Abstract
Laut ICD-10 und DSM-IV-TR muss für die Diagnose einer Dyskalkulie eine Diskrepanz zwischen der allgemeinen kognitiven Leistungsfähigkeit und der tatsächlichen oder vorhergesagten Leistung in einem Mathematiktest vorliegen. Diese Definition impliziert, dass sich rechenschwache Kinder, die diese Diskrepanz aufweisen, von rechenschwachen Kindern ohne Erfüllung des Diskrepanzkriteriums unterscheiden. Die vorliegende Arbeit hatte zum Ziel, mögliche Unterschiede zwischen dyskalkulischen und rechenschwachen Kindern sowie einer Kontrollgruppe in der basisnumerischen Verarbeitung zu prüfen. Zur Identifikation einer Dyskalkulie bzw. Rechenschwäche wurden entweder (a) ein Testverfahren mit basisnumerischem Schwerpunkt (ZAREKI-R) oder (b) Tests zur Erfassung von Rechenfertigkeiten (ZAREKI-R Kopfrechnen und Textaufgaben, HRT 1 – 4 Addition und Subtraktion, WISC-IV rechnerisches Denken) verwendet. Insgesamt bearbeiteten 68 Kinder (Dyskalkulie: Na = 27/Nb = 11, Rechenschwäche: Na = 21/Nb = 18, Kontrollgruppe: Na = 20/Nb = 39) eine Batterie von basisnumerischen Aufgaben: Simultanerfassung, Abzählen, Mengenvergleich, Transkodieren, Number sets und Zahlenstrahl (0 – 100). Zusätzlich wurde die Arbeitsgedächtniskapazität mit einer visuell-räumlichen Aufgabe (Matrixspanne) überprüft. Laut Klassifikation nach ZAREKI-R unterschieden sich rechenschwache und dyskalkulische Kinder in fast allen basisnumerischen Aufgaben klar von der Kontrollgruppe, jedoch nicht untereinander. Bei Klassifikation nach Rechenfertigkeiten konnten rechenschwache und dyskalkulische Kinder ebenfalls nicht differenziert werden, allerdings unterschieden sich nur rechenschwache Kinder von der Kontrollgruppe (bei den Aufgaben Simultanerfassung, Abzählen, symbolischer Mengenvergleich, Transkodieren, Zahlenstrahl). Die Befunde werden vor dem Hintergrund der Verwendung basisnumerischer Fertigkeiten für die Diagnose und Therapie von Dyskalkulie diskutiert.
Background: The current international classification systems ICD-10 and DSM-IV show a large conceptual overlap in their criteria for defining dyscalculia: Firstly, a child's score in a standardized mathematics assessment should fall below a pre-specified percentile rank cutoff (usually 10 %). Secondly, intelligence is required to be in the normal range (i. e., IQ ≥ 70 or 80). And thirdly, a substantial discrepancy between the mathematics and intelligence standard scores should be observed. In this paper, we will define children that conform to all three criteria as having developmental dyscalculia (DD), whereas children who fulfill only the first two criteria are defined to show severe math difficulties (MD).
This discrepancy definition of DD, along with similar definitions for other learning disorders, has long been criticized in the scientific literature on methodological, conceptual, and ethical grounds. Further, at least for dyslexia, a substantial body of research has been published, showing that no reliable differences between high-IQ and low-IQ poor readers can be found on relevant core markers (e. g., word recognition) and that no different treatments are required for high-IQ and low-IQ readers (Stanovich, 2005). However, there are still only few studies comparing DD and MD children. One of the few studies (Gonzalez & Espinel, 2002) reported no substantial differences between these groups with respect to solving algebra word problems and using solution strategies. Ehlert et al. (2012), using a criterion-based test, did not find any differences in mathematical concept comprehension between DD and MD children.
Aims: The main goal of this study was to compare DD and MD children with each other and with an unimpaired control group on a battery of tasks tapping basic numerical capacities. The tasks were chosen such that they broadly covered key aspects of numerical cognition, i. e., dot enumeration (subitizing and counting), number and magnitude comparison, transcoding, number sets and number line. A second goal pertained to a comparison of MD/DD classification strategies. The first strategy (A) used a well-established psychometric test that broadly covered basic numerical capacities and, to a lesser degree, arithmetic ability to identify DD/MD. As a second strategy (B), we used arithmetic ability exclusively to identify children with MD/DD.
