Qualitative or quantitative differences? Latent class analysis of mathematical ability for special education students.

The current article investigates whether the mathematics achievement of students in special education can be used to identity those who share common cognitive skills that may not be in concordance with their disability labels. Latent class analysis of a comprehensive test of mathematics taken by fourth-grade students with various disabilities reveals that a model with 2 latent classes is adequate to characterize the latent structure of the data. A parallel relationship of response profiles across the 2 classes suggests differences in the levels of mathematical ability (quantitative), rather than differences in the type of mathematical ability (qualitative), between the 2 latent classes in terms of generic mathematical proficiency. Cross-validation on a separate data set with careful matching of content areas within the math test verified this conclusion. Although a significant relationship exists between the identified latent classes and various disabilities, the analysis also found common mathematical problem-solving behaviors across disability categories. Implications for intervention and limitations of the content study are discussed.

**********

Students with disabilities comprise a protected population for which particular educational and instructional interventions are expected to be provided. Ideally, remedial interventions should be individualized to each student's need. This is the goal of the Individuals with Disabilities Education Act of 1990 (IDEA) and embodied in the Individualized Education Program (IEP) process, but complete individualization is usually impossible due to limited resources and the need for teachers trained to provide services tailored to specific groups, such as mobility training or Braille for students with visual impairment. As a result, students who meet categorical eligibility' criteria specified in IDEA and further delineated by relevant state special education laws and procedures are sometimes grouped for instruction rather than receiving individual attention.

The disability categories defined by IDEA and state laws were formed on historical and political, rather than empirical, foundations. Categories are typically defined on the basis of such characteristics as type of impairment (e.g., perceptual, language, physical, cognitive) or etiology (e.g., traumatic brain injury). Emotional/behavioral disorders are defined by maladaptive behaviors and moods that differ from the norm for age peers, and specific learning disabilities are defined by differences among abilities and achievement within the individual. Some categories (e.g., perceptual impairments, autism, specific learning disabilities, traumatic brain injury) match medical or psychological diagnoses, whereas others (e.g., other health impairment, physical disabilities) cover a range of diagnostic conditions. Several broad categories defined by IDEA and state law (e.g., noncategorical identification, developmental delay, other health impairment) are used as catch-alls for students who clearly display impairment but cannot be assigned a definitive label because age makes specific diagnosis unreliable or because the nature of the problem does not fit into another category.

Because most of the disability categories are broadly defined, characteristics often intersect, and a student may meet the criteria for more than one type of disability. For example, students with behavioral/emotional impairment and specific learning disabilities frequently have dual diagnoses. Students with perceptual or physical impairments span the normal range of cognitive and learning abilities. Although below-average cognitive functioning is an exclusionary factor to identify specific learning disabilities, studies investigating special education placement decision-making have revealed failures to differentiate students with mental retardation and a consequent overuse of the learning disabilities category as more socially acceptable for students who show cognitive deficiencies (Gottlieb, Alter, Gottlieb, & Wishner, 1994; MacMillan, Gresham, & Bocian, 1998). Recent research also has identified significant overlaps between students with language impairments and reading disabilities (McArthur, Hogben, Edwards, Heath, & Mengler, 2000), language impairments and emotional/ behavioral disorders (Caulfield, Fischel, DeBaryshe, & Whitehurst, 1989; Griffith, Rogers-Adkinson, & Cusick, 1997: Toppelberg & Shapiro, 2000), and comorbidity among these three disabilities (Tomblin, Zhang, & Buckwalter, 2000).

In one well-investigated area. the distinction between special education and general education has been called into question. In reading research spanning the past 2 decades, repeated studies have attempted to distinguish between students with specific reading disabilities who have IQ-achievement discrepancies and those without discrepancies. The reading skills profiles of these two groups are remarkably similar, as are the interventions that are most effective (Francis, Shaywitz, Stuebing, Shaywitz, & Fletcher, 1996; Shaywiiz, Fletcher, Holahan, & Shaywitz, 1994; Siegel, 1989; Stage, Abbott. Jenkins, & Berninger, 2003; Stuebing et al., 2002). A general conclusion of this body of research is that reading disabilities occur in children at all levels of cognitive functioning who may or may not meet special education eligibility criteria. Instructional interventions should target the specific reading difficulties or deficiencies that students experience, regardless of their eligibility for special education services.

Special education categories, then, are likely to be heterogeneous rather than homogenous, and the dividing lines are blurred even between some special education and general education groups. According to Brinker (1990), special education categories are primarily administrative units: "We have persisted in our maintenance of categories of exceptionality that encompass individual differences that vary as greatly as the differences between categories" (p. 182). Further, special education classification does not stipulate services or placement options for individual students (Brinker, 1990). If special education classification does not provide sufficient reliable information to prescribe effective services, what does?