Arithmetic Fact Retrieval
Are There Differences Between Children With Developmental Dyscalculia and Those With Mathematical Difficulties?
Abstract
When diagnosing children with learning disorders (as per ICD-10), their scholastic performance has to be significantly below the level of intelligence. Although this discrepancy criterion has received much criticism in the field of literacy, few researchers in mathematics have examined it. We used a two (mathematical performance) by two (intelligence) factorial design to analyze the arithmetic fact retrieval of low-performing children in mathematics who met the criterion (developmental dyscalculia) or did not (mathematical difficulties) and of two groups of average-achieving children matched for intelligence. The four groups (each n = 27 third-graders) were matched for their attention span and their literacy skills. Children solved addition verification tasks with numbers up to 10 and 20 under standard and under dual task conditions requiring further working memory capacity to evaluate the potential use of counting strategies. Performance in addition tasks proved to be associated with mathematical achievement especially in the higher number range, whereas dual task performance did not point to the use of counting strategies among low performers in mathematics. No interaction between mathematics and intelligence was identified, which would have confirmed the discrepancy criterion. These results illustrate that stable knowledge of arithmetic facts is essential for mathematical achievement, regardless of whether the discrepancy criterion is met.
Current studies on the mathematical performance of primary school children (TIMSS; Mullis, 2012) and on the prevalence of developmental dyscalculia (DD; Butterworth, 2005b; Devine, Soltesz, Nobes, Goswami, & Szücs, 2013; Fischbach et al., 2013) reveal a mentionable percentage of low-performing children. Approximately 9% of students in Europe do not “have some basic mathematical knowledge” (Mullis, 2012, p. 95): they do not reach the lowest benchmark. A similar picture is shown by the prevalence of DD, which affects 6.5% of children in Europe (Butterworth, 2005b). As mathematics skills are required to process information in our digital age, lack thereof may lead to serious individual (e.g., limited chances on the labor market) and social (e.g., loss of tax revenues) outcomes (Gross, 2009).
Empirical evidence suggests that various cognitive processes are involved in the acquisition and application of knowledge of mathematics. Domain-general factors such as working memory components (Baddeley, 1986) are believed to be related to mathematical performance (David, 2012; Friso-van den Bos, van der Ven, Kroesbergen, & van Luit, 2013; Kyttälä & Lehto, 2008; Raghubar, Barnes, & Hecht, 2010; Toll, Van der Ven, Kroesbergen, & Van Luit, 2011) as well as (in)attention (Martin et al., 2013; Passolunghi, Cornoldi, & De Liberto, 1999).
Domain-specific factors such as number sense, basic understanding and processing of numbers and magnitudes (Kuhn, Raddatz, Holling, & Dobel, 2013; Landerl & Kölle, 2009; Noël & Rousselle, 2011; Price & Ansari, 2013; Von Aster & Shalev, 2007), and strategies while performing mathematical tasks (e.g., the retrieval of arithmetic facts) (Grube, 2006; Jordan & Hanich, 2003; Landerl & Kölle, 2009; LeFevre, DeStefano, Coleman, & Shanahan, 2005) have been found to contribute to achievement in mathematics. Owing to consistent empirical evidence for the relevance of domain-specific factors to learning outcomes in mathematics, the American Psychiatric Association (APA) has included these factors in its revised definition of “specific learning disorder with impairments in mathematics” – often referred to as DD. In the Diagnostic Statistical Manual of Mental Disorders (DSM-5; American Psychiatric Association, 2013) DD is characterized as a “neurodevelopmental disorder with a biological origin that is the basis for the abnormalities at a cognitive level that are associated with the behavioral signs…” (p. 68) and is specified by difficulties in mastering number sense or memorizing arithmetic facts (DSM-5; 2013, p. 67).
