mathematics instruction and learning disabilities || algebra progress monitoring and interventions...
TRANSCRIPT
Hammill Institute on Disabilities
Algebra Progress Monitoring and Interventions for Students with Learning DisabilitiesAuthor(s): Anne FoegenSource: Learning Disability Quarterly, Vol. 31, No. 2, Mathematics Instruction and LearningDisabilities (Spring, 2008), pp. 65-78Published by: Sage Publications, Inc.Stable URL: http://www.jstor.org/stable/20528818 .
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ALGEBRA PROGRESS MONITORING AND INTERVENTIONS FOR STUDENTS WITH
LEARNING DISABILITIES
Anne Foegen
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 learn
ing 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 strate
gies for algebra are summarized. Implications for practitioners and future research are discussed for both progress monitoring assessment tools and algebra instructional practices.
ANNE FOEGEN, Ph.D., Iowa State University, Ames, Iowa.
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 stan dards for mathematics, and proficiency in algebra in
particular, is evident in changing graduation require ments. 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 multi
ple years have highlighted the desperate need for more
effective teaching and learning of mathematics in gen eral, and algebra in particular (Carpenter et al., 1981; Silver & Kenney, 2000; U.S. Department of Education, 1997). For students with disabilities, reports of mathe
matics 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 individu
ally 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 nondis abled students performed at this level (Perie, 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
Volume 31, Spring 2008 65
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that no consensus exists on the components that con
tribute to mathematics difficulties. In an effort to iden
tify 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, L?pez Rol?n, & Delazer, 2007), and cognitive science
(Butterworth & Reigosa, 2007). Proposed mechanisms include deficits in general cognitive abilities (e.g., work
ing 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 ele
mentary 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 exist
ing literature does suggest potential avenues for future
investigation. For example, Hecht, Vagi, and Torgesen (2007) described a line of research investigating stu
dents' understanding and computational skill with frac tions. 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 proposi tion is consistent with research by Siegler (1996) illus
trating 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 stu
dents with and without disabilities. Thus, when sur
veyed 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 increas
ing the interest level of the instruction as teacher strate
gies 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 stu dents with learning disabilities are receiving their math ematics 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 sur
vey 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, gen eral 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 consis tent 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/ques tionnaires. 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 assess ment 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 pro
viding supplemental instruction in algebra when stu dents 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 dis abilities.
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
Learning Disability Quarterly 66
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that serve as indicators of general proficiency in a con
tent 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 impor tant, 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 accept able 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 ele
mentary grades (Foegen, Jiban, & Deno, 2007). Although some extensions of this work exist at the pre school 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
Table 1
Algebra Progress Monitoring Measures
Conceptual
Underpinnings
Basic Skills
Automaticity; skills in which proficient students
should be fluent
Algebra Foundations Translations
Type of Task Production
Number of
Items/Duration
Scoring
60/5 minutes
1 point per response/correct or
incorrect
Core concepts in
algebra and pre-algebra
Production
50/5 minutes
1 point per response/correct or
incorrect
Minimal symbolic manipulation; ability to move fluently between multiple
representational modes
Selection (4 item
multiple choice)
42/7 minutes
1 point per response; penalty for
guessing
Content Analysis
Sampling of key concepts and skills from a traditional
Algebra 1 text; assesses mainte
nance and
generalization
Selection (4-item
multiple choice) with option to show work
16/7 minutes
3 points per problem; rubric for
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.
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Impecoven-Lind (in press), a brief summary of these measures is presented in Table l.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 profi ciency 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 alge bra. 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 com
prehensive 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 con
ducted in general education classes that included stu
dents 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 pre sented 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 disabili
ties, it is not possible to differentiate students by specific
disability type. Our experiences working in participat
ing 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 dis tricts that participated in Project AAIMS. More complete reports of the studies and their results are reported in
Perkmen et al. (2006a, 2006b). The participants, meas
ures, 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 last
ing 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 place
ment 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 com
munity of 26,000 students. The four-year high school enrolls approximately 1,300 students; 82% of these stu dents 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 demo
graphic characteristics of the participants in the two studies are reported in Table 2.
