Learning disabilities in mathematics: a review of the issues and children's performance across mathematical tests.

Abstract

In recent years, the research literature on mathematics disabilities (MD) has shown relative growth. The characteristics of children with MD have thus been investigated from different perspectives. The purpose of this paper was to review one aspect of this growing literature: the performance of children with MD on arithmetic facts and word problems. Alongside this primary concern, the paper examined some of the issues surrounding MD. The results suggest that because of the unresolved issue on definition, investigators have used different operational definitions for MD. Despite this variation, studies were consistent in showing that the MD/RD children's performances on both number facts and word problems were significantly worse than the performances of NA children. Results were partly inconsistent when it comes to differences between MD-only and NA children, however. Whereas most studies documented better performance for the NA children, some showed that the two groups had comparable performances on both number facts and word problems particularly when these tasks were not complex or timed. The implications for further research are discussed.

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Introduction

For some years now, mathematics disabilities (MD) have been recognized as a type of learning disabilities (LD), as evidenced by the inclusion of mathematics in LD definitions (Bryant, Bryant, & Hammill, 2000). There is also general consensus among professionals in the field that MD is widespread in young children and that it has serious educational consequences (e.g., Bryant et al., 2000; Ginsburg, 1997; Jordan & Hanich, 2000; Jordan & Montani, 1997; Ostad, 1998). Despite this condition, discussions and research on LD have mostly been limited to difficulties in the area of reading and spelling for many years (Ginsburg, 1997; Hitch & McAuley, 1991; Jordan & Hanich, 2000; Jordan & Montani, 1997; Rourke & Conway, 1997). Little attention has thus been given to MD and there exists a relative lack of research in the area (Badian, 1983). The limited effort in research on MD is particularly evident when one examines the research efforts devoted to understanding poor reading achievement and reading disabilities (RD; Dockrell & McShane, 1993; Geary, Hoard, & Hamson, 1999).

However, in an attempt to redress this condition, researchers in the last decade have explored characteristics of children with MD from several perspectives. Consequently, the research literature is increasing year by year, although relatively slowly. The main purpose of this paper was to review this growing literature with a specific focus on comparing the mathematics performance profiles of children with MD and their normally achieving (NA) peers. The performance of these children could be assessed on different mathematical subtests such as simple computations, quantitative concepts, and word problem solving. Because such subtests are considered different at least in their difficulty (see Jordan & Montani, 1997; Ostad, 1998), examining between-group and within-group differences could provide useful information. The examination could raise the following important questions: Do students with MD differ from NA children in their performance? If so, on which subtests? Does their performance vary with difficulty of the subtests? Or do they perform worse than NA children on all subtests? Answers to these questions could provide useful directions for work with MD children as well as for further research. Among other things, the answers could help plan and develop appropriate intervention/remedial programs for children with MD. But before reviewing this literature, the paper first explores issues surrounding the MD concept. The rationale for raising these issues when the primary concern is examination of group differences derives from the belief that exploring group differences in an area like MD could not be complete without a discussion of the controversial issues.

Issues Surrounding MD: An Overview

Issues on Definition

Definition is central in the identification and provision of services for children with LD (Shaywitz, Fletcher, Holahan, & Shaywitz, 1992). However, there is no unequivocally accepted, precise definition of LD (e.g., Kavale & Forness, 2000). Because complete agreement on definition is nonexistent, researchers use different methods and/or criteria to identify students with MD from a student population. These methods may be generally classified into two categories. The first category comprises methods that employ discrepancy between achievement and ability as a major criterion. The second includes different forms that basically employ low achievement as a major criterion.

The ability-achievement discrepancy formulas differ from each other in terms of the magnitude of the discrepancy (e.g., the commonly used discrepancies have been 1 and 1 1/2 standard deviations between achievement and ability scores). The basis for establishing the discrepancy could also be one source of variation; whereas some determined the discrepancy by simply comparing standard scores, others employed regression methods (MacMillan, Gresham, & Bocian, 1998). Another variety involves a calculation of the discrepancy between mental age taken from an IQ test and grade-age equivalent obtained from a standardized achievement test. A discrepancy of two years was often considered enough to indicate MD/LD (Hallahan & Kauffman, 2000). The variation in the formulas suggests the possibility that students who would be identified as having MD/LD might differ across studies depending on the specific discrepancy formula employed.