In its ICD-10 Classification of Mental and Behavioural Disorders: Clinical Descriptions and Diagnostic Guidelines (ICD-10; World Health Organization, 1992), the World Health Organization (WHO) demands further criteria in its definition of DD, which it calls “specific disorder of arithmetical skills.” Consulting the WHO’s diagnostic criteria (WHO, 1992), which are applied in many countries, “the child’s arithmetical performance should be significantly below the level expected on the basis of his or her age, general intelligence, and school placement…” (p. 248). Also assuming “some type of biological dysfunction” (WHO, 1992) this definition requires further a substantial discrepancy between intelligence and mathematical achievement (IQ–achievement discrepancy or discrepancy criterion). Consequently, children underachieving in mathematics can be classified in two groups when applying the WHO’s criteria: (a) those reaching the IQ–achievement discrepancy and (b) those failing to reach that intelligence – achievement gap. In accordance with Mazzocco (2005), we classify in this study the second group (not reaching the discrepancy) as children with mathematical difficulties (MD) and refer to a broader group of children as a result of less strict (discrepancy) criteria.
The question arises as to whether there are differences between children with DD and those with MD as postulated by the WHO according to its diagnostic criteria or whether they have the same deficit(s) as assumed by the APA. As “we need to establish agreed upon diagnostic criteria to discover differences in the structure and function of DD brains” (Butterworth, 2005b, p. 465), further evidence for relevant criteria defining DD, especially for the discrepancy criterion, shall be provided in the course of this study.
Criticism of the IQ–Achievement Discrepancy
The application of the discrepancy criterion and consequently the categorization of children underachieving in mathematics as having DD or MD have received a lot of criticism for statistical reasons or due to missing empirical evidence (Dyck et al., 2004; Ehlert, Schroeders, & Fritz-Stratmann, 2012; Grube, 2008; Kuhn et al., 2013; Stanovich, 1999, 2005). Most criticism founded on statistical reasons relates to the size of the IQ–achievement gap. The WHO (World Health Organization, 1993) suggests a discrepancy of at least two standard deviations as a rigorous criterion for research. As the probability of reaching the discrepancy depends on the relationship between mathematics and intelligence, the probability simply increases or decreases as a result of increasing or decreasing intelligence (Dyck et al., 2004). Therefore, children who (slightly) underachieve in intelligence measures (e.g., who show an IQ of 80) are more likely to be rejected from the diagnosis of DD even though they might not differ in the cause of their mathematical underachievement from those obtaining the diagnosis as a result of higher intelligence (e.g., children who show an IQ of 100). Consequently, most researchers and practitioners refer to discrepancies varying from 1.2 to 1.5 standard deviations in diagnosing DD (Jiménez González & García Espinel, 1999; Kuhn et al., 2013) or to regression models of IQ–achievement (Ehlert et al., 2012). Nevertheless, using regression models also may exclude children from the diagnosis of DD for statistical reasons as the diagnosis depends on the assumed correlation between mathematics and intelligence varying from r = .4 to r = .6 across grades (Cattell, Weiß, & Osterland, 1997; Grube & Hasselhorn, 2006) and across level of ability (Ehlert et al., 2012).
Determining the relevance of the discrepancy criterion in defining DD (as done by the WHO) could be done in comparisons of underachievers in mathematics who reach a certain IQ–achievement discrepancy and those who do not. Potential group differences may shed light on various causes of DD and MD and could highlight the importance of the IQ–achievement discrepancy. However, no differences were found between children with DD and those with MD regarding strategies used to solve word problems (Jiménez González & García Espinel, 1999, 2002), working memory performance (Mähler & Schuchardt, 2011), the understanding of basic mathematical concepts (Ehlert et al., 2012), and basic mathematical capacities (Kuhn et al., 2013). Such results suggest equal cognitive patterns in children with DD and in those with MD, thus indicating no important role of the discrepancy criterion.
Research using different control groups – each corresponding to a level of intelligence of DD and MD – could add to rare empirical evidence of the importance of the discrepancy criterion in defining DD and MD, and it could refute statistic-based criticism. As children with MD probably have a lower IQ than children with DD, differences should be considered in the context of two control groups matched for intelligence.
