Children With Learning Difficulties in Mathematics

Increasingly, state and local agencies are using high-stakes assessments to make important educational decisions. Although there are serious consequences for children who fail them, requires that children with learning disabilities be included in the assessments. Typically high-stakes testing in mathematics begins at the end of third grade. However, little is known about the early achievement growth trajectories of children who are behind in this subject area. For example, are the growth rates for children who show mathematics difficulties (MD) in early primary school different from those for children who show no difficulties? Do some children with MD catch up in their achievement levels while others stay behind? We addressed these issues by examining children’s early achievement in mathematics longitudinally. We identified two groups of children with MD, that is, children with difficulties in mathematics but not in reading (MD only) and children with difficulties in mathematics and difficulties in reading (MD–RD). For contrast, we also identified two groups of children without MD, namely children with difficulties in reading but not in mathematics (RD only) and children with normal achievement in mathematics and in reading (NA).

Although many studies define children with MD as a single group of low achievers (e.g., Geary, 1990; Geary, Brown, & Samaranayake, 1991; Ostad, 1997; Russell & Ginsburg, 1984), young children with MD only show a different pattern of cognitive deficits than do children with MD–RD. Recently, Hanich, Jordan, Kaplan, and Dick (2001) found that second graders with MD, regardless of whether they were MD only or MD–RD, performed worse than NA children in most areas of mathematical cognition. However, children with MD only outperformed children with MD–RD on orally presented arithmetic combinations (e.g., “How

much is 8 7”) and on story problems (e.g., “Jen had 7 pennies. Then she gave some pennies to Joe. Now Jen has 2 pennies. How many pennies did she give to Joe?”). The two MD groups did not differ on tasks assessing approximate arithmetic or estimation (e.g., 50 – 9 11 or 40), place value, rapid fact retrieval, calculation principles, and written computation with multidigits. Children with MD only seem to have an advantage over children with MD–RD on mathematical tasks that may be mediated by language but not on ones that depend more on understanding of numerical magnitudes, visuo-spatial processing, and automaticity.

Geary, Hamson, and Hoard (2000), Geary, Hoard, and Hamson (1999), Jordan and Hanich (2000), and Jordan and Montani (1997) report similar findings with subgroups of primary school children  with MD.

To date, few studies have examined mathematics difficulties from a longitudinal developmental perspective. Most approaches to studying children with MD assess outcomes at a single time point (Jordan, Hanich, & Uberti, in press). However, measurement with only one time point cannot determine a child’s growth rate, which is fundamental to understanding learning and learning difficulties (Francis, Shaywitz, Stuebing, Shaywitz, & Fletcher, 1994). Measurement of growth through longitudinal investigations has been a primary interest in the study of reading difficulties (e.g., Blachman, 1994; Byrne, Freebody, & Gates, 1992; Francis, Shaywitz, Stuebing, Shaywitz, & Fletcher, 1996). For example, growth curve modeling has been used to examine whether children with reading difficulties are characterized by developmental lags or by cognitive deficits (S. E. Shaywitz, Shaywitz, Fletcher, & Escobar, 1990) and to examine reading skill development in children who have received early interventions and later remediation (Foorman et al., 1997). Much less work, however, has been done in mathematics difficulties, although in a few studies these difficulties have not been stable over time. For example, Silver, Pennett, Black, Fair, and Balise (1999) found that a classification of isolated arithmetic weaknesses is less stable over a year and a half period than is a classification of pervasive weaknesses in arithmetic, reading, and spelling. Moreover, Geary (1990), Geary et al. (1991), and Geary et al. (2000) identified a group of “variable” children who showed mathematics difficulties on an achievement test in first grade but not in second grade. Unlike children with persistent MD, variable children do not show underlying deficits in numerical or arithmetical cognition. They appeared to have outgrown their early developmental delays or have been misidentified.

In the present state, we have to built on previous research in mathematics difficulties in several ways. First, we have to use educationally relevant classifications of children (i.e., MD only vs. MD–RD). Second, we  have to measure children’s growth rates in reading and mathematics within a longitudinal framework. We began our study in second grade because this is the earliest point at which mathematics difficulties can be reliably identified, at least with current testing instruments (Geary et al., 1991). In addition to looking at overall mathematics performance, our data allowed us to look at performance in mathematics calculation and applied problem solving separately. Likewise, in reading we examined overall reading performance as well as performance in word decoding and passage comprehension. Our longitudinal data has to be analyzed by means of growth curve modeling, a procedure that has been used extensively by researchers in numerous fields for the study of intraindividual differences in change (Bryk & Raudenbush, 1992; Rogosa, Brandt, & Zimowski, 1982; Willett, 1988; Willett & Sayer, 1994). Third, we have to balance our subgroups as closely as possible with respect to ethnicity, income level, and gender, allowing us to use these variables as controls in the growth models.

We should use IQ data on all children (when they were in third grade) to determine the effects of underlying cognitive abilities on children’s growth. Finally, we have to record whether children were receiving specialized interventions in reading and mathematics, whether children were retained at the end of second grade, and what type of mathematics program children were taught with to determine the impact of these factors on achievement growth. Regarding the latter variable, we have to follow with one of two distinct approaches:

A problem-centered mathematics approach, with little emphasis on facts or algorithms, or a more traditional approach, which provides more explicit instruction in mathematics skills. We predicted that children with MD–RD would grow at a slower rate than children with MD only in mathematics achievement because of their lack of compensatory strengths in reading. We did not predict that children with MD–RD would grow more slowly than children with RD only in reading achievement, particularly in word decoding, because “garden-variety” poor readers develop in a fashion similar to children with specific reading problems (e.g., Shaywitz, Escobar, Shaywitz, Fletcher, & Makuch, 1992). 

We  can also predict that achievement growth rates would be affected by income level and intellectual ability. That is, holding achievement group constant, we expected low-income children and children with lower intellectual functioning to grow at a slower rate than middle-income children and children with higher intellectual functioning. Finally, we can predict that instruction in mathematics might have a differential effect on children’s growth, according to their achievement group. That is, children with MD might benefit from more explicit instruction in skills more than children without MD.