Many researchers have criticized the use of the ability-achievement discrepancy criterion in the identification of students with LD/MD for several conceptual and statistical reasons (for a review, see Aaron, 1997; Swanson, 2000). Specifically, opposing the use of discrepancy formulas and presenting evidence that IQ is irrelevant to the LD definition, Siegel (1989, 1999) and Stanovich (1989, 1999), among others, have suggested that our focus be changed to poor performance on specific tests of achievement. However, even with the poor or low achievement criterion, studies may not come up with identical sample, given a population, because there seems to be no agreed upon cutoff point to distinguish MD and NA children. In other words, researchers often employ different operational definitions of low or poor achievement.

There is, however, some consensus that the 25th percentile point on an achievement measure be used as a cutoff point to distinguish students with MD/LD from NA students (e.g., Siegel, 1994; Siegel & Ryan, 1989; see also Geary, Hamson, & Howard, 2000). Practically, however, researchers employed cutoff points ranging from a low of 14th percentile (Ostad, 1997, 1999, 2000) to a high of 46th percentile (Geary, 1990). With cutoff points as variant as these, the comparability of the MD sample across studies is questionable. In connection with this, it should be noted that studying the mathematical deficits of MD students calls for a comparison of their performance with that of NA children. But the criteria used to select these NA children likewise vary across studies. Particularly when the low achievement criterion is used, the cutoff points are as variant as (if not more variant than) those used to identify MD students. One could easily sense this by simple inspection of the cutoff points employed across studies. These included a score on a mathematics achievement test at or above the 14th percentile (Ostad, 1999, 2000), the 25th percentile (e.g., Wilson & Swanson, 2001), the 30th percentile (e.g., Jordan & Montani, 1997; Siegel & Ryan, 1989), the 40th percentile (e.g., Jordan & Hanich, 2000; Hanich et al., 2001; Swanson & Sachse-Lee, 2001), the 46th percentile (e.g., Geary, 1990), and the 66th percentile (Geary et al., 2000). In addition, while some researchers randomly selected the comparison group from NA children (e.g., Badian, 1999; Badian & Ghublikian, 1983), others employed a comparison group matched with the MD group in terms of one or more variable (e.g., Ostad, 1997).

Thus, one challenge in studying the characteristics of students with MD is lack of consensus on definition. The absence of an unequivocally accepted definition is indeed the principal problem in the field. As a result of the definitional issue, no uniform procedure is employed to identify MD students and given a student population, it is possible that different researchers select different samples of MD students. To make things worse, there is no agreement on how to select the non-MD, comparison group from general education students.

Prevalence of MD

MD often occurs along with reading and/or spelling difficulties. But it can also occur independent of these language-based difficulties. Regardless of its occurrence with other problems or alone, a number of investigators have argued that MD is relatively common in young children (e.g., Dockrell & McShane, 1993; Ginsburg, 1997; Jordan & Hanich, 2000; Jordan & Montani, 1997; Ostad, 1998). But there is no definitive figure that indicates how common the problem is among the student population.

Bryant et al. (2000) estimated the prevalence of mathematics weaknesses among identified LD students. The investigators asked professionals who were instructors of the LD students to identify whether each of the LD students who participated in their study had weaknesses in any of six areas (i.e., listening, speaking, reading, writing, mathematics, and reasoning). According to the ratings, 870 of the 1724 students with LD (i.e., about 50%) had weaknesses in mathematics. While the results suggest that MD is indeed very common among the LD population, they could not help to estimate the prevalence rate in the student population as a whole.

Badian's (1983) and Kosc's (1974) studies are among the most frequently cited investigations in the literature in relation to estimating the prevalence of MD. It is interesting to note that the two studies were undertaken in different countries apparently at different times. Nonetheless, they came up with the same figure (i.e., 6.4%). Gross-Tsur, Manor, and Shalev (1996) have also reported a similar prevalence figure (i.e., 6.5%) for developmental dyscalculia. In her recently published longitudinal study, Badian (1999) again revealed with a large sample (over 1000 students) that the prevalence of MD--including both MD only and MD with comorbid RD--was 6.9 percent.

