Improving Math Programming for Students at Risk: Introduction to the Special Topic Issue
By Crawford, Lindy Ketterlin-Geller, Leanne R
Response to Intervention (RTI) is a process of systematically integrating assessment and instruction to evaluate and address student needs through the use of research-based instructional practices. RTI is based on a three-tiered prevention model. Tier 1 consists of all students; they are provided with general instruction and assessment. Students who find success at Tier 1 are students who respond well to the general education curriculum and learn through typically delivered instruction without additional intervention, support, or monitoring. Tier 2 consists of some students who receive supplementary instruction and assessment. Students in this category often encounter learning challenges that cannot be addressed fully within the general education curriculum. And finally, a limited number of students receive Tier 3 support in the form of specialized instruction and assessment (Brown-Chidsey & Steege, 2005). Students who receive instruction at Tier 3, the smallest group of students, are those who require additional and/or intensive instruction, frequent monitoring, and the support of an intervention team. Depending on how the RTI model is implemented in a particular school or school district, this group of students may or may not qualify for special education services. In the past, our educational system had essentially two tiers: general and special education. Students who did not do well in the general education curriculum were referred to, and oftentimes qualified for, special education. In light of this traditional two-tiered model, much has been published about curriculum, instruction, and assessment at Tier 1, including research on core curricula (see for example the What Works Clearinghouse, 2002); differentiation of large-group instruction (see for example the work of Carol Tomlinson and, in particular, Tomlinson & McTighe, 2006); and whole group, classroom-based assessment systems (see for example the work of James Popham and, in particular, Popham, 2003). Similarly, much attention has been focused on educating students who receive special education-those students requiring intensive and specialized instruction to reach their educational goals. Federal laws have been written mandating appropriate instruction for these students, professional research journals have published thousands of articles on effective curricula, instruction and assessment for this group of students, and private and federal funding has supported empirical investigations into effective programming for students with disabilities. As such, support is generally available for students in Tier 1 and Tier 3.
RTI, with its three-tiered approach, has created a need for validated curriculum, instruction, and assessment practices for students in Tier 2. Moreover, as schools focus increasingly on students who are on the bubble in terms of passing statewide tests, there is a renewed interest in meeting the needs of students who may be at risk for failing to pass these accountability measures. A limited number of research studies in the domain of reading have shown increased achievement gains for students at Tier 2 when they are exposed to a comprehensive, standards-based curriculum; provided with evidence-based instruction; and monitored closely for achievement gains (Denton, Fletcher, Anthony, & Francis, 2006; Speece, Case, & Molloy, 2003). Additional research is needed, however, to verify these findings for students who are struggling in the area of mathematics. We must ensure that mathematics teachers have, at their hands, the same quality and quantity of evidence- based practices for students receiving Tier 2 support as for all other students.
Students needing supplemental supports through Tier 2 interventions are the focus of this special topic issue. In this issue of Remedial and Special Education, we explore curriculum, instruction, and assessment systems in mathematics designed to address the needs of students who are at risk of being referred for special education services if not provided with educational opportunities reaching beyond the core curriculum and instruction provided to all students. Specifically, we highlight needed supports including a standards-based supplemental curriculum, targeted and effective instructional interventions, and systematic screening and progress monitoring. Education professionals must proactively design curricula, instruction, and assessments that meet the needs of all learners; no longer can we use post hoc methods to modify curricula, adjust instructional programs, and provide test accommodations in our attempts to meet the needs of diverse students. Student diversity is too prevalent and student needs too extensive to continue to “patch together” our educational programming. One proactive model designed to meet the needs of all students is RTI.
RTI is not a new concept. For decades, successful schools and effective teachers have been screening children upon entrance to school or upon transition to a new grade, using data provided by these screenings to identify students who need extra support, providing them with the instruction they need to be successful, and monitoring the effects of this instruction on student learning. What is new is federal support for this model implicitly provided in the No Child Left Behind (NCLB) Act of 2001 and the reauthorization of the Individuals With Disabilities Education Act (IDEA; 2004). Both laws require use of evidence-based instruction, and IDEA further stipulates that teachers collect and analyze data on student performance in the general education curriculum as part of the referral process for a student suspected of having a learning disability. In addition, the model has strong face validity; it makes sense. In practice, however, many challenges at many levels impede full and successful implementation. For example, little is known about effective practices in content areas other than reading. Moreover, limited research exists about meeting the academic needs of older students who receive Tier 2 support. In response to these needs, this special topic issue focuses exclusively on mathematics curriculum, instruction, and assessment across a range of grade levels (including three articles focusing on students in intermediate grades or middle school).
In this special topic issue, we report on research and development efforts aligned with RTI components within mathematics. Articles focus on creating a supplemental mathematics curriculum, delivering supplemental mathematics interventions, and employing math assessment strategies designed primarily (although not exclusively) for Tier 2 students. In the first article, Freeman and Crawford discuss how to design a mathematics curriculum to supplement the general education core curriculum. They describe the development of a supplemental (not remedial) curriculum for middle school English-language learners, but their focus on vocabulary support, multiple opportunities to respond, and consistent and corrective feedback highlight the program’s usefulness for any students requiring additional supports to achieve in mathematics.
