Learning Disabilities Practice

A review of popular approaches to teaching mathematics that emphasize nonverbal skills, such as using manipulatives or problem-solving schema, shows that they are often not immediately applicable to some important areas of secondary algebra. However, graphic organizers in various forms have been widely suggested and researched as an intervention approach to improve reading comprehension. In this article, suggestions for modifying graphic organizers to make them more applicable to teaching higher-level mathematics concepts and procedures are proposed. Using an appropriately modified graphic organizer to teach higher-level mathematics skills may help students with relatively weak verbal skills and strong nonverbal reasoning skills to be more successful in mathematics. This subgroup of students with learning disabilities has been consistently identified in several schemes for subtyping mathematical disorders developed from empirical evidence. Excerpts of two sample scripted lessons based on this approach, one for the solving of systems of linear equations in three variables and one for the concept of negative integer exponents, are included. Yesterday, the first author received an e-mail from a former student. The note reminded us why we wanted to write this article. It wasn’t the content of the note as much as the student himself that was important. For six years the first author taught secondary mathematics courses at a small private school dedicated to students with learning disabilities; this student had been in several of his classes. As a high school senior the student took an introduction to differential calculus course with Requests for reprints should be sent to Bob Ives, Department of Curriculum & Instruction, University of Nevada at Reno, Mail Stop 282, Reno, NV 89557. the first author and had also demonstrated some talent as an artist, particularly in oil painting. At the same time, he was reading at about a fifth-grade level despite several years of specialized reading instruction, and he found memorization of basic mathematics facts nearly impossible. This student was one of many students who seemed capable of understanding the complex content of advanced secondary mathematics in spite of substantial reading and language deficits. While traditional classroom instruction relies heavily on reading and language-based instruction, my challenge with these students was finding ways to give these students access to the information without relying so heavily on these reading and language skills. The point here is that students come to class with a wide range of relative strengths and weaknesses in cognitive abilities as well as in achievement. This is particularly true of students with learning disabilities. It becomes especially important for us, as teachers, to understand and respond to the relationships between these differences and various approaches to instruction. However, instruction at higher levels of secondary mathematics is particularly dependent on language skills, suggesting that students with weak verbal abilities and strong nonverbal abilities are being underserved in this area. The purposes of this article are to provide a rationale for applying graphic organizers to instruction in secondary mathematics, particularly algebra, and to make recommendations for the creation and application of graphic organizers for this material. More specifically, the article provides a rationale for using graphic organizers based on the fact that students with language disorders often struggle with conventional mathematics instruction that places large demands on language skills. Some current practices in mathematics instruction reduce this emphasis on language skills, but are not readily applicable to some higher-level mathematics skills. Following this discussion, review of the use of graphic organizers to improve reading comprehension LEARNING DISABILITIES PRACTICE 37 is used to identify features of graphic organizers that contribute to their effectiveness. These important features of graphic organizers are then applied to mathematics content. Two specific examples of direct instruction in secondary algebra using graphic organizers are described. Finally, some general recommendations for both teachers and researchers conclude the article. RATIONALE FOR THE USE OF GRAPHIC ORGANIZERS Language skills and mathematics skills are not mutually exclusive. Mathematics instruction often assumes that the learner has typical language skills. However, this is not necessarily the case for students with learning disorders. Students with language disorders can be at a disadvantage when mathematics instruction relies heavily on language skills. Some approaches to mathematics instruction, such as the use of manipulatives and graphs, offer these students an alternative to dependence on receptive language skills. However, these approaches have generally been applied only to basic mathematics skills, and they may not be applicable to some more advanced mathematics skills. The literature on reading comprehension has shown that a variety of graphic organizers have been effective in improving student performance. This section ends with some suggestions for modifying the definition and use of these graphic organizers so that they may be applied to mathematics instruction. Relationships Between Language Skills and Arithmetic Skills Deficits Mainstream classroom instruction in mathematics assumes adequate language and reading competence (Bley & Thornton, 1995; Moses & Cobb, 2001; Rivera, 1998). This reliance presents a challenge for students who demonstrate learning disorders. Students who have difficulties with language can struggle with mathematics instruction in a variety of ways (Bley & Thornton, 1995; Miller & Mercer, 1998). For example, they may have difficulty following or understanding instructions, understanding mathematical terms, reading or understanding word problems, recognizing variations on tasks, verbalizing what they know, or identifying irrelevant information. These students are found in classrooms along with other students who do not have to face these language challenges. Over the years, teachers and researchers have developed several techniques for illustrating mathematics concepts that are not dependent solely on reading and language comprehension (Harris, Miller, & Mercer, 1995; Jitendra & Hoff, 1996). These techniques include the use of concrete objects and spatial representations of relationships that may not rely as heavily on language and reading skills as conventional instruction does. Little empirical evidence for aptitudetreatment interactions yet exists to support the claim that these approaches would benefit students with language disorders more than approaches that are more dependent on language skills. However, this approach is consistent with a long tradition in special education of modifying instruction to take advantage of relative cognitive strengths. Further, efforts to subtype developmental arithmetic disabilities based on empirical evidence are at least consistent with this view. Several authors have proposed subtypes of developmental arithmetic disabilities that are derived from empirical evidence (Geary, 1993, 2000; Kosc, 1974; Rourke, 1989, 1993; Rourke & Conway, 1998). A tentative comparison of these approaches is outlined in Figure 1. The figure indicates ways subtypes from one approach might be comparable to, subsumed by, or overlap with subtypes from other approaches. Entries in the figure indicate the name assigned to each subtype and indicate areas of relative weakness for each of the subtypes. A brief review of each of these subtyping approaches follows. Rourke has suggested that students with specific learning disabilities (LD) may be categorized into two general groups and, further, that the arithmetic difficulties of each group may have different origins (Rourke, 1989; Rourke & Conway, 1998). Rourke’s Group R-S was substantially deficient in reading and spelling but not as weak in arithmetic performance (although not performing up to age norms in this area). Rourke suggested that their weakness in arithmetic might be secondary to their reading problems. This group was described as having verbal learning disabilities (VLD).