foegen et al., 2007; Fuchs, 2004). Three of the four measures (Basic Skills, Algebra Foundations, Translations) represent "robust indicators," or more general representations of proficiency in algebra. In addition, the difficulty level of the items in these measures is more closely aligned with pre-algebra content than with formal high school algebra. The fourth measure (Content Analysis) was designed by sampling key skills and concepts from the chapters of a commonly used algebra textbook. The items on this measure provide a more direct and comprehensive representation of the instructional content of most Algebra 1 courses. Further, three of the four measures (all but Translations) tend to emphasize the manipulation of algebraic symbols that is commonly associated with traditional approaches to algebra instruction. The Translations measure was designed to avoid this emphasis on symbolic manipulation and, instead, to emphasize more conceptual understandings of algebra by requiring students to translate between alternative forms of representing the relationships between two variables (equations, data tables, graphs, and story scenarios).

Results from Two Progress Monitoring Studies: Findings for Students with Disabilities

The bulk of the research for Project AAIMS was conducted in general education classes that included students with disabilities, and the research reports completed thus far have not specifically disaggregated results for these students. Although a comprehensive research report is beyond the scope of this article, two of the studies completed as part of the project will be presented along with two sets of results for each study: those obtained for the full sample and those obtained for students with disabilities.

Because Project AAIMS took place in a state that uses non-categorical identification of students with disabilities, it is not possible to differentiate students by specific disability type. Our experiences working in participating schools over three years have led us to conclude that the majority of the students with disabilities who were enrolled in algebra courses would likely be labeled as having a learning disability and/or a behavior disorder in a categorical state. The two studies took place during the 2005-06 academic year, each in one of the three districts that participated in Project AAIMS. More complete reports of the studies and their results are reported in Perkmen et al. (2006a, 2006b). The participants, measures, procedures, and results for the two studies are described concurrently in the sections that follow.

Context and Procedures

District A serves students from four small midwestern towns and the rural agricultural areas between them. The student population of the district is predominantly White (97%); 18% of the district's students are eligible for free/reduced-cost lunch. The junior/senior high school enrolls 600 students in grades 7 to 12 and has a seven-period day, with each instructional period lasting approximately 45 minutes. Nine of the 11 students with disabilities were enrolled in general education algebra classes taught by general education teachers; 2 students with more intensive needs were enrolled in an algebra course taught by a special education teacher. The IEP teams for these students had decided that a small-group setting permitting more individualized instruction in algebra was the most appropriate placement for these students, who required instruction at a slower pace than could be accommodated in general education. (2)

District B serves students from a midwestern community of 26,000 students. The four-year high school enrolls approximately 1,300 students; 82% of these students are White, and 47% are eligible for free/reduced-cost lunch. All 15 students with disabilities were taking general education algebra classes, some of which were co-taught by special education teachers. The demographic characteristics of the participants in the two studies are reported in Table 2.

Three of the four algebra progress monitoring measures mentioned earlier were used in the studies. In both districts, teachers administered two types of measures each month (with minor deviations for school holidays), using two forms of one measure during the middle of the month and two forms of the other measure at the end of the month. Students' scores from the two forms were averaged for the analyses. General education teachers in District A collected progress monitoring data from September through April, alternating between the Algebra Foundations and the Content Analysis measures. The special education teacher alternated between the Algebra Foundations and the Basic Skills measures. In District B, general education teachers alternated between the Basic Skills and the Content Analysis measures, gathering data from September through mid-January, when their block-scheduled courses were completed.

Data were also collected on criterion measures, including both classroom-based indicators and standardized test scores. Classroom-based measures included teacher ratings of student proficiency in algebra and course grades in algebra. The single-item teacher rating scale required teachers to rate each student's proficiency in algebra in comparison to that of typical peers using a five-point scale; we administered this scale about one month into the school year. Course grades represented the student's final grade in the course and was converted from a letter grade scale (e.g., A, A-, B+) to a four-point numerical scale (e.g., A = 4.0. A- = 3.67).