Arithmetic Facts and Mathematics
Arithmetic facts describe a declarative knowledge network that contains results of simple arithmetic problems (e.g., additions up to 20; Ashcraft, 1982; Kaye, 1986). Children first solve arithmetic problems by counting (Butterworth, 2005a). Once they acquire such declarative knowledge, they rely on it, that is, they retrieve or memorize arithmetic facts (Geary, 2004; Geary & Hoard, 2002; Shrager & Siegler, 1998). Retrieving arithmetic facts frees up working memory (WM) capacity (Baddeley, 1986), that was formerly dedicated to counting or to other mentally effortful processes (Kaye, 1986). WM is a multicomponent system that builds an interface between perception and long-term memory. A central executive system coordinates two subsystems storing different information by controlling attention (Baddeley, 1986). While solving mathematical problems, the visual spatial sketchpad stores visual input (e.g., digits and operators) whereas the phonological loop focuses on verbal information (e.g., verbal counting). Due to the limited capacities of WM, memorizing arithmetic facts is an important strategy, as it frees up capacity needed for tasks of higher complexity, such as word problems, or tasks requiring use of higher numbers (Geary, 2004; Grube, 2006). Employing this strategy would be especially important for children with DD or MD, as they have deficits in WM (Dowker, 2005; Landerl & Kaufmann, 2008; LeFevre et al., 2005). However, evidence supports that contrary to average-achieving peers, who switch to fact retrieval in the middle of grade three or earlier, underachieving children in mathematics continue (verbal) counting (Ashcraft, 1982; Geary, Hoard, & Hamson, 1999; Hecht, 2002; Kaye, 1986; Ostad, 1997). As counting strategies are more time-consuming and vulnerable to errors, especially when crossing decimal boundaries (Geary et al., 1999; Grube, 2006; Kaye, 1986), relying on a stable knowledge network of arithmetic facts is important for mathematical achievement. Accordingly, evidence supports difficulties in retrieving arithmetic facts as being strongly related to DD and to MD (Butterworth, 2005b; Geary, Hamson, & Hoard, 2000; Jordan & Hanich, 2003).
The Present Study
Children with DD and those with MD are compared in their performance in arithmetic fact retrieval to investigate potential differences that could justify the IQ–achievement discrepancy. The performance of two average-achieving groups of children in mathematics is further analyzed to examine whether difficulties in retrieving arithmetic facts are a potential factor for mathematical underachievement. Conditions that enable or disable the use of counting strategies while solving arithmetic problems with a higher (up to 20) and a lower (up to 10) number range are administered to evaluate strategy use and resulting different patterns (in time and accuracy) for tasks of varying complexity. A control group selected for its slightly lower level of intelligence is matched to children with MD, while children with DD and the corresponding control group both have an average level of intelligence. Using two control groups allowed us to determine whether arithmetic fact retrieval was affected by intelligence. Children in all four groups performed at an average level in literacy owing to further criteria of the WHO (ICD-10; 1992) and were further matched according to their ability to pay attention.
Doing this resulted in a two (low and average mathematical achievement) by two (slightly lower and average intelligence) factorial design (see Table 1 ) to test the following hypotheses:
- 1.children with DD and those with MD will not differ in performance (accuracy and latency) in solving simple arithmetic problems assumed to be solved by fact retrieval,
- 2.both underachieving groups in mathematics will perform worse (less accuracy and increased latencies) in solving these problems than average-achieving children, and
- 3.the performance of children with DD and those with MD will deteriorate when counting is prevented.
