Effective Mathematics Instruction for Students with Learning Disabilities: Introduction to the Two-Part Series Diane Pedrotty Rivera

The University of Texas at Austin

Mathematical literacy is the ability to apply skills and concepts to reason through, communicate about, and solve mathematical problems. Mathematics instruction must involve methodology and curricular materials that assist students in mastering instructional objectives that are relevant to the development of mathematical literacy and appear on Individualized Education Programs (IEPs) of students with learning disabilities (LD). Over the past decade, much has been learned about the characteristics of students with LD in mathematics. Additionally, research results in mathematics instructional design and methodology have documented practices that promote success for students with LD at the elementary and secondary level in mastering the mathematics curriculum. These findings hold great promise for teachers who are challenged to provide effective mathematics programs to prepare youngsters with LD for postsecondary and adulthood transitions. Finally, given the findings about students with LD in mathematics, professionals are challenged to identify and implement “best practices” in mathematics instruction. Translating research into practice takes time and careful scrutiny to ensure that practices do indeed meet the individual needs of students with LD in mathematics. This is a two-part series on mathematics instruction. The purpose of this series is to provide practitioners with research-based methodology and suggestions for how best to enable students with math disabilities to master the mathematics curriculum. There are a total of twelve articles in this two-part series: six articles in this issue of the LD Forum and the remaining six articles in the next issue. This article provides an introduction to the series by (a) discussing educational trends that have implications for mathematics instruction; (b) providing an overview of the research in LD and mathematics instruction; and (c) describing the contributions of the authors in Part One of the series. EDUCATIONAL TRENDS It is important to recognize the ramifications of educational trends on the development, implementation, and evaluation of instruction; in this case, mathematics instruction. Teachers must carefully examine trends, and their research base to determine how to embrace new ideas while retaining proven “best practices” for students. In this section, four educational trends are discussed briefly in terms of their impact on mathematics instruction for students with LD: mathematics education reform, cultural and linguistic diversity influences on instruction, inclusion, and “alternative” assessments. It is beyond the scope of this article to provide an indepth discussion of these trends; readers can obtain further information in the articles in this series and elsewhere in the literature. (It should be noted that technology also is an educational trend that has implications for instruction in mathematics. Readers are referred to the LD Forum (1993) Technology Applications special issue.) Mathematics Education Reform The field of mathematics education has undergone reform as a result of current findings, including the recurring poor performance of America's students on national mathematics tests

(e.g., McKnight et al., 1987), changing sociological forces (e.g., technological advancements, cultural and linguistic diversity), mathematical expectations of employers, life skill requirements of adulthood, and interest in alternative" assessment approaches for evaluating students' progress (Rivera, Taylor, & Bryant, 1994-95). At National Council of Teachers of Mathematics have been at the forefront of reform efforts. Mathematics reform efforts have been aimed at redefining mathematics curriculum and instruction in relation to mathematical literacy, instructional goals and objectives, and methodology. Diversity and Mathematics Instruction Our schools are increasingly attended by a diverse student population with a rich cultural and linguistic heritage. Estimates indicate that by the 21st century about one-third of all school-age students will be from diverse backgrounds, primarily of African American, Asian, and Hispanic descent (Smith & Luckasson, 1995) Some of these youngsters will require special education services to meet their individual learning needs; therefore, it is necessary for us to learn about and infuse instructional techniques that accommodate these students' special needs. Mathematics instruction presents special challenges for many culturally and linguistically diverse (CLD) students with LD. For instance, mathematics has its own language system and set of processes Wiig and Semel (1984) called the language of mathematics “conceptually dense” because one cannot gain contextual meaning of mathematical symbols by “reading” the rest of a math sentence. Thus, students must understand the meaning and operation of each symbol encountered in number sentences. Students with LD for whom English is a second language may struggle with the new language system and the interpretation of the symbols. Also, students from diverse cultural backgrounds may have learned mathematics in ways that may be counter to those used in American schools (Scott & Raborn present an interesting discussion about this topic in their article in this series). Thus, sensitivity to cultural and linguistic differences in the mathematical knowledge of students with LD is crucial if these students are mainstreamed into programs where mathematics instruction is presented in English. For additional information (see references) about serving CLD students with LD, readers may want to study the work of Cloud (1993), Cummins (1989), Baca and Harris (1988), and Banks (1988). Inclusion and Mathematics Instruction “lnclusion” is defined and operationalized in different ways across school districts. However, paramount to the inclusion trend is a philosophy that students with disabilities have a right to receive their educational program in the general education classroom. Most professional organizations (e.g., Council for Learning Disabilities, Division for Learning Disabilities of the Council for Exceptional Children, Learning Disabilities Association of America) support an “inclusive philosophy” and also the retention of a continuum of services, recognizing that the individual needs of students with LD vary and may require assistance outside of the general education classroom for some portion of the school day. To ensure success in mathematics (and all areas) instruction for students with LD who spend more and more of their school day in the general education classroom, it is necessary for general and special education teachers to provide individualized instruction tailored to IEP goals and objectives. Such instruction may necessitate curricular and instructional adaptations and the use of different instructional arrangements (e.g., peer tutoring, cooperative learning, one-to-one instruction) in the general education setting. The role of the special education teacher in such learning environments is to provide assistance (e.g., collaboration, resource support) and instruction to ensure that students with LD are indeed progressing in the mathematics curriculum.