The standardized test data included scores from the Iowa Tests of Educational Development (ITED) and the Iowa Algebra Aptitude Test (IAAT). The ITED is administered annually by the district for accountability purposes; we used students' national percentile ranks on the Total Math score in our analyses. Equivalent forms of the IAAT were administered at the beginning and the end of each course to examine students' growth on an external measure relative to their changes on the algebra progress monitoring measures. The IAAT is an aptitude test, rather than an achievement test: we were unable to identify a suitable norm-referenced achievement test of Algebra 1 content.

Results of the Studies: A Focus on Students with Disabilities

Because teachers' use of progress monitoring assumes that these measures reflect important outcomes and represent changes in student learning, we have chosen to limit our focus to results related to criterion validity and growth. Readers interested in a more complete report of the findings are referred to Perkmen and colleagues (2006a, 2006b). Criterion validity analyses included both concurrent validity and predictive validity. The results for all students and for the subgroups of students with IEPs are reported in Table 3.

Concurrent validity coefficients reflect the relations between algebra progress monitoring measures administered in the fall or spring and criterion measures gathered at the same points in time. Predictive validity coefficients represent relations between fall scores on the algebra progress measures and criterion measures obtained in the spring. For the full samples, most coefficients were in the moderate range (r = .4 to .6). Among students with IEPs, most correlations involving the teacher rating scale were not statistically significant. This was not surprising given the small number of cases and the limited range (1 to 5) for this scale. On the remaining criterion variables, concurrent validity coefficients for students with disabilities tended to be comparable to, if not stronger than, those obtained for the full sample of students. Two exceptions to this pattern were the post-IAAT scores for students who had completed the Algebra Foundations measure in District A and the Content Analysis measure in District B.

A different pattern of results emerged from the predictive validity coefficients. Whereas the findings in District B were roughly similar, with many coefficients for students with IEPs comparable to or larger than those for the full sample, none of the coefficients for students with IEPs in District A were statistically significant. Although the criterion validity results must be interpreted with caution given the small sample numbers of students with IEPs, the data support a tentative conclusion that the measures work about as well for students with disabilities as they do for students in general when measuring outcomes at the same point in time. Further research is needed before the measures may be used with confidence in a sample of students with disabilities to predict future performance.

In addition to serving as static indicators of performance, algebra progress monitoring measures must also be sensitive to student learning changes and reflect varying levels of performance over time. Because our studies involved gathering multiple data points across time, we were able to examine the slopes produced by these data and estimate the typical amount of growth students demonstrated on each type of measure. Estimated weekly rates of growth for the full samples and for students with IEPs are reported in Table 4. For each type of algebra measure, the rates of growth are listed for students by class type, reflecting the four types of classes represented in the study.

Students in eighth-grade Algebra were advanced students who had been selected to enroll in algebra one year ahead of the typical mathematics sequence; all eighth-grade algebra students were in District A. Algebra 1 was a traditional high school algebra class, whereas Algebra 1A and 1B represented options available to students who wanted to complete algebra at a slower pace. The Algebra 1A course included the first half of the content typically taught in Algebra 1, but extended the teaching of this content over the full span of a course. After completing Algebra 1A, students typically enrolled in Algebra 1B, which represented the second half of Algebra 1 content, again taught as a full course (year-long in District A's traditional schedule and semester-long in District B's block schedule). These data reflect typical instruction. Teachers had only limited access to student progress monitoring data, and no attempts were made to impose interventions in response to student data.

For the full samples, weekly rates of growth ranged from .32 to .87 points correct. In most cases, the rates of growth corresponded to the difficulty level of the course, with eighth-grade Algebra having the highest rates of growth, followed by Algebra 1, then Algebra 1B and 1A. The one exception to this pattern was the Content Analysis measure in District B, where Algebra 1A students demonstrated substantially higher rates of growth than their Algebra 1B peers. As might be expected, without any instructional interventions, students with IEPs had comparable or lower rates of growth in most cases the Basic Skills growth rates of most (Algebra 1B in District B was the lone exception). The data suggest that the measures are likely to be useful for monitoring student progress among typical students, as well as those with disabilities. At a minimum, all students were improving (even without targeted interventions) at approximately .25 points correct per week, which would allow teachers to detect monthly improvements in progress.