Whereas Freeman and Crawford outline the components of a successful supplemental curriculum at middle school, the next two articles in the special topic issue focus on the effectiveness of supplemental interventions designed to target the challenges faced by students identified as needing additional support in mathematics. Although research on effective interventions in mathematics is limited, recent meta-analyses examining instructional approaches for students with disabilities (Gersten, Chard, Jayanthi, Baker, & Lee, 2006) and low-achieving students (Baker, Gersten, & Lee, 2002) provide several promising findings. Instructional strategies such as providing visual and graphic depictions of mathematical concepts, explicit instruction, verbalization of mathematical processes, peer- assisted learning, and using assessment data to inform instruction are emerging as effective for students needing supplemental interventions. Additional research points to the use of graduated instructional sequences as effective for students who struggle in mathematics (Witzel, 2005). Instruction begins with concrete representations of the mathematical concepts and progresses to more abstract representations as students become more fluent with the applications. These and other features of interventions for students struggling in mathematics are discussed in the two articles focusing on supplemental instructional practices.
First, Bryant, Bryant, Gersten, Scammacca, and Chavez contribute to a limited research base in the field of primary mathematics instruction (Gersten, Jordan, & Flojo, 2005) by designing a mathematics intervention for first- and second-grade students who qualified for Tier 2 supplementary instruction. They report successful outcomes for second-grade students exposed to number sense lessons and lessons focused on fluency building with arithmetic combinations. Although they do not report similar results for first-grade students, they propose that additional time for the intervention may be needed to demonstrate a difference in outcomes (i.e., 20 minutes per session as opposed to 15 minutes). They hypothesize that this additional time would provide students more opportunities to practice, possibly resulting in greater gains. The authors share implications of their findings, perhaps the most important being that teachers who employ small group interventions designed to improve students’ performance on number, operation, and quantitative reasoning tasks while employing evidence-based effective teaching behaviors have a moderately good chance of seeing improvements in student performance. The importance of intervening with students in primary grades to ensure their success in later grades is accentuated by Ketterlin-Geller, Chard, and Fien, who focus on math interventions for students in intermediate grades. These authors studied two supplemental interventions: (a) a method for reteaching critical mathematics concepts to students who may not have acquired them as younger students-or if they were acquired were not retained; and (b) extended time learning and practicing material presented in the general education core curriculum. KetterlinGeller et al. stress the importance of evidence-based interventions for students struggling in mathematics, and they place this discussion within the need for an integrated system of instruction and assessment, a key feature of an RTI model. These authors found that intervening with students who are underperforming in math can improve their mathematics performance on various achievement measures. Whereas Bryant et al. hypothesized that additional time would have resulted in improved outcomes, KetterlinGeller et al. found evidence to support this assertion. On a state accountability test in mathematics, they report effect sizes favoring an “extended core” intervention that provides students with preteaching, reteaching, and extra practice in the grade-level curriculum over a concept-based curriculum that focuses on reteaching critical mathematical concepts.
To have a truly integrated instruction and assessment system, teachers and other educational professionals must use data to guide instructional decision making. Examining data is the best way to ensure that students receive targeted instruction based on their instructional needs. Although teachers have access to a variety of data sources, screening and progress monitoring data may be the most useful within an RTI framework. Screening tests are administered to all students to predict future performance on the benchmark goal, typically a state accountability test. Screening data helps teachers identify students who may be at risk for failure and may need additional instructional supports. Progress monitoring tests are administered frequently to students who may be at risk for failure to determine if they are making adequate progress toward reaching the instructional goal. Progress monitoring data allow teachers, administrators, students, and parents to evaluate the effects of instruction on student learning. Appropriate instructional decisions cannot be made in the absence of valid and reliable data; thus, we conclude the special issue with articles about screening and progress monitoring.
Clarke, Baker, Smolkowski, and Chard describe screening measures in mathematics for identifying primary-grade students who may be at risk for later math difficulties. They extend their previous work on early childhood mathematics measures to investigate the unique contribution (over and above other measures) of slope of growth on four early numeracy curriculum-based measures (CBMs) in predicting kindergarten students’ performance on a comprehensive measure of mathematics achievement at the end of the school year. Clarke et al. report evidence that four early numeracy measures-oral counting, number identification, quantity discrimination, and missing number- are effective in screening students for possible problems with mathematics in the future. They note that schools can employ these four measures to identify students who need additional interventions in mathematics.