Three of the four algebra progress monitoring meas ures mentioned earlier were used in the studies. In both
districts, teachers administered two types of measures each month (with minor deviations for school holi
days), using two forms of one measure during the mid dle 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 educa tion teachers in District A collected progress monitor
ing data from September through April, alternating between the Algebra Foundations and the Content
Analysis measures. The special education teacher alter nated between the Algebra Foundations and the Basic Skills measures. In District B, general education teach ers 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 stan dardized test scores. Classroom-based measures in cluded 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 pro
ficiency in algebra in comparison to that of typical peers using a five-point scale; we administered this
Learning Disability Quarterly 68
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Table 2
Demographic Characteristics of Participants in Each Study
Total Number of Participants
Gender
Boys Girls
Ethnicity White
Black
Hispanic Asian American
Free/Reduced-Cost Lunch
Mean ITED Math Score3
District A
Full IEP
Sample Subsample
78
27
51
76
2
0
0
11
60.73
11
2
9
10
1
0
0
2
31.15
District B
Full IEP
Sample Subsample
102
63
39
83
10
6
2
38
46.85
15
13
2
13
1
1
0
__b
37.40
Note. The full sample includes students with IEPs. IEP = individualized education program.
aITED = Iowa Test of Educational Development Total Math score, reported as a national percentile rank. Scores for 18 students in grade 8 are not included in the data for District A, as these students completed a different district-wide achievement measure.
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.
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 admin
istered annually by the district for accountability pur
poses; 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 achieve ment 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 col
leagues (2006a, 2006b). Criterion validity analyses included both concurrent validity and predictive valid
ity. 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 admin
istered in the fall or spring and criterion measures gath ered at the same points in time. Predictive validity coefficients represent relations between fall scores on
the algebra progress measures and criterion measures
Volume 31, Spring 2008 69
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Table 3 Concurrent and Predictive Validity Results for the Full Sample and for the IEP Subsample
Algebra Measure/ Criterion Measure
District A
Full
Sample (N = 62-71)
IEP
Subsample (N=9-ll)
District B
Full
Sample (N = 79-97)
IEP
Subsample (N= 11-15)
Concurrent Validity Basic Skills
Teacher Proficiency
Rating (fall) Pre-IAAT (fall) Post-IAAT (spring)
Algebra Foundations
Teacher Proficiency
Rating (fall) Pre-IAAT (fall) Post-IAAT (spring)
Content Analysis Teacher Proficiency
Rating (fall) Pre-IAAT (fall) Post-IAAT (spring)
.61*
.73
.57
ns
.60
.73
ns
.74
ns
ns
.70*
.82
.48
.55
.60
.41
.59
.60
ns
.65
.89
.79
ns
.54 p = .08
Basic Skills
Algebra Grade
Post-IAAT
ITED Total Math
Algebra Foundations
Algebra Grade
Post-IAAT
ITED Total Math
Content Analysis
Algebra Grade
Post-IAAT
ITED Total Math
Predictive Validity
.38
.60
.54
.54
.62
.52
ns
ns
ns
ns
ns
ns
.32
.60
.39
.41
.60
.26*
.46 p = .08
.89
.73
.75
.54p = .07
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.
Learning Disability Quarterly 70
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obtained in the spring. For the full samples, most coef ficients were in the moderate range (r = .4 to .6).
Among students with IEPs, most correlations involv
ing the teacher rating scale were not statistically sig nificant. 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 excep tions to this pattern were the post-IAAT scores for stu
dents who had completed the Algebra Foundations measure in District A and the Content Analysis meas ure 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 coeffi cients 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 per formance.
In addition to serving as static indicators of perform ance, 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
Table 4 Mean Weekly Rates of Growth by Class Type and IEP Status
Basic Skills
Eighth-Grade Algebra
Algebra I
Algebra IB
Algebra IA
Algebra Foundations
Eighth-Grade Algebra
Algebra I
Algebra IB
Algebra IA
Content Analysis
Eighth-Grade Algebra
Algebra I
Algebra IB
Algebra IA
District A
Full IEP
Sample Subsample
.54
.35
.32
.87
.47
.37
.28
.24
.37
.37
District B
Full IEP
Sample Subsample
.65
.48
.44
.64
.79
.52
.29
.29
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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 stu
dents 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 IA and IB represented options avail able to students who wanted to complete algebra at a slower pace. The Algebra IA 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 IA, students typ ically enrolled in Algebra IB, 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 lim ited 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 IB and IA. The one exception to this pattern was the Content Analysis measure in District B, where Algebra IA students demonstrated substantially higher rates of
growth than their Algebra IB peers. As might be
expected, without any instructional interventions, stu dents with IEPs had comparable or lower rates of
growth in most cases the Basic Skills growth rates of most (Algebra IB in District B was the lone exception). The data suggest that the measures are likely to be use ful for monitoring student progress among typical stu
dents, 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 stu dents. 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 moni
toring system in algebra comes the professional obliga tion 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 per formance and learning trajectories; they also need evi dence-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 interven tions specific to algebra, presented by the type of instructional approach used. Approaches related to
cognitive strategy instruction are described first, fol lowed 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 disabili ties 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 suc
cessfully 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 spe cific 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 stu dents 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
Learning Disability Quarterly 72
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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 representa tional problems).