Method
Procedure
This study was part of a longitudinal study investigating potential differences between children with DD and those with MD in domain-general and domain-specific factors. In order to recruit children who meet the specific criteria of DD, MD, and control groups, 1,913 children were screened at the end of grade two and at the beginning of grade three for intelligence and performance in reading, spelling, and mathematics. Only children whose parents consented took part. The screening lasted four lessons and the standardized classroom tests were conducted over 2 days (with a maximum break of 1 week). Intelligence and spelling were evaluated on the first day and reading and mathematics on the second day. The 50 participating regular elementary schools were located mostly in rural (8 schools) and urban (42 schools) areas in northern Lower Saxony and in Bremen. The population of these areas was predominantly middle class. Of the children, 3.7% met criteria for MD and 5.3% for DD (see below). These children were invited to take part in the longitudinal study. The propensity score was estimated on logistic regression (West et al., 2014) to select average-achieving peers (using the one-by-one nearest neighbor method) with regard to intelligence and performance in mathematics for both control groups. The children were tested individually in the middle of grade three on their performance in arithmetic fact retrieval and attention. The experiment and all tests were introduced and supervised by research assistants trained in the procedures.
Participants
As only 27 children of the MD group consented, children of the larger sampled DD (n = 48) as well as Control Groups A (n = 54) and B (n = 48) were further matched via propensity score (using the same methods mentioned above) for the subsequent study with children with MD in literacy and attention. Consequently, data of 108 children (8–10 years of age, 62% girls) were analyzed. The children were assigned to one of the four groups (each n = 27) according to the two (mathematical performance) by two (intelligence) factorial design.
Low-performing students in mathematics (T < 40) were classified as follows:
- 1.children with DD showing at least an average intelligence (IQ > 90) and an IQ–achievement discrepancy (of at least 1.2 SD), or
- 2.children with MD not meeting the discrepancy and having a slightly lower level of intelligence (70 < IQ < 90).
Their average-achieving peers in mathematics (T > 43) were assigned to one of two control groups:
- 1.children with at least an average level of intelligence (IQ > 90) to Control Group A (matched with children with DD), or
- 2.children with a slightly lower level of intelligence (matched with children with MD) to Control Group B.
All children performed at least at an average level in literacy (T > 40) and showed no sensory impairments. Approximately one fifth of the children claimed to speak with at least one parent a language other than German, but there was no relation between the language spoken at home and group membership (χ2(3, N = 108) = 1.99, p = .574). Though there were more girls (39 girls) in the low-performing groups in mathematics than in the control groups (28 girls), no significant differences could be shown in the distribution of gender among groups (χ2(3, N = 108) = 5.78, p = .123). Details of the sample are shown in Table 2 .
Materials
Scholastic Performance, Intelligence, and Attention
Mathematical achievement was assessed with a standardized paper-pencil test based on the curriculum in Germany (DEMAT2+, Krajewski, Liehm, & Schneider, 2004), and the 36 tasks were subsumed under three domains: (a) 24 under arithmetic (e.g., addition and subtraction up to 100), (b) four tasks under geometry (e.g., determination of the number of cubes in a figure), and (c) eight tasks under numerical sizes (e.g., comparison of lengths). The T-value (class-specific norms for the end of grade two or the beginning of grade three) of the total score was a criterion. As class-specific norms reported for second- and third-graders overestimated children’s performance (M = 53, SD = 9.7), they were adapted for the purpose of this study. Another standardization was applied simultaneously (M = 50, SD = 9.6, in the sample) with data from the screening and data from another study evaluating second- and third-graders’ performance in mathematics (further N = 1,594) in different parts of Germany.
To evaluate literacy skills, a mean T-value was calculated for performance on a reading comprehension test and on a spelling test. A standardized paper-pencil test (ELFE1-6; Lenhard & Schneider, 2006) was administered to measure reading performance and rated the children’s comprehension of words, sentences, and short texts (120 items). T-values (class-specific norms) based on the total score were criteria. A standardized dictation task (WRT2+, Birkel, 2007) was conducted to assess knowledge of the spelling of basic German words (43 items), also providing class-specific norms.
Fluid intelligence was measured with a nonverbal, culture-fair intelligence test (CFT1; Cattell et al., 1997) consisting of figural material (e.g., identification of similarities, figure classifications) without any numerical items (total 108). The criterion was an IQ based on age-specific norms (standardized in 1995).