The importance of early screening is made apparent by Clarke et al., but once students are screened and those at risk for failure are identified, we must use technically adequate measures to monitor their progress over time. Progress monitoring is a process that provides students, teachers, and other members of the educational community with information necessary to support student learning. Progress monitoring involves gathering baseline data, setting goals, administering brief assessments frequently, and evaluating performance over time. The information gathered in progress monitoring has the potential to improve student outcomes by providing an empirical database that can be used to improve instructional programs. Moreover, progress monitoring is a critical component of an RTI model in that it provides teachers with evidence related to the effectiveness of interventions and provides data to support an argument for changing educational supports or educational classifications.
In a subsequent issue continuing the special topic, Anne Foegen provides readers with a thorough description of how to investigate the technical adequacy of progress monitoring measures in mathematics and, in particular, the technical adequacy of middle school progress monitoring measures. Without technically adequate measures, teachers may make erroneous conclusions about student growth, thereby compromising implementation of the RTI model. Foegen researches technical features of curriculum-based mathematics measures for middle school students while providing a clear description of why these features are important for teachers to understand. Technical features such as normal distribution, alternate form and test-retest reliability, and criterion validity are examined and discussed, as is the “extent to which the measures reflect changes in student performance that correspond to student learning” (forthcoming).
Foegen reports on two technically robust measures for 6th-grade students, including one that requires students to solve conceptual/ application problems and one that requires students to solve computation problems. Quantity discrimination (also discussed by Clarke et al.) was found to be the most sensitive to growth in Grades 7 and 8. Additionally, at Grade 7, the conceptual/ application measure demonstrated strong technical adequacy. Studies such as those conducted by Foegen and Clarke et al. are critical when attempting to design “an integrated instruction and assessment system” (Ketterlin-Geller et al., p. 44). An integrated system relies on technically adequate mathematics measures sensitive to student growth. Measuring growth reliably is critical as schools begin (or refine) implementation of RTI. “Rate of growth over time, as an index of student learning, should be a central feature of evaluating the effectiveness of an instructional program” (Clarke et al., p. 54).
In summary, three elements contribute significantly to the successful implementation of an RTI model: (a) a standards-based curriculum, (b) evidence-based interventions, and (c) data-based evaluation systems. In this special topic issue, we share information and original research in each of these three areas, with the goal of accelerating students’ learning of mathematics within an RTI model.
References
Baker, S., Gersten, R., & Lee, D. (2002). A synthesis of empirical research on teaching mathematics to low-achieving students. The Elementary School Journal, 103, 51-73.
Brown-Chidsey, R., & Steege, M. W. (2005). Response to intervention: Principles and strategies for effective practice. New York: Guilford.
Denton, C. A., Fletcher, J. M., Anthony, J. L., & Francis, D. J. (2006). An evaluation of intensive intervention for students with persistent reading difficulties. Journal of Learning Disabilities, 39, 447-466.
Gersten, R., Chard, D. J., Jayanthi, M., Baker, S., & Lee, D. (2006). A meta-analysis of research on mathematics interventions for elementary students with learning disabilities. Manuscript under review.
Gersten, R., Jordan, N. C., & Flojo, J. R. (2005). Early identification and intervention for students with mathematics difficulties. Journal of Learning Disabilities, 38, 293-304.
Individuals With Disabilities Education Act, 20 U.S.C. [section] 1414 (2004).
No Child Left Behind Act of 2001, 20 U.S.C. [section] 6301 et seq. (2001).
Popham, W. J. (2003). Classroom assessment: What teachers need to know (3rd ed.). Boston: Pearson.
Speece, D. L., Case, L. P., & Molloy, D. E. (2003). Responsiveness to general education instruction as the first gate to learning disabilities identification. Learning Disabilities Research & Practice, 18, 147-156.
Tomlinson, C. A., & McTighe, J. (2006). Integrating differentiated instruction and understanding by design. Alexandria, VA: Association for Supervision and Curriculum Development.
What Works Clearinghouse. (2002). Retrieved July 12, 2007, from http://www.wharworks.ed.gov/
Witzel, B. S. (2005). Using CRA to teach algebra to students with math difficulties in inclusive settings. Learning Disabilities: A Contemporary Journal, 3(2), 53-64.
Lindy Crawford
University of Colorado at Colorado Springs
Leanne R. Ketterlin-Geller
University of Oregon, Eugene
Authors’ Note: Address correspondence to Lindy Crawford, PhD, University of Colorado at Colorado Springs, College of Education- COH6, 1420 Austin Bluffs Parkway, P.O. Box 7150, Colorado Springs, CO 80907-7150; e-mail: mcrawfor@uccs.edu.
Lindy Crawford, PhD, is an assistant professor in and chair of the Department of Special Education at the University of Colorado at Colorado Springs. Her research interests include meaningful inclusion of students with disabilities in statewide test programs and effective curriculum and instruction for English language learners.
Leanne R. Ketterlin-Geller, PhD, is an assistant professor of educational leadership at the University of Oregon. Her interests focus on the development of effective assessment procedures in mathematics and valid decision-making systems for students with diverse needs.
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