Acting on Progress Monitoring Data

The research conducted to date on the algebra progress monitoring measures suggests that these tools may have sufficient technical adequacy to be used as indicators of student development in algebra. Clearly, more research evidence is needed for the measures to be appropriate for high-stakes decisions regarding students. As research continues and the algebra progress monitoring measures are refined, teachers will be able to use them to monitor the progress of students with learning disabilities and make timely changes to their instruction when the data reveal that students are not making acceptable levels of progress.

However, with implementation of a progress monitoring system in algebra comes the professional obligation to take action when the data suggest that a student is not making sufficient progress. Teachers need more than measures to serve as indicators of student performance and learning trajectories; they also need evidence-based practices to implement when current instructional methods are not producing desirable results.

ALGEBRA INTERVENTIONS FOR STUDENTS WITH DISABILITIES

The section that follows focuses on the published peer-reviewed research conducted to date on interventions specific to algebra, presented by the type of instructional approach used. Approaches related to cognitive strategy instruction are described first, followed by those involving a concrete-representational-abstract progression. The section concludes with strategies involving classwide peer tutoring and graphic organizers.

Cognitive Strategy Instruction

Hutchinson conducted some of the earliest work on algebra instruction for students with learning disabilities in the late 1980s and early 1990s. Hutchinson's (1987) approach drew from work in special education on cognitive strategy instruction (Deshler, Alley, Warner, & Schumaker, 1981) with adolescents and from Montague and Bos' (1986) approach to teaching the solution of two-step word problems.

In addition, Hutchinson (1993) asserted that solving complex problems in algebra requires students to successfully complete two phases of activity. First, students must represent the problem, translating the information given in written format into a mental structure or idea that holds mathematical meaning for the individual student. In this phase, Hutchinson taught students to attend to the mathematical structure of three types of problems (relational, proportion, and two-variable two-equation), rather than the "surface structure," or specific context of the problem. The second phase of instruction centered on problem solution, which included both planning how to solve the problem and executing the procedures necessary to do so. Hutchinson imbedded instruction in each phase within a context of cognitive strategy instruction in which students were taught to use self-questioning to guide them through the process. Instruction began with teacher modeling and think-alouds, followed by guided practice with teacher support, assistance, and feedback. As students gained proficiency in using the strategy, they completed independent practice activities and received feedback on their performance.

In her 1993 study, Hutchinson used cognitive strategy instruction to teach 12 students with mathematics learning disabilities to solve the three types of algebra word problems, meeting with students for individual, 40-minute sessions every other day for four months. A control group of eight students received conventional algebra instruction. Students in the treatment group moved through the instructional program at their own pace and were required to meet a proficiency criterion in one phase (i.e., representation of relational problems) before moving to the next (i.e., solution of representational problems).

The results of the study revealed positive improvements in problem representation and solution on the problem types for which students had received instruction. Analysis of think-aloud data from students as they solved problems revealed their use of the instructed strategy. Comparisons of students in the treatment and comparison group revealed significantly higher scores for the cognitive strategy instruction group on the posttest.

Methods Incorporating a Graduated Instructional Sequence

Subsequent work in algebra instruction for students with disabilities involved cognitive strategy instruction, but also incorporated the use of a graduated teaching sequence that proceeded along a continuum from using concrete materials for solving problems to using representational formats (i.e., drawings) to, finally, using more abstract or symbolic mathematical representations. This approach, which is often referred to as CSA (concrete-semiconcrete-abstract), or CRA (concrete-representational-abstract), was used successfully by Miller and Mercer (1992, 1993) to teach basic math facts and associated problem-solving strategies to elementary-level students with learning disabilities.