Methods: Overall, N = 68 children participated in this study. The following criteria for DD were used: IQ ≥ 80 (based on the WISC-IV perceptual reasoning index and vocabulary; Petermann & Petermann, 2011), standard reading score ≥ 80 (SLS 1 – 4; Mayringer & Wimmer, 2003), standard math score ≤ 80 (strategy A: ZAREKI-R; von Aster, Weinhold Zulauf & Horn, 2006; strategy B: mean of five arithmetic subtests, two from ZAREKI-R [mental calculation, word problems]; two from HRT 1 – 4 [addition, subtraction], Haffner, Baro, Parzer, & Resch, 2005; and the arithmetic subtest from WISC-IC), discrepancy between IQ and standardized math score > 1.5 standard deviations. For MD, the only difference was that the discrepancy between IQ and math assessment was required to not exceed 1.5 standard deviations, while all other criteria applied. Children in the control group conformed to the following criteria: IQ ≥ 80, standard math score ≥ 90, standard reading score ≥ 80.
For strategy A (B), these criteria resulted in a selection of N = 27 (11) children with DD, N = 21 (18) children with MD, and N = 20 (39) children in the control group. All basic numerical capacities were administered on a computer with headphones. The first task was dot enumeration, where children had subitize one to three dots (6 items) or count four to nine dots (12 items) as fast as possible. The second task assessed symbolic magnitude comparison, where children had to indicate the larger of two single-digit numbers (24 items). Next, a mixed magnitude comparison task was administered in which the larger magnitude (single digit or number of dots) had to be selected (24 items). A subsequent transcoding task required children to type the numbers they had just heard (8 items). A discrete trial variant of the number set task (Geary et al., 2009) was presented next, in which children were supposed to quickly judge whether a target number on top of the screen corresponded to a number set shown below (140 items, speed test). Then, a number line task (Booth & Siegler, 2006) ranging from 0 – 100 was administered in which children had to indicate where a number shown on top of the screen was located on the number line (23 items). Two additional tasks were administered, a binary-choice reaction time task (20 items) to assess processing speed and a visual matrix span task (up to 16 items) to assess visuo-spatial working memory.
Results: Computing multiple ANOVAs with post-hoc comparisons, for classification strategy A we found statistically significant differences between the control group and MD on all subtests except three (reaction time, subitizing, mixed magnitude comparison), with a very similar result for the comparison of DD and the control group (in this comparison, however, the difference for counting was insignificant, whereas the difference for mixed magnitude comparison was substantial). We were unable to find any differences between DD and MD children. For strategy B, a different picture emerged; here, only MD children differed from the control group on five subtests (subitizing, counting, symbolic magnitude comparison, transcoding, number line), with MD and DD groups not showing any differences either. Next, we took a closer look at some tasks, and found two key results of interest. First, although mostly, reliable main effects of group (MD vs. control/DD vs. control) could be found for subitizing, counting, and symbolic magnitude comparison, we did not find any interaction effects between task properties (e. g., numerical distance) and group. This provides evidence for the fact that MD or DD children do not show qualitatively different cognitive patterns in counting, subitizing, or magnitude comparison. Second, when analyzing the number line task using three different regression models (linear, logarithmic, proportional judgement model; Spence, 1990), we found that even though linear regression was the best-fitting model overall, data from a substantial proportion of MD and DD children could be fit best using a logarithmic model, even up to the third grade.
Discussion: This study was conducted to investigate differences in basic numerical capacities between DD and MD children in order to evaluate the discrepancy criterion in defining this learning disorder. When using a classification strategy based on a broad sampling of basic numerical capacities, reliable differences between the control group and DD/MD groups was found, although the groups did not substantially differ. When using arithmetic tests only for classification, a much more diverse picture emerged, resulting in more severe problems in children with MD, whereas children with DD could not be separated from the control group on statistical grounds. We assume that these differences can at least partly be explained by the low classification reliability using a double discrepancy criterion (Francis et al., 2005). However, in order to screen for DD/MD and to design appropriate interventions, basic numerical capacities need to be a core part of assessment. Future studies are required that investigate whether different interventions are needed for DD and MD children, thus enabling the evaluation of the discrepancy criterion from an additional point of view.
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