To assess attention, the T-value of performance on a speed paper-pencil discrimination test was determined. This test is similar to the d2 test of selective attention (Brickenkamp & Zillmer, 2010) and was derived from an intelligence and development assessment tool designed for children (IDS; Grob, Meyer, & Hagmann-von Arx, 2010).
Experiment: Arithmetic Fact Retrieval
To assess arithmetic fact retrieval, children worked on simple computer-based verification tasks (addition up to 20) in three conditions. Items were presented as black digits (each 6.5 cm large) in the middle of a gray 17″ laptop screen. Children were positioned approximately 50 cm in front of the screen and instructed to judge as quickly as possible if the equation was right or wrong by pressing certain keys on a keyboard.
Items varied systematically in correctness of the equation, position of the larger addend (first/second in equation), and number range (up to 10/up to 20). Ties (equations with the same addends) were excluded, as evidence suggests less time to solve them (Blankenberger, 2003). The single digits “0” and “1” also were omitted, as these problems are considered to be stored rules instead of arithmetic facts (Domahs & Delazer, 2005; Landerl & Kaufmann, 2008). Solutions to incorrect equations differed between “1” and “2” from the correct result. Two seconds after the answer to the previous item was given the next item was presented. To prevent repetition of items, three sets of equal problems with regard to number range, correctness, and position of larger addends were defined. This resulted in 24 items per series, as three items were presented for each of the eight combinations. Each child performed two experimental series on two different dual tasks and a standard condition. In the first series 24 items were answered without any additional tasks (standard condition). Another 24 items were judged afterwards while performing the first of the two dual tasks: (a) repeating “deedeedee” as specific verbal content (articulation) – or (b) tapping continuously the space bar.
The articulation task was used to prevent counting and to force arithmetic fact retrieval (Jordan & Hanich, 2003), as it causes interference with the phonological WM involved in verbal counting. The tapping task was used as a motor control condition that also requires WM capacity without strongly interfering with the phonological subsystem, thus allowing counting. The rhythm of articulation and tapping (130 beats/min) was presented beforehand by a metronome. An hour later children answered another 24 items in the standard condition and then answered an additional set of 24 items on the second dual task. The dual task to be completed first was randomly selected. The number of correct answers (accuracy) and the average time needed to respond to an item (response latency) in each condition were dependent measures. A mean was calculated for both standard conditions. Latencies in each of the eight combinations were trimmed to a value of ± 2 standard deviations from the group mean (separately for each group and condition) resulting in 207 of 1296 latencies that were Winsorized.
Results
To evaluate group differences in the experimental baseline, two separate two-way analyses of variance (ANOVA) were conducted. Mathematical achievement (low/average) and intelligence (low/average) as between-subject variables were examined for effects on latency and accuracy (within-subject variables) in fact retrieval with repeated measures on the factor number range (up to 10/up to 20) in the standard condition.
Furthermore, two (mathematical achievement) by two (intelligence) by two (dual task) by two (number range) ANOVAs with repeated measures on the last two factors were calculated to examine accuracy and latency (within-subject variables) in the two dual task conditions (articulation and tapping) and number ranges.1 The repeated measures design was implemented to evaluate the potential use of counting strategies by the different groups as well as their performance on the tasks requiring use of different number ranges.
Consulting the a priori conducted power analysis (G*Power; Faul, Erdfelder, Lang, & Buchner, 2007) data of 108 children proved to be sufficient for the repeated measures ANOVA design (between-within interaction; 1 − β = .95; correlations among repeated measures set to r = .05; α = .05) with a medium effect size (f = .25; Cohen, 1988). Mean scores and standard deviations for accuracy are shown in Table 3 and for latency in Figure 1 .
Analyses of Standard Condition
An ANOVA revealed no significant effects for the between-group factors on accuracy, thus no effect occurred for intelligence, F(3, 104) = 0.65, p = .421, and mathematics, F(3, 104) = 0.29, p = .592, whereas a significant interaction effect was found for mathematics and number range, F(3, 104) = 11.08, p = .001, η2 = .10, but not for intelligence and number range (F(3, 104) = 0.23, p = .880). Descriptive data (Table 3) revealed that correct answers of low performers in mathematics were affected (with a mean loss of 1.28) by the higher number range more than children in the control groups (with a mean loss of 0.46).