Maccini and her colleagues (Maccini & Hughes, 2000; Maccini & Ruhl, 2000) provided individual instruction to students using an algebra problem-solving strategy for problems involving subtraction of integers. Based on the mnemonic STAR, the strategy guides students to first Search the word problem by reading it carefully, and then Translate the words into an equation in picture form, choose the correct operation, and represent the problem in an appropriate format (concrete objects when in the concrete phase of instruction, drawings when in the semi-concrete phase, and algebraic symbols when in the abstract phase). In the concrete and semi-concrete phases, students represent the problems, but are not required to obtain solutions. When students have successfully demonstrated their ability to translate algebra problems into equation formats at the concrete and semi-concrete formats, they next learn to Answer the problem using rules for addition and subtraction of integers and to Review the solution by checking their answer.

Maccini and Ruhl (2000) noted that the STAR strategy was taught using a process consisting of teacher modeling, guided practice with feedback, and independent practice similar to that described above in Hutchinson's (1987, 1993) application of cognitive strategy instruction. Specifically, in the concrete phase, students used Algebra Lab Gear (Picciotto, 1990), a commercially available algebra instructional program that incorporates manipulatives. During the semi-concrete phase, students drew pictures of the Algebra Lab Gear tiles to represent the problems.

Maccini and Ruhl (2000) used a single-subject, multiple-probes-across-subjects design to evaluate the effects of STAR instruction on three eighth-grade male students with mathematics learning disabilities. Using 1:1 instructional sessions of 20-30 minutes per lesson, all students learned to accurately complete computation and word problems involving integers. Students demonstrated limited generalization to near- and far-transfer tasks, but obtained scores of 90% accuracy or better on maintenance probes administered up to six weeks after the intervention ended.

In a similar study, Maccini and Hughes (2000) provided individual instruction in computation and problem solving with integers across all four operations (addition, subtraction, multiplication, division). Instructional sessions were again 20 to 30 minutes long, with total instructional time of 4 to 6 hours. All six participating students learned to solve integer word problems using the operation of addition. Five of the six learned to solve integer problems requiring subtraction, multiplication, and division. Study data revealed that students used the strategy to accurately represent and solve integer word problems; maintenance measures administered up to 10 weeks following the intervention revealed students had a 91% accuracy rate on problem solution.

Other researchers have examined the merger of strategy instruction and a CRA teaching sequence for use in general education classrooms. For example, Witzel and his colleagues (Witzel, 2005; Witzel, Mercer, & Miller, 2003) have used researcher-designed manipulatives (string, cups, toothpicks) for the concrete phase and simple drawings of the same materials for the representational phase. Witzel et al. (2003) worked with 10 general education teachers and over 350 sixth- and seventh-grade middle school students in inclusive mathematics classes. Teachers delivered 19 fifty-minute lessons on algebra concepts, including reducing expressions, inverse operations, negative and divisor variables, transformations on one side of the equal sign, and transformations across the equal sign. Classes were assigned to either the treatment or the comparison group. The instructional curriculum and pacing were identical across conditions. The only difference was that the comparison group's lessons were conducted at the abstract level, while teachers in the treatment group provided instruction first at the concrete level and then at the representational level before moving to abstract representations. Analyses of data drawn from 34 matched pairs of students from the treatment and comparison group revealed that students who received CRA instruction achieved posttest scores that were significantly higher than those of their matched peers in the comparison group.

In a subsequent analysis of the full set of data (not limited to the 34 matched pairs reported in the earlier paper), Witzel (2005) examined pretest, posttest, and follow-up test scores of 231 students enrolled in inclusive middle school Algebra 1 courses. He found that students in the experimental condition using a CRA sequence outperformed peers in the comparison condition in which all instruction was provided at the abstract, or symbolic, level on both the posttest and the follow-up test.

Classwide Peer Tutoring

Several forms of classwide peer tutoring (CWPT) systems have been reported in the literature. Despite differences in structure, the use of classwide peer tutoring has been recognized as an evidence-based practice for improving students' academic outcomes in both special and general education settings (Fuchs, Fuchs, Phillips, Hamlett, & Karns, 1995; Greenwood, Delquadri, & Hall, 1989; Maheady, Harper, & Mallette, 1991).