Many investigators appeared to have accepted that between six and seven percent of elementary and junior high school children suffer from a deficit that interferes with their ability to acquire grade-level (or age-level) mathematical competencies (e.g., Fuchs & Fuchs, 2002; Geary, 1993; Geary et al., 1999; Ginsburg, 1997; Rourke & Conway, 1997). Some studies, however, have reported somewhat higher incidence rates. For example, a study conducted in Norway (Ostad, 1998) indicated that the schools' support services had selected about ten percent of primary school children as having learning problems in mathematics when the children were in grade two. The study did not indicate the criteria employed by the schools' support services in selecting the children, however. Share, Moffitt, and Silva (1988) reported even higher incidence rate among a sample of 459 New Zealand 11 year-old children followed from birth. Specifically, the study revealed that 8.5 percent of the children had MD/RD whereas 6.5 percent had a specific MD, giving a total of 15 percent who were poor in mathematics.

With regard to the incidence rate, one final point is noteworthy. As with other investigators, one can tentatively accept the above incidence rate for MD. But it is important to note that this is not something final. Like the larger LD field, estimating the prevalence of MD is dependent on several unresolved issues. The principal issue is on definition. All other issues will be settled more or less easily once there is an agreed upon definition. For instance, where there is no consensus on what constitutes MD, investigators will continue to disagree on how to identify children who have this disability. If there is no agreement on how to identify the children in question, it is obviously difficult to agree on the incidence rate. The above estimates could not therefore be taken seriously until these differences are resolved.

MD Subtypes and their Stability

LD represents a heterogeneous group of disorders rather than a unitary phenomenon. Although subdividing these disorders along some dimensions and determining distinct subtypes is not an easy task, researchers have attached great value to conducting studies in the LD field based on the subtypes. In general, two methods have been used to subdivide a heterogeneous LD sample into smaller, more or less homogeneous subgroups, namely, the clinical inferential approach and multivariate classification techniques (McKinney, 1984; Silver et al., 1999). The clinical inferential approach classifies LD children into fairly homogeneous groups based on a priori criteria that consider the students' patterns of intellectual abilities (verbal and performance IQ) or patterns of achievement (in arithmetic and reading). On the other hand, the multivariate classification techniques, namely Q-factor analysis and cluster analysis, categorize LD children based on patterns of test scores but do not employ a priori criterion (McKinney, 1984; Silver et al., 1999). Regardless of the specific method employed, many studies determined MD subtypes on the basis of children's mathematics and reading achievement patterns. Three subtypes are frequently adopted in studying differences among children with and without LD. These are disabilities in mathematics without accompanying reading problems (MD-only), disabilities in both mathematics and reading (MD/RD), and disabilities in reading but not in mathematics (RD-only; e.g., Geary et al., 1999; Jordan & Hanich, 2000). Since the third group of children does not have difficulty in mathematics, investigators (e.g., Jordan & Montani, 1997) often prefer to omit it whenever they deal with MD. As a result, they often use only two MD subtypes with somewhat different terminology: MD-specific and MD-general. While MD-specific is another name for the MD-only subtype, MD-general and MD/RD are used interchangeably to connote difficulties in both mathematics and reading.

In a similar vein, Rourke and his associates (e.g., Rourke, 1993; Rourke & Conway, 1997) identified two MD subtypes (group R-S and group A) based on a series of studies. According to Rourke and Conway, whereas the R-S group exhibited a pattern of verbal IQ that is less than performance IQ, group A children exhibited the opposite pattern (that is, performance IQ less than verbal IQ). The authors characterized the two groups as exhibiting equally impaired levels of arithmetic achievement for very different reasons. Rourke and Conway further illustrated that children in the R-S group were apparently experiencing MD due to verbal deficiencies, whereas group A children appeared to be encountering greater difficulty with the visual-spatial and nonverbal reasoning dimensions of arithmetic performance. It is difficult to ascertain whether Rourke's two MD subtypes match the MD-only and the MD/RD subtypes mentioned earlier. But if they do, Rourke's group A children are likely to be similar to children with MD-only while the R-S children are probably similar to the MD/RD children.

