|
Article Excerpt
Abstract. The focus of this article is intervention for third-grade students with serious mathematics deficits at third grade. In third grade, such deficits are clearly established, and identification of mathematics disabilities typically begins. We provide background information on two aspects of mathematical cognition that present major challenges for students in the primary grades: number combinations and story problems. We then focus on seven principles of effective intervention. First, we describe a validated, intensive remedial intervention for number combinations and another for story problems. Then, we use these interventions to illustrate the first six principles for designing intensive tutoring protocols for students with mathematics disabilities. Next, using the same validated interventions, we report the percentage of students whose learning outcomes were inadequate despite the overall efficacy of the interventions and explain how ongoing progress monitoring represents a seventh, and perhaps the most essential, principle of intensive intervention. We conclude by identifying issues and directions for future research in the primary and later grades.
**********
Approximately 5-9% of the school-age population may be identified with mathematics disability (e.g., Badian, 1983; Gross-Tsur, Manor, & Shalev, 1996). Although this estimate is similar to the prevalence of reading disability, less systematic study has been directed at mathematics disability (Rasanen & Ahonen, 1995), even though poor mathematics skills are associated with life-long difficulties in school and in the workplace. For example, mathematics competence accounts for variance in employment, income, and work productivity even after intelligence and reading have been explained (Rivera-Batiz, 1992).
Some research illustrates how prevention activities at preschool (e.g., Clements & Sarama, 2007), kindergarten (e.g., Griffin, Case, & Siegler, 1994), or first grade (e.g., Fuchs, Fuchs, Yazdian, & Powell, 2002) can substantially improve math performance. For example, at the beginning of first grade, Fuchs and colleagues (2005) identified 169 students in 41 classrooms as being at risk for math difficulties based on their low initial performance. These children were randomly assigned to a control group or to receive small-group tutoring three times per week for 16 weeks. Results showed that math development across first grade was significantly and substantially superior for the tutored group than for the control group on computation, concepts, applications, and story problems. In addition, the incidence of students with mathematics disability was substantially reduced at the end of first grade, and this reduction in mathematics disability remained in the spring of second grade, one year after tutoring ended (Compton, Fuchs, & Fuchs, 2006).
Nevertheless, despite the efficacy of tutoring, students were not universally responsive. A subset of the tutored students, approximately 3-6% of the school population (depending on the measure and cut-point for the severity of mathematics performance used), manifested severe mathematics deficits. As this example illustrates, the need for intensive remedial intervention persists even when prevention services are generally effective.
In this article, we focus on serious mathematics deficits at third grade. In third grade, serious mathematics deficits are clearly established, and identification of mathematics disabilities typically begins (Fletcher, Lyon, Fuchs, & Barnes, 2007). We provide background information on two aspects of mathematical cognition that present major challenges for students in the primary grades: number combinations and story problems. We describe validated, intensive interventions for number combinations and another for story problems and use these interventions to illustrate six principles for designing interventions for students with mathematics disabilities. Next, using the same validated interventions, we report the percentage of students whose learning outcomes were inadequate despite the overall efficacy of the interventions and we how ongoing progress monitoring therefore represents a seventh and perhaps the most essential principle for intensive intervention. We conclude by identifying issues and directions for future research in the primary and later grades.
NUMBER COMBINATIONS AND STORY PROBLEMS
Number combinations refer to problems with single-digit operands, which can be solved by counting or committing to long-term memory for automatic retrieval. Widespread agreement exists that number combination skill is essential for all children to acquire. The National Research Council (Kilpatrick, Swafford, & Findell, 2001), for example, identified number combination skill as a key component of math proficiency. Moreover, research shows that number combination skill is a significant path to procedural computation and story-problem performance (Fuchs et al., 2006).
To answer number combination problems, typical children gradually develop procedural efficiency in counting. First they count the two sets (e.g., 2+3) in their entirety (i.e., 1, 2, 3, 4, 5); then they count from the first number (i.e., 2, 3, 4, 5); and eventually they count from the larger number (i.e., 3, 4, 5). Also, as their conceptual knowledge about number becomes more sophisticated, children develop additional back-up strategies for deriving answers (e.g., decomposition: [2+2] = 4 + 1 = 5). As increasingly efficient counting and back-up strategies help children consistently and quickly pair problems with correct answers in working memory, associations become established in long-term memory, and children gradually favor memory-based retrieval of answers, although most continue to use a mix of strategies over time.
Students with mathematics disabilities, on the other hand, manifest greater difficulty with counting (e.g., Geary, Bow-Thomas, & Yao, 1992; Geary, Hoard, Byrd-Craven, Nugent, & Numtee, in press). They also persist with immature back-up strategies. Not surprisingly, therefore, they fail to make the shift to memory-based retrieval of answers (Fleishner, Garnett, & Shepherd, 1982; Geary, Widaman, Little, & Cormier, 1987; Goldman, Pellegrino, & Mertz, 1988). When children with mathematics disabilities do retrieve answers from memory, they commit more errors and manifest unsystematic retrieval speeds compared to younger, typically developing counterparts (e.g., Geary et al., in press; Geary, Brown, & Samaranayake, 1991; Gross-Tsur et al., 1996; Ostad, 1997). In fact, number combinations constitute a signature deficit of students with mathematics disabilities (e.g., Fleishner et al., 1982; Geary et al., 1987; Goldman et al., 1988).
Conventionally, number combinations are incorporated into the curriculum at kindergarten through second grade, although most general education programs do not allocate systematic attention to developing strategies for fluent number combination performance. Nevertheless, most typically developing students are well on their way toward automatic retrieval of number combinations as their major solution strategy by the beginning of third grade. Therefore, when students still manifest deficiencies involving number combinations at the beginning of third grade, a pressing need exists for remediation.
In contrast to number combinations and other aspects of calculation skill, story problems incorporate linguistic information that requires children to construct a problem model. Whereas a calculation problem is already set up for solution, a story problem requires students to use text to identify missing information, construct the number sentence, and derive the calculation problem for finding the missing information. This transparent difference would seem to alter the nature of the task and, in fact, some research suggests that computation and math problem solving may be distinct aspects of mathematical cognition (Fuchs et al., in press).
Four large-scale studies have addressed indirectly the distinction between computation and problem solving by examining the cognitive correlates of these domains of mathematical cognition in large representative samples.
In a study of 353 first through third graders, Swanson and Beebe-Frankenberger (2004) identified working memory as an ability that contributed to strong performance across both areas of mathematical cognition, but some unique cognitive abilities were also important, including phonological processing for computation and fluid intelligence as well as short-term memory for story problems.
In the second study, Swanson (2006) followed these students' development of calculation and problem-solving skill over one year. He identified predictors of computation (inhibition or controlled attention, vocabulary knowledge, visual-spatial working memory) that differed from problem solving (working memory's executive system; i.e., listening span, backward digit span, and digit/sentence span).
The third study involved a sample of 312 third graders. Here Fuchs et al. (2006) examined the concurrent cognitive correlates of computation versus story problems, this time controlling for the role of arithmetic skill within story problems. Teacher ratings of inattentive behavior emerged as a correlate common to both domains of math, but the remaining abilities differed: for computation, phonological decoding and processing speed; and for story problems, nonverbal problem solving,...
|