Allsopp (1997) described an instructional program for middle school algebra that incorporated classwide peer tutoring along with cognitive strategy instruction and a CRA instructional sequence. Allsopp's (1997) study included 262 students and 4 teachers. All teachers provided instruction using a 12-lesson, researcher-developed curriculum designed to help students understand and solve teaching division equations and algebra word problems. The 12 teacher-directed lessons, which occurred across 16 to 18 class days within a 5-week period, taught students to use three learning strategies (with associated mnemonics) to organize and remember the steps to specific types of problems. In addition, the instruction began with concrete materials and progressed to abstract representations.

The experimentally manipulated element of the study was the format for students' practice activities following the teacher-directed lessons. In the comparison group, students completed independent practice activities using worksheets. Students in the treatment group were assigned to pairs and used CWPT to engage in practice activities. While CWPT students used the same worksheets as the independent practice students, one student served as the "player," completing problems with the assistance of a "coach," who had an answer key and provided guidance and modeling to assist the player in completing the problems correctly. After a period of practice, the students reversed roles, with the former "coach" now acting as a "player" and completing problems. Prior to the initiation of the algebra unit, both students and teachers received instruction in implementation of CWPT in the classroom.

No differences were found (Allsopp, 1997) between the groups on posttest performance or on a maintenance test administered one week after the conclusion of the intervention. All students improved with the base instructional program (structured curriculum using strategy instruction and a CRA instructional sequence), but differential gains were not obtained for students who practiced using CWPT. Students reported they enjoyed CWPT and believed it helped them learn algebra problem solving, but teachers expressed concerns about the amount of time necessary to organize practice activities and to document individual and team points earned during CWPT for purposes of providing group and individual reinforcement.

Graphic Organizers

A final algebra instructional strategy reported in the literature is the use of graphic organizers. Ives (2007) hypothesized that the use of graphic organizers, demonstrated to be effective in reading comprehension instruction, would serve a valuable function for instruction in advanced mathematical concepts, particularly those for which a CRA sequence cannot be easily developed. Working with secondary students (grades 6 to 12) in a private school for students with learning disabilities, Ives conducted two studies addressing the solution of systems of linear equations.

In the first study, he taught two groups of students (14 experimental, 16 comparison) to solve systems of two linear equations with two variables. Students in both groups used the same instructional materials, received the same amount of instruction, and completed the same practice activities. The only difference for the experimental group was the use of a graphic organizer (a matrix of cells designed to provide non-verbal structure to the problem solution process). Ives found that the experimental group's scores on a teacher-developed assessment and a researcher-developed test of conceptual understanding of the procedures were statistically significantly higher than the scores of the comparison group that did not use the graphic organizers. The results for a researcher-created test of problem solving revealed no statistically significant differences between groups.

A second study (Ives, 2007) was conducted as a systematic replication, using different students and instruction on solving linear systems with three equations with three variables. A similar graphic organizer (with an expanded matrix of cells to address the more complex linear systems) was used. Experimental and comparison groups each consisted of 10 students; parallel researcher-developed assessments of conceptual understanding of procedures and problem-solving accuracy were administered.

The results of the second study differed from those of the first; scores from the two groups on the problem-solving test were not significantly different, but scores on the conceptual understanding test favored students in the graphic organizer group. Ives noted that while the mean differences were in the same direction across studies, the smaller sample size in the second study may have influenced statistical significance.

THE FUTURE OF ALGEBRA INSTRUCTION AND ASSESSMENT FOR STUDENTS WITH LEARNING DISABILITIES

The research reported and reviewed here suggests that there is a growing need for mathematics assessment and intervention tools for secondary students with learning disabilities. In particular, there is a critical need for work in areas that address more advanced mathematical topics such as algebra.