One related issue when discussing subtypes of MD, or that of LD in general, is their stability. The issue of stability is concerned with the question: Do children who have been identified as having MD based on characteristics assessed at a single point in time continue to display the same deficits and remain in the same MD classification if assessed at a later time? The issue of stability of subtypes in the LD field is of paramount importance. For example, if a child is labeled as having MD for a temporary learning problem in mathematics, the label could have detrimental effects associated with the stigma (see Leondari, 1993; Higgins, Raskind, Goldberg, & Herman, 2002). To avoid such negative effects, we need to avoid a one-time assessment, which in essence means changing the bases of identification from cross-sectional data to longitudinal data. In relation to this, some investigators have argued against identifying MD students on the basis of outcomes assessed at a single point in time (Ostad, 1997; Silver et al., 1999). These investigators asserted that if children are identified as having MD based on tests administered only once, it is possible that a sample so chosen will be heterogeneous in the sense that it may be composed partly of children with temporary difficulties and partly of children with difficulties of a permanent nature (Ostad, 1997).

Apart from the logical argument advanced by researchers, empirical support is limited because only a small number of studies have investigated stability of LD classifications. The few available studies, however, came up with findings that called into question the validity of LD classification based on measurements obtained at a single point in time. For example, Silver et al. (1999) demonstrated that only one-third of the children originally classified as having MD-only continued to display this pattern of isolated MD at 19-month retesting. But considering all children with MD (i.e., MD-only, MD/RD, MD with comorbid spelling difficulties, and MD with comorbid RD and spelling difficulties), the study found approximately 50 percent of the participants to have MD after the 19-month interval. It was also interesting to note from the study that the proportions of students who continued to display the same deficits after the indicated time interval were approximately the same across three operational definitions of LD. Likewise, Ostad (1997, 1999) disclosed that about 22 percent of MD students in his study had improved their mathematics performance over a two-year period, no longer satisfying the operational definition of MD (i.e., score below the 14th percentile). Furthermore, about 45 percent of MD children in Geary's (1990) study improved their mathematics performance over a one-year period, showing no apparent deficits in mathematics. These findings suggest that not all students who are identified as having MD at one point in time continue to have MD.

All in all, identifying students with MD based on measurement of the children's characteristics at a single point in time carries a risk. The alternative that minimizes the problem of misidentification, among other things, is to base the identification process on longitudinal data and to examine achievement gains, if any, across time (Geary, 1990). That is, gathering data at least twice on a student who is suspected to have MD is necessary, if not sufficient, to determine whether or not that student has MD.

The Mathematics Performance of MD and NA Children

Numerous studies have found that students with MD experience difficulties in mathematics more frequently than their peers without MD (see Miller & Mercer, 1997). Children with MD encounter challenges in a range of mathematical problems including mathematics computations, concepts, and problem solving (Zentall & Ferkis, 1993; Miller & Mercer, 1997). Research also indicates that students with MD not only have lower performance on mathematics tests relative to age- or grade-matched NA peers but function at an alarmingly lower level. For example, Cawley and Miller (1989) reported that eight- and nine-year olds with LD performed at about a first-grade level on computation and application problems. Fleischner, Garnett, and Shepherd (1982) likewise disclosed that the performance of sixth graders with LD on basic addition facts was no better than that of third graders without LD. The following subsections separately examine performance on arithmetic facts and word problems.

Performance on Simple Arithmetic (or Number-Fact Problems)

The phrases number-fact problems, simple arithmetic, and simple computations have been used in the literature to represent simple problems based on arithmetic operations (e.g., addition, subtraction). Despite differences in terminology, the tasks represented by these problems are generally more similar than different. This section reviews differences, if any, between MD and NA children on these tasks.

Research indicates that there are significant differences between the performance of students with MD and their NA peers on simple arithmetic problems. The performance of the former group has been reported to be inferior (e.g., Algozzine, O'Shea, Crews, & Stoddard, 1987; Garnett & Fleischner, 1983; Ostad, 2000). But as we will see shortly, such a conclusion oversimplifies the issue because it does not consider other pertinent variables. We need to take these variables into account because the group differences sometimes vary as a function of the variables.