Alternative Assessment of Mathematical Skills and Concepts Different terms (authentic, performance-based, portfolio) are being used to label assessment approaches that purport to examine student performance in relation to curricular goals and instructional techniques. Linked to the school reform movement, these “alternative” assessments are gaining in popularity as a result of interest in (a) identifying assessment options to traditional, standardized testing procedures; (b) evaluating youngsters’ work in relation to their ongoing performance rather than to a “normed group”; (c) examining more directly student achievement as it relates to the school’s curriculum; and (d) focusing more on analyzing processes rather than just products. In mathematics instruction, “alternative” assessment approaches are being used to describe performance and to measure mastery of instructional objectives. The intent is to evaluate more directly the strategies students use to solve problems, students’ ability to communicate effectively with mathematical language, and students’ understanding of mathematical skills and concepts. Thus, “alternative” assessment approaches such as clinical interviews, portfolios, criterion-referenced assessment, curriculum-based assessment, and teacher observations are being used to evaluate students’ mastery of the mathematics curriculum. The role of these “alternative” assessment approaches must be carefully examined in the instructional programming and decision-making process. Educators should determine reasons for assessment and the techniques that are most appropriate, including both traditional and “alternative” assessments. Four trends have been briefly introduced to provide a contextual base for information about mathematics instruction for students with LD. In the next section of this introductory article, an overview of research on the characteristics of students with math disabilities and effective math instruction for this population is presented. CHARACTERISTICS OF STUDENTS WITH

MATHEMATICS LEARNING DISABILITIES AND THE

COMPONENTS OF EFFECTIVE MATHEMATICS INSTRUCTION

Much has been learned about the characteristics of students with LD in mathematics and the components of effective mathematics instruction. The following is a brief overview of characteristics and instructional components. Characteristics of Students with LD in Mathematics We now have a better understanding of the educational aspects of LD in math. For example, we know that elementary and secondary students with math LD may demonstrate developmental delays in acquiring and applying various mathematical skills and concepts (Cawley & Miller, 1989; Fleischner, Garnett, & Shepherd, 1982) and limited mathematics achievement compared to their typical peer group (Cawley & Baker-Kroczynski, 1992). Thus, students with math LD may exhibit difficulties using (a) effective cognitive and metacognitive strategies (Montague & Applegate, 1993), memory and retrieval processes (Bley & Thornton, 1995), and generalization skills (Rivera & Smith, 1987; Woodward, 1991). These difficulties could affect, for instance, computational proficiency (Kirby & Becker, 1988; Pellegrino & Goldman, 1987); story problemsolving abilities (Montague, 1992); and mastery of other skills, such as fractions, decimals, measurement, and algebra. We also know that mathematical difficulties may stem from LD but may also be attributable to ineffective instructional components. Components of Effective Mathematics Instruction