The results of the initial research on algebra progress monitoring are encouraging. Positive findings have been obtained when the measures have been used as static indicators of student performance levels and as dynamic indicators of student learning over time. The measures appear to hold promise for identifying students who are likely to experience difficulty with algebra and for monitoring the progress of these students as educators strive to implement more effective instructional programs that meet students' individual needs. A major limitation in the research on algebra progress monitoring to date is that it has been conducted entirely by one research group and in a single midwestern state with students representing limited diversity in race/ethnicity, language, and socioeconomic backgrounds.

The most prominent approaches in the research on algebra instructional programs have involved cognitive strategy instruction and the use of a concrete-representational-abstract (CRA) teaching sequence. Findings from studies using these techniques have been positive, but are limited by the use of relatively simple algebraic content (e.g., integer operations, solving simple one-variable equations). The literature on algebra instructional techniques offers encouragement that these methods will provide teachers with tools to develop students' initial and basic understandings of algebraic concepts and problem solving, but it is less clear that the methods (particularly the CRA sequence and the use of generic problem solving strategies) will be sufficiently powerful to support instruction of more complex (and abstract) algebraic concepts and problems.

Two exceptions to the general pattern are evident in the work of Hutchinson (1993) and Ives (2007). Hutchinson's version of strategy instruction included specific attention to the mathematical structure and relationships represented in problems and explicitly taught students to differentiate them from surface-level (or story line) characteristics. This focus on helping students acquire mathematical schema for specific problem types is similar to the work of Jitendra, whose middle school mathematics problem-solving research emphasizes schema-based instruction (Jitendra, Hoff, & Beck, 1999; Xin, Jitendra, & Deatline-Buchman, 2005). Ives' use of graphic organizers allows for a dramatic expansion of the types of algebraic topics that can be addressed using the instructional strategy.

The majority of the algebra instruction research has been conducted in contexts more typical of intensive intervention (e.g., special classes or individualized tutoring) than of general education instruction. The strategies imbedded within the instructional research hold promise for improving core instruction for all students. Beginning with concrete objects to develop conceptual understanding before progressing to abstract symbolic representations is likely to benefit students across a range of ability levels and is consistent with the recommendations of the National Council of Teachers of Mathematics (NCTM; 2003). Likewise, cognitive strategy instruction seems to be a promising means of helping students guide themselves through the procedural aspects of algebra problem solving.

Implications for Future Research

The possibilities for additional research on algebra progress monitoring and instructional strategies are vast, but particular issues merit more immediate attention. First, research is needed to determine whether the algebra measures developed thus far will be effective indicators of performance and progress in other regions of the country, with more diverse student groups, and under varying curriculum contexts. In addition, research needs to be conducted to determine whether teachers' use of the algebra progress monitoring data to inform their instructional decisions results in increased achievement among their students. Given growing interest in multi-tier intervention models, future research should also examine the use of these measures for screening and assistance with selection of appropriate algebra course options for students of varying proficiency levels.

Research related to instructional strategies in algebra should focus on expanding the range of instructional contexts in which the methods can be effectively used and extending the complexity of the instructional content. Of particular interest is the application of methods found to be successful with students with learning disabilities to a broader range of students, including those considered at risk for mathematics difficulties and typical general education students. The conceptual foundations that are emphasized in the C-R-A studies and Ives' graphic organizer work may potentially benefit students in more heterogeneous settings, regardless of their disability status. In addition, research on instructional strategies that encompass a wider range of algebra concepts and problem types will provide more robust methods for teachers and greater benefit to students.

Implications for Practice

Although the research base on algebra progress monitoring measures and instructional strategies is not extensive, it offers some guidance for practitioners. With respect to progress monitoring, the algebra measures described in this article have demonstrated sufficient technical adequacy for use by teachers to monitor students' growth. Given that the existing data are drawn from samples in a single state and with limited diversity, caution must be advised if the data are to be used for high-stakes decisions. Until research has determined the generalizability of the findings, practitioners must be aware of the importance of verifying that students' scores on the algebra progress monitoring measures are related to important indicators of algebra proficiency in their respective districts and states.