Concerning latency, a significant effect could be observed for mathematics F(3, 104) = 22.66, p < .001, η2 = .18, whereas intelligence proved not to be significant, F(3, 104) = 1.01, p = .316 nor did the interaction of both factors, F(3, 104) = 1.51, p = .221. Figure 1 shows that low-performing children in mathematics needed significantly more time to judge an equation than their average-achieving peers. All groups were significantly affected by number range F(3, 104) = 229.8, p < .001, η2 = .68, thus all children needed more time for additions up to 20 than for those up to 10, whereas no significant number range by mathematics nor intelligence by mathematics interaction effect (both Fs(3, 104) ≤ 0.66, p ≥ .42) pointed to a group-specific effect on latency.
Analyses of Dual Task Conditions
A two-way repeated measures ANOVA revealed a significant main effect of mathematical achievement on accuracy in the dual task conditions, F(3, 104) = 7.63, p = .007, η2 = .07. Intelligence was not significant F(3, 104) = 0.87, p = .353, nor was the interaction of the between-group factors, F(3, 104) = 1.24, p = .269. This shows that both groups of low-performing children in mathematics lagged similarly behind their peers irrespective of their level of intelligence. A significant number range by dual task by mathematics by intelligence interaction, F(3, 104) = 11.86 p < .001, η2 = .10, occurred. Separate for each group, additional repeated ANOVAs were conducted for number range and dual task as within-subject factors to break down this interaction. These analyses showed significant dual task by number range effects on accuracy for Control Group A, F(1, 26) = 5.28, p = .030, η2 = .17, and Control Group B, F(1, 26) = 6.35, p = .018, η2 = .20. Performance of both groups deteriorated only in one of the two conditions when doing addition crossing boundaries of 10. This effect could not be seen for children with MD or with DD (both Fs(1, 26) ≤ 3.81, p ≥ .06). Low-performing children in mathematics were affected in both conditions by number range, as a significant main effect was found for children with MD, F(1, 26) = 16.73, p < .001, η2 = .39, and DD, F(1, 26) = 19.31, p < .001, η2 = .43, resulting in less accuracy in additions up to 20 (Table 3).
In the analyses for latency a significant effect of mathematical achievement was observed, F(3, 104) = 11.20, p = .001, η2 = .10. Intelligence had no significant effect on latency, F(3, 104) = 0.19, p = .662, nor did the interaction of mathematics and intelligence, F(3, 104) = 0.13, p = .567. Children with DD as well as those with MD needed significantly more time (Figure 1) to judge an equation than those performing at an average level, which was not attributable to intelligence. The dual task had no effect on latency, F(3, 104) = 2.46, p = .120, nor did the mathematics by dual task or intelligence by dual task interaction (both Fs(3, 104) ≤ 3.61, p ≥ .06), thus there was no statistical evidence for the use of more time-consuming counting strategies in a certain dual task condition among children with DD or those with MD. A main effect was found for number range, F(3, 104) = 124.89, p < .001, η2 = .55, which was irrespective of the dual task, mathematics and intelligence, as no interactions were further significant (all Fs(3, 104) ≤ 2.26, p ≥ .14). Hence, all children needed more time to judge additions up to 20 than those up to 10 (Figure 1).
Discussion
Children with DD and those with MD were compared in order to evaluate the relevance of the IQ–achievement discrepancy for defining DD as postulated by the WHO (ICD-10; 1992). Taking into account that children not reaching the discrepancy were more likely to show a lower IQ than those reaching it, data of two control groups matched for intelligence also were analyzed. This design was implemented to determine whether probable differences between DD and MD are attributable to intelligence.