Much research has been carried out to identify those components that contribute to successful instruction in mathematics for students with LD. For example, we know that students learn mathematics skills when teachers use direct instruction (eg., advance organizers, presentation of subject matter, guided and independent practice, modeling, examples) and that some students can benefit from working in various instructional arrangements (e.g., peer tutoring, cooperative learning). Research findings support instruction in cognitive strategies that are specific to the content area and help students generalize learning. Additionally, we have learned that students with math LD must be taught the language of mathematics and that textbooks should be used judiciously for instructional purposes. Our knowledge base about LD in mathematics has broadened over the years. The challenge for educators is how to identify practices that are deemed most effective for their students and to apply instruction systematically to enable students with math LD to master curriculum objectives. TRANSLATING RESEARCH INTO PRACTICE This two-part series on mathematics instruction for students with LD addresses a variety of topics pertinent to current trends in the field of education (mathematics education, general and special education) and to teaching youngsters with LD at the elementary and secondary level. The omission of an article on teaching computational skills is intentional due to the numerous articles already available. The articles in this series include content related to the phases of teaching including design, implementation, and evaluation. Specifically, the authors have presented information that translates special education research on effective instruction in mathematics into practical ideas and suggestions for practitioners. The articles in Part One focus on diversity, adapting math instruction for the general education classroom, assessment, cognitive strategy instruction, and instructional materials. The next issue of the LD Forum will contain Part Two of the math series, which includes articles on word problem-solving; teaching decimals, fractions, and percent; integrated curricular programming; peer tutoring; cooperative learning; and functional curriculum. To begin in Part One, Scott and Raborn offer information about mathematics instruction for youngsters from culturally and linguistically diverse (CLD) backgrounds who have LD. These authors provide insight into the students’ mathematics achievement and the linguistic, cultural, and cognitive influences on students’ abilities to learn mathematics. Of particular interest is a section devoted to potential trouble spots students may encounter as they learn mathematics and recommendations for addressing these problems. Lock provides suggestions and guidelines for adapting mathematics instruction when teaching students with LD in the general education classroom. Thus, her article conveys information for general education teachers who are seeking instructional ideas to accommodate the needs of students with LD and for special education teachers who may consult with these teachers. In addition to suggestions for adapting instruction in computation and story problemsolving, Lock also provides guidelines for effective mathematics instruction. The article on “alternative” assessment by Bryant and Maddox includes information about assessment techniques that can be used to monitor student progress on a regular basis as it relates to the curriculum. Bryant and Maddox offer a variety of techniques that educators can use to make decisions about students’ progress on IEP mathematics goals and objectives and the instructional objectives that relate specifically to the school's curriculum. Miller, Strawser, and Mercer discuss ways to improve mathematics performance through cognitive strategy instruction (a popular instructional approach). These authors offer tips for