As practitioners consider the research evidence for different types of algebra interventions, they must evaluate the context in which the research was conducted (e.g., small groups vs. general education classrooms) and the degree to which a particular method can effectively encompass the range of instructional content in a particular algebra course. As noted above, much of the research to date has been conducted with relatively simple concepts and problem types in algebra. In addition, the concrete models that have been studied vary in complexity. My intent is not to question the use of concrete representations for algebra problems that are traditionally taught entirely on a symbolic level. Instead, I urge practitioners to consider the degree of flexibility within the different systems and the degree to which they may be used effectively with the range of problem types represented in the curriculum.

Another consideration for practitioners, particularly within the context of this special series of the Learning Disability Quarterly, is the use of the literature to inform decisions about core instruction and supplemental interventions. With regard to supplemental instruction, the methods investigated to date offer practitioners a range of proven options to consider for students who require additional assistance. The greatest challenge in supplemental instruction will be the development of strategies that are amenable to more advanced topics. Ives' graphic organizer study illustrates the potential of this approach, but practitioners will need other graphic organizers that address additional advanced topics. Until such materials are developed, practitioners may consider developing their own graphic organizers to support algebra learning.

The existing work in algebra instruction and progress monitoring for students with learning disabilities is promising and provides direction for future efforts. Particularly important is the need to expand the scope of research examining the technical characteristics of the algebra progress monitoring measures with larger and more diverse samples and to investigate the effects of teachers' use of the progress monitoring data on student achievement. The instructional methods examined to date also require further examination. Some approaches, used previously only in individual tutoring contexts, should be explored in more typical classroom settings. Others must be studied further to see if they can be extended to more complex algebraic concepts. As this work continues, the literature will provide a more comprehensive evidence base to support teachers' efforts to improve their students' learning in algebra.

AUTHOR NOTE

Project AAIMS was funded by the U.S. Department of Education, Office of Special Education Programs (Award H324C030060), and partially supported the completion of this work.

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ANNE FOEGEN, Ph.D., Iowa State University, Ames, Iowa.

FOOTNOTES

(1) Sample copies of each of the algebra progress monitoring measures may be downloaded from the Project AAIMS web site (www.ci.hs.iastate.edu/aaims) on the Resources page.

(2) Readers should note that following the year in which these studies were conducted, District A has discontinued the option for "Special Education Algebra" courses (algebra courses taught by special education teachers entirely for students with disabilities). In part, these changes were a result of No Child Left Behind mandates related to requirements for highly qualified teachers.

Please address correspondence to: Anne Foegen, Iowa State University, N162D Lagomarcino Hall, Ames, IA 50011; [e-mail]

Table 1
Algebra Progress Monitoring Measures

                 Basic Skills          Algebra Foundations

Conceptual       Automaticity;         Core concepts in
Underpinnings    skills in which       algebra and pre-algebra
                 proficient students
                 should be fluent

Type of Task     Production            Production

Number of        60/5 minutes          50/5 minutes
Items/Duration

Scoring          1 point per           1 point per
                 response/correct or   response/correct or
                 incorrect             incorrect

                 Translations             Content Analysis

Conceptual       Minimal symbolic         Sampling of key
Underpinnings    manipulation; ability    concepts and skills
                 to move fluently         from a traditional
                 between multiple         Algebra 1 text;
                 representational modes   assesses mainte-
                                          nance and
                                          generalization

Type of Task     Selection (4 item        Selection (4-item
                 multiple choice)         multiple choice)
                                          with option to
                                          show work

Number of        42/7 minutes             16/7 minutes
Items/Duration

Scoring          1 point per              3 points per
                 response; penalty for    problem; rubric for
                 guessing                 partial credit

Note. The Basic Skills, Algebra Foundations, and Translations
measures all represent indicators of general proficiency in
algebra; the Content Analysis measure reflects key skills and
concepts in the instructional curriculum.