Performance in the retrieval or memorization of arithmetic facts, which is suggested by the APA (DSM-5; 2013) and researchers (Butterworth, 2005b; Geary et al., 2000; Jordan & Hanich, 2003) to be a determining factor for DD, was analyzed to identify potential differences in DD and MD that could justify the discrepancy criterion. Corresponding to the assumptions of the first hypothesis, the results revealed no differences between children with DD and those with MD in arithmetic fact retrieval. Consistent with other findings (Ehlert et al., 2012; Jiménez González & García Espinel, 1999, 2002; Kuhn et al., 2013; Mähler & Schuchardt, 2011), the definition of two mathematical underachieving groups – (a) reaching the discrepancy (DD) and (b) not reaching the discrepancy (MD) – that differ in mathematics-related cognitive variables is doubtful according to our findings. Nevertheless, the chosen IQ–achievement gap (1.2 SD) in this study was smaller than recommended for research by the WHO (1993) in order to reach sufficient sample sizes. Consequently, the WHO’s criterion of IQ–achievement discrepancy can be only mildly criticized based on the results obtained, although it could be argued that many practitioners and researchers (Jiménez González & García Espinel, 1999; Kuhn et al., 2013) apply smaller discrepancies (1.2–1.5 SD).
Less than one percent of children who participated in the screening (N = 1,913) reached the required IQ–achievement gap of two standard deviations without co-occurring reading and spelling difficulties. The question arises as to whether more common and less strict criteria for research and intervention would be preferable when defining DD concerning the demands of a relatively high percentage of children who actually underachieve in mathematics (TIMSS; Mullis, 2012).
As a consequence of the ongoing theoretically- and empirically-based criticism (Scanlon, 2013), the APA (DSM-5; 2013) has removed the discrepancy criterion in identifying children with DD in its revised definition of learning disorders.
With regard to the second hypothesis, average- and low-performing children in mathematics were compared in order to evaluate arithmetic fact retrieval as a mathematics-related cognitive variable and a potential factor for underachievement. Findings suggest that children with DD and those with MD lag behind average-achieving peers in simple addition tasks thought to be solved via fact retrieval in grade three (Ashcraft, 1982). Without any additional WM load of a dual task, differences became apparent only in greater response times and error rates on tasks requiring use of the number range of up to 20. According to evidence that children with DD and those with MD do not switch from counting to fact retrieval (Ashcraft, 1982; Geary et al., 1999; Hecht, 2002; Kaye, 1986; Ostad, 1997), increased latency could be the result of time-consuming counting. As counting occupies WM capacity, it results more often in errors during more complex tasks. Consequently, the reduced accuracy on tasks requiring use of the higher number range also could point to the use of counting in children with DD and those with MD. Therefore, a dual task paradigm was realized to evaluate this potential counting strategy use in low-performing children in mathematics.
An articulation dual task requiring greater verbal cognitive WM load was administered in order to prevent verbal counting (Grube, 2006; Jordan & Hanich, 2003). In that condition children with DD and those with MD needed more time and showed less accuracy than children in the control groups. No trade-off between latency and the error rate pointed to different confident criteria in fact retrieval (Geary & Hoard, 2002), as thought to be used by “perfectionists” (Siegler, 1988), who retrieve only answers that are undoubtedly correct.
The tapping dual task was realized as a control condition requiring less verbal WM capacity, permitting counting and needing supervision similar to articulation by the central executive WM system. Despite consistent evidence for the continuous use of counting strategies (Ashcraft, 1982; Geary et al., 1999; Hecht, 2002; Kaye, 1986; Ostad, 1997), a dual task effect that could identify verbal counting was observed for children of control groups only by a dual task by number range interaction. The third hypothesis – that the performance of children with MD and those with DD deteriorates when they are not able to compensate for their deficit in fact retrieval by counting – has to be rejected because of the obtained results.