developing a “strategic” classroom environment that fosters effective mathematics instruction. By combining the “strategic” tips and learning strategies, teachers can implement research-based instructional practices to promote mathematics achievement of students with LD. Lambert presents information about mathematics textbooks, materials, and manipulatives, which are commonly a part of mathematics instruction. Important strengths and weaknesses of these instructional tools when teaching students with LD in mathematics are emphasized. Lambert also offers instructional considerations and specific evaluation scales forcareful selection of textbooks, materials, and manipulatives to support mathematics instruction. As editor of this series, I hope you find the ideas, techniques, and suggestions presented in these articles useful as you design, implement, and evaluate instructional programs in mathematics for students with LD. I would like to thank the authors of the papers in this series for their time and expertise in the development of their articles, Kirsten McBride for her excellent editorial skills and feedback, Rich Wilson for his support and feedback in this project, and Terry Zimmerman (our technology specialist in the College of Education at The University of Texas) for his invaluable assistance in converting documents. REFERENCES Baca, L., & Harris, K. C. (1988). Teaching migrant exceptional students. Teaching Exceptional Children, 20, 32-35. Banks, J. A. (1988). Multiethnic education: Theory and practice (2nd ed.). Boston: Allyn & Bacon. Bley, N., & Thornton, C. (1995). Teaching mathematics to students with learning disabilities (3rd ed.). Austin, TX: PRO-ED. Cawley, J. F., Baker-Kroczynski, S., & Urban, A. (1992). Seeking excellence in mathematics education for students with mild disabilities. Teaching Exceptional children, 24, 40-43. Cawley, J. F., & Miller, J. H. (1989). Cross-sectional comparisons of the mathematical performance of children with learning disabilities: Are we on the right track toward comprehensive programming? Journal of Learning Disabilities, 22, 250-259. Cloud, N. (1993). Language, culture, and disability: Implications for instruction and teacher preparation. Teacher Education and Special Education, 16, 60-72. Cummins, J. (1989). A theoretical framework for bilingual special education. Exceptional children, 56, 111-119. Fleischner, J. E., Garnett, K., & Shepherd, M. (1982). Proficiency in arithmetic basic fact computation by learning disabled and nondisabled children. Focus on Learning Problems in Mathematics, 4, 47-55. Hofmeister, A. M. (1993). Elitism and reform in school mathematics. Remedial and Special Education, 14(6), 8-13. Hutchinson. N. L. (1993). Students with disabilities and mathematics education reform Let the dialogue begin. Remedial and Special Education, 14(6), 20-23. Kirby, J., & Becker, L. (1988). Cognitive components of learning problems in arithmetic. Remedial and Special Education, 9(5), 7-16. McKnight, C., Crosswhite, F., Dossey, J., Kifer. E., Swafford. J., Travers, K., & Cooney, T. (1987). The underachieving curriculum: Assessing U.S. school mathematics from an international perspective. Champaign, IL: Stipes. Mercer, C. D., Harris, C. A., & Miller, S. P. (1993). Reforming reforms in mathematics. Remedial and Special Education, 14(6), 14-19. Montague, M. (1992). The effects of cognitive and metacognitive strategy instruction on the mathematical problem solving of middle school students with learning disabilities. Journal

of Learning Disabilities, 25, 230-248. Montague, M., & Applegate, B. (1993). Middle school students' mathematical problem solving: An analysis of think-aloud protocols. Learning Disability Quarterly, 16, 19-30. National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author. Pellegrino, J., & Goldman. S. (1987). Information processing and elementary mathematics. Journal of Learning Disabilities, 20, 23-32. Rivera, D. (1993). Examining mathematics reform and the implications for students with mathematics disabilities. Remedial and Special Education, 14(6), 24-27. Rivera, D., & Smith, D. D. (1987). Influence of modeling on acquisition and generalization of computational skills: A summary of research findings from three sites. Learning Disability Quarterly, 10, 69-80. Rivera, D. P., Taylor, R. L, & Bryant, B. R. (1994-95). Review of current trends in mathematics assessment for students with mild disabilities. Diagnostique, 20 (1-4), 143-174. Smith, D. D., & Luckasson, R. (1995). Introduction to special education: Teaching in an age of challenge (2nd ed.). Boston: Allyn & Bacon. Thornton, C. A., Langrall, C. W., & Jones, G. A. (1997). Mathematics instruction for elementary students with learning disabilities. Journal of Learning Disabilities, 30, 142-150. Wiig, E. H., & Semel, E. H. (1984). Language assessment and intervention for the learning disabled (2nd ed.). New York: Merrill/Macmillan. Woodward, J. (1991). Procedural knowledge in mathematics: The role of the curriculum. Journal of Learning Disabilities, 24, 242-251.