Hitch and McAuley (1991) employed visually presented calculations and oral problems (involving reference to diagrams) in comparing the performance of children with MD and NA. The study found the MD group's performance to be significantly lower than the control group on both types of problems. In a similar manner but using two MD groups rather than one (i.e., MD-only and MD/RD), Jordan and Hanich (2000) investigated performance of second-grade children on number facts and written calculation. In this study, whereas the MD/RD group performed significantly worse than the NA group on number facts and written calculation, the MD-only group performed as well as the NA group on both tasks. Another study with second graders (Hanich, Jordan, Kaplan, & Dick, 2001) replicated the above findings for the MD/RD group. Nevertheless, the study did not replicate the findings for the MD-only group; that is, the MD-only group performed worse than the NA group on both number facts and written computations.

Despite possible differences of the measures the first studies employed (Hitch & McAuley, 1991; Jordan & Hanich, 2000), the inconsistent findings would seem to hint the value of dealing with two MD subgroups rather than one. Yet, the examination of the findings of the latter two studies (Jordan & Hanich, 2000; Hanich et al. 2001) that employed two MD subgroups also produced results that were partly inconsistent. Whereas the studies produced consistent results for the MD/RD group, this was not so for the MD-only group.

The above three studies did not give any indication about the relative difficulty of the problems in each section of the mathematics tests they employed or as to which section contained more difficult problems. But some investigators (see Ostad, 1998) have formulated a model that classifies number-fact problems into simple and complex. Accordingly, when some combinations of an unknown and numbers represent a simple algebraical problem, for example, of a sum or a difference, the relative position of the unknown in the equation determines the difficulty of the problem. More specifically, number-fact problems in which the unknown is placed either in the first ([+ or -] b = c) or the second position (a [+ or -] = c) in the equation are more difficult than problems where the unknown is the sum (a [+ or -] b =) or generally the result of a linear calculation.

Using this model with longitudinally followed samples of MD and NA children in grades two, four, and six, Ostad (1998) investigated, among other things, differences in the performance of the two groups on number-fact problems. The results indicated that the NA children's performance was significantly better than the MD group. However, this was not true for all kinds of items. The MD group performed as well as the NA group on simple problems (of the type a [+ or -] b=) but significantly worse on more difficult or complex problems (of the type a [+ or -] = c and [+ or -] b = c). Ostad's (1998) findings suggest that difficult number-fact problems, but not easier ones, are likely to differentiate the two groups in a manner that is consistent with their achievement. Thus, if Ostad's results are generalizable across samples of MD students, then the difficulty level of the problems would be a moderator variable that should be taken into account in future research. Nevertheless, Jordan and Montani's (1997) study did not support Ostad's findings. In Jordan and Montani's study, unlike Ostad's, the NA children performed significantly better than both the MD-only and the MD/RD children on both kinds of problems. Also, the MD-only group performed significantly better than the MD/RD group.

Although they used the same model in classifying the problems as simple and complex, the above two studies revealed results that were partly inconsistent. While Ostad's (1998) study revealed significant differences between the two groups in complex problems only, Jordan and Montani (1997) found differences in both simple and complex problems. Along with the inconsistent findings, there were other notable differences between the studies, however. First, though both studies employed the same operational definition for MD (that is, low achievement), they used different cutoff scores. Whereas Jordan and Montani used the 30th percentile as a cutoff point to distinguish MD and NA children, Ostad used the 14th percentile. Second, the MD children in Jordan and Montani's study were selected from general education classes that received no special education services. In contrast, Ostad's MD children received remedial instruction in mathematics although they remained in regular classes for most of the school day. Two important factors are noteworthy in this latter difference: the MD label and the remedial instruction. It is widely believed that labeling and remedial instruction do affect children's academic performance in opposite ways. That is, while labeling seems to affect performance negatively, remedial instruction appears to improve children's performance. Third, the studies administered mathematics tests that were not the same. Overall, these differences could possibly explain the inconsistent results reported in the two studies although one cannot be certain whether or not these differences fully explain the discrepant findings.