The results of the study revealed positive improve ments in problem representation and solution on the
problem types for which students had received instruc tion. 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 instruc
tion, 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 success
fully 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 pic ture 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 model
ing, guided practice with feedback, and independent practice similar to that described above in Hutchinson's
(1987, 1993) application of cognitive strategy instruc tion. Specifically, in the concrete phase, students used
Algebra Lab Gear (Picciotto, 1990), a commercially available algebra instructional program that incorpo rates 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, multi
ple-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) pro vided individual instruction in computation and
problem solving with integers across all four opera tions (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 subtrac
tion, multiplication, and division. Study data revealed that students used the strategy to accurately represent and solve integer word problems; maintenance meas ures administered up to 10 weeks following the inter vention revealed students had a 91% accuracy rate on
problem solution. Other researchers have examined the merger of strat
egy 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 represen tational phase. Witzel et al. (2003) worked with 10 gen eral 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 expr?s
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sions, 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 inclu
sive middle school Algebra 1 courses. He found that stu
dents in the experimental condition using a CRA
sequence outperformed peers in the comparison condi tion 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) sys
tems have been reported in the literature. Despite dif
ferences 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 pro vided instruction using a 12-lesson, researcher-devel
oped 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 pro
gressed 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 work sheets as the independent practice students, one stu dent 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 prob lems. Prior to the initiation of the algebra unit, both students and teachers received instruction in imple
mentation of CWPT in the classroom. No differences were found (Allsopp, 1997) between
the groups on posttest performance or on a mainte nance 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 alge bra 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, demon strated to be effective in reading comprehension instruction, would serve a valuable function for instruc tion in advanced mathematical concepts, particularly those for which a CRA sequence cannot be easily devel
oped. Working with secondary students (grades 6 to 12) in a private school for students with learning disabili
ties, 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 com
pleted 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-devel
oped test of conceptual understanding of the proce dures were statistically significantly higher than the
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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 signifi cant differences between groups.
A second study (Ives, 2007) was conducted as a sys tematic replication, using different students and instruction on solving linear systems with three equa tions 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; paral lel researcher-developed assessments of conceptual understanding of procedures and problem-solving accu
racy 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 top ics 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 stu dents who are likely to experience difficulty with alge bra and for monitoring the progress of these students as
educators strive to implement more effective instruc tional 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 midwest ern state with students representing limited diversity in
race/ethnicity, language, and socioeconomic back
grounds. The most prominent approaches in the research on
algebra instructional programs have involved cognitive strategy instruction and the use of a concrete-represen tational-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 instruc tional techniques offers encouragement that these
methods will provide teachers with tools to develop stu dents' initial and basic understandings of algebraic con
cepts 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 stu dents acquire mathematical schema for specific prob lem 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 tutor
ing) than of general education instruction. The strate
gies 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 recom
mendations of the National Council of Teachers of Mathematics (NCTM; 2003). Likewise, cognitive strat
egy instruction seems to be a promising means of help ing 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 atten tion. 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
Volume 31, Spring 2008 75
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interest in multi-tier intervention models, future research should also examine the use of these measures
for screening and assistance with selection of appropri ate algebra course options for students of varying profi ciency 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 con tent. Of particular interest is the application of methods found to be successful with students with learning dis abilities to a broader range of students, including those considered at risk for mathematics difficulties and typi cal general education students. The conceptual founda tions 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 dis
ability status. In addition, research on instructional
strategies that encompass a wider range of algebra con
cepts and problem types will provide more robust meth ods for teachers and greater benefit to students.
Implications for Practice
Although the research base on algebra progress mon
itoring measures and instructional strategies is not
extensive, it offers some guidance for practitioners. With respect to progress monitoring, the algebra meas ures described in this article have demonstrated suffi cient 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 lim ited 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, practi tioners must be aware of the importance of verifying that students' scores on the algebra progress monitor
ing measures are related to important indicators of
algebra proficiency in their respective districts and states.
As practitioners consider the research evidence for dif ferent 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 par ticular algebra course. As noted above, much of the research to date has been conducted with relatively sim
ple 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 con crete representations for algebra problems that are tra
ditionally 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 stu dent achievement. The instructional methods exam
ined 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.
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FOOTNOTES Sample copies of each of the algebra progress monitoring meas
ures may be downloaded from the Project AAIMS web site
(www.ci.hs.iastate.edu/aaims) on the Resources page.
2Readers should note that following the year in which these stud
ies 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 man
dates related to requirements for highly qualified teachers.
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.
Please address correspondence to: Anne Foegen, Iowa State
University, N162D Lagomarcino Hall, Ames, IA 50011;
Learning Disability Quarterly 78
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