Table 2
Demographic Characteristics of Participants in Each Study

                               District A           District B

                                Full       IEP       Full       IEP
                               Sample   Subsample   Sample   Subsample

Total Number of Participants     78        11        102        15
  Gender
  Boys                           27         2         63        13
  Girls                          51         9         39         2
Ethnicity
  White                          76        10         83        13
  Black                          2          1         10         1
  Hispanic                       0          0         6          1
  Asian American                 0          0         2          0
Free/Reduced-Cost Lunch          11         2         38      -- (b)
Mean ITED Math Score (a)       60.73      31.15     46.85      37.40

Note. The full sample includes students with O's. IEP = individualized
education program.

(a) TIED for 18 students in 8 are not= Iowa Test of Educational
Development Total Math score, reported as a national percentile rank.
Scores grade included in the data for District A, as these students
completed a different district-wide achievement measure.

(b) Counts of lunch program participants in District B were provided
by the school for the entire sample without identifying individual
students; as a result, the number of participants among the IEP
subsample cannot be determined.

Table 3
Concurrent and Predictive Validity Results for the Full Sample
and for the IEP Subsample Algebra Measure/Criterion Measure

                              District A

                           Full          IEP
                          Sample      Subsample
                        (N = 62-71)   (N = 9-11)

                          Concurrent Validity
Basic Skills
  Teacher Proficiency
  Rating (fall)
  Pre-IAAT (fall)
  Post-IAAT (spring)
Algebra Foundations
  Teacher Proficiency      .61 *         ns
  Rating (fall)
  Pre-IAAT (fall)          .73           .74
  Post-IAAT (spring)       .57           ns
Content Analysis
  Teacher Proficiency      ns            ns
  Rating (fall)
  Pre-IAAT (fall)          .60           .70 *
  Post-IAAT (spring)       .73           .82

                          Predictive Validity
Basic Skills
  Algebra Grade
  Post-IAAT
  ITED Total Math
Algebra Foundations
  Algebra Grade            .38           ns
  Post-IAAT                .60           ns
  ITED Total Math          .54           ns
Content Analysis
  Algebra Grade            .54           ns
  Post-IAAT                .62           ns
  ITED Total Math          .52           ns

                               District B

                           Full          IEP
                          Sample       Subsample
                        (N = 79-97)   (N = 11-15)

                               Concurrent Validity

Basic Skills
  Teacher Proficiency      .48            ns
  Rating (fall)
  Pre-IAAT (fall)          .55           .65
  Post-IAAT (spring)       .60           .89
Algebra Foundations
  Teacher Proficiency
  Rating (fall)
  Pre-IAAT (fall)
  Post-IAAT (spring)
Content Analysis
  Teacher Proficiency      .41           .79
  Rating (fall)
  Pre-IAAT (fall)          .59            ns
  Post-IAAT (spring)       .60        .54 p = .08

                             Predictive Validity
Basic Skills
  Algebra Grade            .32        .46 p = .08
  Post-IAAT                .60           .89
  ITED Total Math          .39           .73
Algebra Foundations
  Algebra Grade
  Post-IAAT
  ITED Total Math
Content Analysis
  Algebra Grade            .41           .75
  Post-IAAT                .60        .54 p = .07
  ITED Total Math          .26 *          ns

Note. p < .01 for all correlation coefficients unless marked.
* = p < .05. ns = coefficient was not statistically significant.
IAAT=Iowa Algebra Aptitude Test; IEP = individualized education
program; ITED = Iowa Tests of Educational Development.

Table 4
Mean Weekly Rates of Growth by Class Type and IEP Status

                         District A           District B

                          Full       IEP       Full       IEP
                         Sample   Subsample   Sample   Subsample
Basic Skills
  Eighth-Grade Algebra
  Algebra I
  Algebra IB                                   .65        .79
  Algebra IA                                   .48        .52
Algebra Foundations
  Eighth-Grade Algebra    .54
  Algebra I               .3S        .28
  Algebra IB
  Algebra IA              .32        .24
  Content Analysis
Eighth-Grade Algebra      .87
  Algebra I               .47        .37
  Algebra IB                                   .44        .29
  Algebra IA              .37        .37       .64        .29