Another variable which could be a source of variation between MD and NA children, as Jordan and Montani's (1997) findings suggest, is mathematics performance in timed and untimed conditions. According to Jordan and Montani, relative to the performance of the NA group, the performance of the MD-only group varied as a function of time condition. Specifically, the MD-only group performed as well as the NA group in untimed conditions but worse than the same group in timed conditions on both simple and complex number-fact problems. In contrast, the MD/RD group performed worse than the NA group in all conditions. Furthermore, although the MD-only group's performance on simple number-fact problems was better than the MD/RD group in both timed and untimed conditions, this was not the case for complex problems. On the latter problems, the MD-only group performed better in untimed but not in timed conditions.

In summary, studies that employed only one MD group in their comparison consistently revealed better performance for the NA group on number-fact problems. However, one study (Ostad, 1998) that categorized the number-fact problems into simple and complex revealed better scores for the NA group only in complex number facts. In this study, the MD group performed as well as the NA group in simple number facts. On the other hand, studies that compared NA children with two MD subgroups disclosed somewhat different results. These studies showed that the MD/RD group performed significantly worse than the NA group on number facts irrespective of conditions (timed versus untimed problems, simple versus complex problems). But results regarding the MD-only group's performance on number facts were mixed. Whereas some investigators found the MD-only group's performance to be comparable with that of the NA group (Jordan & Hanich, 2000), others revealed that this is so only when these children did perform without time limits (Jordan & Montani, 1997). The Hanich et al. (2001) study, however, reported significantly worse performance for the MD-only group compared to the NA group.

Performance on Word Problems

It is widely held that solving arithmetic word problems is different from solving number-fact problems. Most word problems require analysis and interpretation of the given information before deciding on solutions (Parmar, Cawley, & Frazita, 1996). Also, apart from basic ability in calculation, solving word problems requires sufficiently developed verbal skills (Cummins, 1991; Jordan, Levine, & Huttenlocher, 1995). Furthermore, most word problems involve more than one step to reach a solution. Hence, the literature on word problem performance needs a separate analysis.

Research that compared the performance of MD and NA children on arithmetic word problems indicates that the NA children's performance is superior. Ostad's (1998) longitudinal study with second-, fourth-, and sixth-grade children, for example, supported this conclusion. Using two MD subgroups and a comparison group of NA children in the second grade, Jordan and Hanich (2000) and Hanich et al. (2001) produced a result that was generally consistent with that of Ostad (1998). That is, the MD children (both MD-only and MD/RD) performed significantly worse than the NA group. But comparison of the two MD groups revealed better performance for the MD-only group. Several other studies have supported the result that students with MD have a significantly poorer word problem (or applied problem) performance compared to their NA peers (e.g., Algozzine et al., 1987; Englert, Culatta, & Horn, 1987; Jordan & Montani, 1997; Montague & Applegate, 2000; Parmar et al., 1996). Nevertheless, like the findings for number-fact problems, the examination of other important variables would extend our knowledge of the group differences in word problem performance. Of importance, for example, is the examination of between- and within-group differences across grade or age levels, problem types, problem structures or formats, and time conditions.

Following this line of inquiry, Ostad (1998) analyzed performance of the two groups across grades (2, 4, and 6). According to the author, whereas the NA children's performance improved gradually across grades, the performance of the MD children reached its maximum by grade two and was almost flat after that. The study, however, indicated that the mean scores for the MD children were not the same across problem types; that is, they scored relatively higher on easier arithmetic word problems than on more difficult arithmetic word problems.

Cawley and Miller (1989) conducted a within-group analysis of LD students' word problems (or applied problem) performance across 10 different age levels (ages 8 through 17, N = 220). Consistent with Ostad's (1998) findings, the study disclosed that the LD students' performance was far below grade level. For example, 16- and 17-year olds' performance on applied problems was almost equivalent to that of an average fifth grader. Unlike Ostad, however, they reported a steady increase in mean percentage correct across ages (38.44% at age 8 to 70.58% at age 17). Explaining this result, Cawley and Miller indicated that students with LD could become more proficient with certain items and their mean score might increase as a result although the discrepancy between age-in-grade and level of achievement widened.

As in the above study, Jordan and Montani (1997) found that the MD/RD group performed worse than the nonimpaired group across time conditions and problem types. However, differences between the NA and MD-only groups tended to vary as a function of the time condition. The MD-only group performed as well as the NA group on both problem types in untimed but not in timed conditions. On the other hand, although the MD-only group performed significantly better than the MD/RD group on simple word problems in both timed and untimed conditions, the former group outperformed the latter on complex problems in untimed but not in timed conditions. Overall, the results show that the MD-only group needs longer time when it comes to solving complex word problems and the authors linked the relatively lower performance of this group in timed conditions to basic problem in fact retrieval.