As children with DD and those with MD performed worse than their peers in both conditions, it is questionable what strategies – other than verbal counting – they applied which required more time but resulted in less accuracy. Instead of direct retrieval, other memory-based strategies (Dowker, 2009; Geary & Hoard, 2002) such as decomposition (i.e., solving unknown facts by retrieving partial sums) could provide an alternative explanation. Using such strategies could have resulted in greater response time, irrespective of the different extents of phonological interference activated by dual tasks.
Less accuracy could be attributable to unstable fact knowledge, that is, to a weak association between problem and answer due to less experience with that problem, resulting in use of error-prone strategies other than retrieval (Ashcraft, 1982). As a consequence of repeated exposure to children, incorrect results also might be stored in memory (Ashcraft, 1982). Alternatively, De Visscher and Noël (2014) suggest a hypersensitivity to interference in the process of storing fact knowledge as being related to knowledge of incorrect facts.
Despite application of counting strategies, unstable fact knowledge is associated with mathematical underachievement. This pattern occurs whether or not the IQ–achievement discrepancy is met. As with other studies of cognitive processes in children with DD and those with MD (Ehlert et al., 2012; Jiménez González & García Espinel, 1999, 2002; Kuhn et al., 2013; Mähler & Schuchardt, 2011), we do not support the application of the discrepancy criterion in identifying children with DD.
Therefore, in the course of establishing “agreed upon diagnostic criteria, to discover differences in the structure and function of DD brains” (Butterworth, 2005b, p. 465) we suggest focus be on the presence of core deficits, such as deficits in basic numerical processing or fact retrieval, in line with the APA’s new definition of a LD with impairment in mathematics (DSM-5; 2013), which does not consider an IQ–achievement discrepancy a prerequisite.
Limitations and Conclusion
There are some limitations that have to be considered in our study. We did not assess information on children’s socioeconomic status, which has been linked with achievement in mathematics (Jordan & Levine, 2009). Consequently, we cannot provide any information as to whether children developed numeracy and literacy skills at home, which would contribute to subsequent achievement in mathematics (LeFevre et al., 2009). Furthermore, intelligence was measured on scales (CFT1; Cattell et al., 1997) which were standardized in 1995. Due to the Flynn effect (Flynn, 1987), children’s intelligence could be overestimated, resulting in a higher probability of reaching the demanded discrepancy and thus of obtaining the diagnosis of DD. This also could be true for the groups assigned in our study. Because we focused on potential differences between children with DD and those with MD, it could be argued that the sufficient size of the groups (each n = 27), the realized gap in intelligence between children with MD and those with DD (20 IQ-points), and the obtained IQ – mathematics achievement discrepancy for the DD group (mean SD = 1.7) tolerate this slight overestimation in intelligence.
Further, children’s accuracy in addition tasks requiring use of numbers up to 10 in the standard condition was at ceiling. Therefore, probable differences between low-performing and average-achieving children in mathematics could have been concealed, although latencies pointed to differences in that condition. Nevertheless, children with MD and those with DD lag behind their peers. They exhibit increased latencies and produce fewer correct answers on more complex tasks (higher number range or secondary tasks) requiring further WM load. Overall, performance on complex tasks as well as analyses for latencies should be used instead of the IQ – mathematics achievement gap to evaluate knowledge of facts and to identify children who underachieve in mathematics.
1We tested and met the assumption of normality for the groups with regard to latencies. Accuracy data, especially in the standard condition, were at ceiling and not normally distributed.
To meet the assumption of normality, we reflected and log transformed the distribution of accuracy data as suggested by Tabachnick and Fidell (2007). The same pattern of results emerged for the transformed as well as for the non-transformed data. For this reason and because the sample size was sufficient concerning the central limit theorem (Field, 2011), we decided to present analyses of non-transformed data to facilitate the description and interpretation of our results.
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This study was supported by the German Federal Ministry of Education and Research (BMBF) (Ref. No. 01GJ1008). We thank all the teachers, children, and parents who participated in this study. We thank the editor and the anonymous reviewers for providing helpful feedback, which strengthened our paper. We also thank Pirjo Aunio and Emilie Prast for their comments on an earlier version of this manuscript.