In a similar manner, Parmar et al. (1996) investigated problem-solving differences between students with and without mild disabilities (the former category comprising children with LD and with behavior disorders) in grades three through eight. The study employed word problems that made comparisons possible across the four arithmetic operations, direct-indirect statement format, presence-absence of extraneous information, and one-versus two-step problems. The results showed that students with mild disabilities performed at a significantly lower level than did the comparison group in each problem category.

Apart from between-group differences, the Parmar et al. (1996) study disclosed some important within-group differences. For example, children with mild disabilities performed worse on both indirect and two-step problems that on those that contained extraneous information. Simple inspection of the descriptive statistics also indicates that children with disabilities particularly those in third and fourth grades were unable to solve even a single indirect or two-step problem. Furthermore, one could notice a sharp decrease in mean scores from addition toward division problems. Thus, the mathematics performance of children with mild disabilities was not only different from their NA peers but also varied across mathematical subtests. For example, they tended to perform better on addition and subtraction problems than on multiplication and division items.

A comprehensive review of research of the word problem performance of LD and comparison students by Zentall and Ferkis (1993) indicates that cognitive ability (or IQ) and reading comprehension do contribute, at least partly, to the group differences. According to the authors, much more specific group differences emerge when both IQ and reading comprehension are controlled. Specifically, they suggest that students with disabilities encounter greater difficulty with problems dealing with abstract mathematical concepts and with those requiring mixed actions, mixed operations, and a mixed order of operations.

In summary, studies have shown that the performance of students with MD on problem solving tasks was significantly lower than that of non- MD students. This result was demonstrated to be true with elementary (e.g., Ostad, 1998; Parmar et al., 1996), 7th- and 8th-grade (e.g., Montague & Applegate, 2000; Parmar et al., 1996), and secondary students (e.g., Algozzine et al., 1987). The group differences were also observed across a variety of problems that emphasized quantitative concepts and application (Jordan & Montani, 1997; Parmar et al., 1996). Furthermore, evidence suggests that the comparison children were faster problem solvers than MD students (Englert et al., 1987). Research has shown that the MD students' performance on mathematical problem solving items does not progress to a level that normally occurs at a certain grade or age (Ostad, 1988; see also Parmar, Cawley, & Miller, 1994; Zentall & Ferkis, 1993). In other words, compared to their NA peers students with MD generally demonstrate a slower growth in mathematical problem solving, their scores showing no significant improvement from year to year.

With respect to within-group differences (or performance variation across different mathematical subtests), on the other hand, research has revealed similar trends for both NA and MD children. Both groups of children performed significantly better on number-fact problems than on word problems (Jordan & Montani, 1997; Ostad, 1998). In other words, number-fact problems were easier than word problems for both groups of children. This finding held true regardless of whether the problems were simple or complex (Algozzine et al., 1987; Jordan & Montani, 1997). In general, MD students demonstrated their highest level of achievement with computation of basic facts (see Zentall & Ferkis, 1993) compared to word problems, and with addition and subtraction relative to multiplication and division problems (Parmar et al., 1996). Finally, studies have shown that the MD-only children were better problem solvers than were MD/RD children. Some researchers have suggested that the latter children have basic difficulties with problem conceptualization, use of calculation procedures, and rapid retrieval (Jordan & Montani, 1997).

Discussion

The review generally indicates that the performance of MD children was significantly worse than that of NA children. Even in mathematical computations on which both groups were found to obtain better scores than they did on word problems, the NA group performed better than the MD group in most studies. Irrespective of the kind of subtest administered, the MD/RD children performed significantly worse than did the NA children. But results regarding differences between the MD-only and NA children were mixed. Whereas most investigators found significant differences on number facts as well as word problems between the groups, some did not find such differences especially when the tasks were simple or untimed.

It is reasonable to assume that good performance on computation and problem solving subtests is dependent at least partly on knowledge of basic quantitative concepts. Alternatively, it is reasonable to assume that deficits in the mentioned areas could be due partly to insufficient knowledge of basic mathematical concepts. Unfortunately, investigations of the mathematics performance of MD students have focused more on number facts and word problem solving than on quantitative concepts. The few studies that examined knowledge of basic mathematical concepts among MD students showed that these students do not possess knowledge of many such concepts (Parmar et al., 1994). This finding was also confirmed by other studies that investigated group differences on knowledge of mathematical concepts (Badian, 1999; Montague & Applegate, 2000; Zentall & Ferkis, 1993). For example, Badian (1999) found significant group differences among MD, RD, MD/RD, and NA children who were followed from grade one to grades seven or eight. Specifically, although students with MD obtained a significantly higher mean score than the MD//RD children, they performed worse than both the NA and RD children. Examination of the quantitative concepts data recently reported by Montague and Applegate (2000) also showed significant group differences among MD, average achieving, and gifted students, MD students receiving significantly lower scores than the other two groups.

After confirming the existence of significant differences between MD and NA children in one set of mathematical problems or another, the reader may want to know why this is so. In response, research suggests that both cognitive and noncognitive factors could contribute for the differential performance of children with MD. Specifically, problem solving strategies, problem solving persistence, attributions, academic self-perceptions, and perceptions of difficulty of tasks could contribute to poor problem solving among students with MD (Montague & Applegate, 2000). Other explanations included problems associated with retrieval of facts (e.g., Geary et al., 1999) and speed of processing (e.g., Bull & Johnston, 1997; Kirby & Becker, 1988; Ostad, 2000). Much more attention in explaining the group differences, however, has been directed toward the investigation of two factors -strategy use and retrieval of facts. As a result, strategy use differences and differences in retrieval of facts between MD and NA children have received focal attention in the MD literature for some time.

Studies suggest that students with MD often approach mathematical problems differently than their nondisabled peers because they lack cognitive strategies that are needed for effective problem solving (Montague & Applegate, 1993). In brief, compared to NA children, MD children are characterized by poor strategy choices (Geary, 1990), frequent use of immature strategies (Geary, 1990; Ostad, 1999), less frequent use of retrieval strategies (Jordan & Montani, 1997; Ostad, 1999), less varied collection of usable strategies (Ostad, 1999), and use of significantly fewer problem solving strategies (Montague & Applegate, 2000).

With regard to memory-based processes, research has revealed consistent differences between NA and MD children. Studies have found children with MD to have impaired performance on a number of tasks tapping different aspects of working memory (e.g., Bull & Johnston, 1997; Geary, Widaman, Little, & Cormier, 1987; Hitch & McAuley, 1991; McLean & Hitch, 1999; Ostad, 1997; Passolunghi & Siegel, 2001; Siegel & Ryan, 1989; Wilson & Swanson, 2001). There is also some suggestion that MD children's difficulty to shift from procedural-based problem solving to memory-based problem solving is due to difficulties in storing or accessing facts in or from long-term memory (Geary & Hoard, 2001; Hitch & McAuley, 1991).

Finally, the results of this review have implications for further research in the area. First, differences between MD and NA children across different areas of mathematical cognition have not yet been exhaustively investigated. Although there are a number of studies in some areas (e.g., counting and arithmetic), there is still a need for more investigations that clearly document the performance of these children across different mathematical subtests. Second, studies have so far focused on counting and simple arithmetic and perhaps on mathematical problem solving as well, almost to the neglect of quantitative concepts. Given their importance in mathematics in general and their contributions in solving computational items as well as word problems in particular, it is unfortunate that quantitative concepts did receive very little attention in the research literature. Future research should investigate the performance of MD students on quantitative concepts and compare this with that of NA children and with their own performance on other subtests like problem solving and computations. Third, it must be reiterated that not all investigations produced consistent results. This was particularly true of results pertaining to children with MD-only. For example, whether the MD-only children's performance on word problems differed significantly from that of NA children varied as a function of the complexity of the word problems presented in some studies. Also, the children's performance on timed and untimed tasks tended to make some difference on the accuracy of their performance and hence on the group differences. However, the number of studies that addressed these issues is too small to warrant firm conclusions. Further research is therefore needed to ascertain the generalizability of these results.

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Seleshi Zeleke

University of Oslo, Norway