Abstract
Defizite in den Bereichen der basisnumerischen Verarbeitung, der Zählfertigkeiten, beim Transkodieren, des numerischen Faktenwissens sowie beim Ausführen arithmetischer Prozeduren gehören zu den typischen Symptomen einer entwicklungsbedingten Rechenschwäche im Grundschulalter. Während für den englischen Sprachraum einige umfassende Förderprogramme evaluiert wurden, liegen für den deutschen Sprachraum bislang kaum wissenschaftlich evaluierte Interventionsprogramme vor, die gezielt auf diese Defizite eingehen. Mit der vorliegenden Studie wird die Wirksamkeit eines modular aufgebauten, neuropsychologisch orientierten numerischen Förderprogramms evaluiert. Dieses Programm fokussiert den Aufbau basisnumerischer und konzeptueller Kompetenzen, wobei prozedurales und arithmetisches Faktenwissen integriert werden. Im Rahmen eines Experimental-Kontrollgruppen-Designs wurden fünf Gruppen von Zweit- und Drittklässlern (geförderte rechenschwache Kinder ohne und mit Leseschwäche, eine Vergleichsgruppe rechenschwacher Kinder mit Leseschwäche, durchschnittliche Rechner ohne und mit Leseschwäche) zu zwei Messzeitpunkten (vor/nach der Intervention) hinsichtlich ihrer arithmetischen Kompetenzen untersucht. Während sich die Kinder der Interventionsgruppen bedeutsam verbessern konnten, waren bei den Kindern der Vergleichsgruppen kaum Leistungszuwächse festzustellen. Der Leistungszuwachs der Kinder mit isolierter Rechenschwäche war deutlich größer als der der Kinder mit einer zusätzlichen Leseschwäche. Die Ergebnisse dieser Studie belegen, dass das eingesetzte Programm zur Verbesserung der arithmetischen Fertigkeiten rechenschwacher Kinder spezifisch wirksam ist. Sie veranschaulichen zudem den Bedarf an Interventionsstudien, die die Wirksamkeit unterschiedlicher Förderansätze für rechenschwache Kinder ohne und mit komorbider Leseschwäche vergleichen.
Background: Primary-school children who suffer from mathematical difficulties (MD) typically show deficits in various number-related domains such as basic numerical processing, counting, transcoding, numerical fact and procedural knowledge. Despite the high prevalence rate of MD in the general population, no empirically evaluated intervention programs that focus on the whole range of potential deficits have been published in German-speaking countries up to date. However, a pilot-study on third-graders with MD (Kaufmann et al., 2003) suggested that a German adaption of the Numeracy-Recovery Program (Dowker, 2001, 2007) may present a promising approach for the remediation of mathematical difficulties. This intervention program for primary schoolers is based on neuropsychological models of number processing and arithmetic (e. g. triple-code model; Dehaene, 1992). In the context of this intervention program, children's understanding and execution of numerical operations are trained by a concurrent tutoring of both basic numerical and conceptual knowledge. Basic numerical knowledge, an important precursor for the acquisition of formal arithmetic operations, is defined as the ability to process and compare numerical magnitudes. Conceptual knowledge, on the other hand, describes an understanding of arithmetic operations and procedures. It enables the child to carry out arithmetic procedures and facilitates the storage of arithmetical facts (Baroody, 2003, 2006).
An investigation into mathematical difficulties and strategies for remediation is, however, complicated by the observation that MD is a rather heterogeneous phenomenon. In fact, there are at least two forms of MD that should be differentiated: an isolated form of MD and a combined form which is accompanied by additional reading problems. A number of studies have demonstrated that these two subgroups differ from each other with respect to defining deficits and residual capacities (e. g. Hanich et al., 2001; Jordan et al., 2003; Jordan et al., 2002). Unfortunately, studies comparing the efficacy of remediation programs for these two subgroups of MD are rare (but see Powell et al., 2009).
Aims: The current study examines the efficacy of an numerical intervention program originally implemented by Handl and Kaufmann (Kaufmann, Delazer et al., 2005; Kaufmann et al., 2003). First of all, we wanted to determine to what extent second and third graders with MD can improve their arithmetic performance with the aid of this intervention program. Secondly, we were interested in finding out whether children with additional reading deficits would be able to benefit from the intervention to the same extent as children with MD only.
Methods: We investigated the efficacy of the intervention program by comparing the development of 46 children with MD with 18 typically achieving control children. Based on their performances in a reading test, these two groups were further subdivided into children with MD only (MD; N = 24), children with MD and additional reading deficits (MD/RD; N = 22), controls without (CC; N = 9) and with reading deficits (RD; N = 9). One part of MD/RD was assigned to a MD control group (CC/MD/RD; N = 5), which was treated with a reading intervention program (Engl, et al., 2009) first and only after the current study was intruduced to the numerical intervention program. The MD group and the remaining MD/RD children were the experimental groups that underwent the numerical training. Prior to remediation onset, the three groups of children with MD were comparable with respect to their arithmetic achievement levels. All groups were matched for age, intelligence and working memory performance.
To assess changes in children's arithmetic performance levels, all children were administered a curricular oriented arithmetic test (Haffner et al., 2005) twice. MD children were further assessed using a neuropsychological dyscalculia test battery (Kaufmann et al., 2008) prior to the onset and after the completion the intervention program.
The numerical intervention program is composed of nine modules focusing on basic numerical and conceptual knowledge and is complemented by procedural and fact knowledge tutoring. Even though the different training modules are arranged in a hierarchical sequence, in practice, they are applied with a certain amount of overlap. The program consists of the following modules:
(1) magnitude representations and relations,
(2) counting knowledge,
(3) written arithmetical symbols and transcoding up to 20,
(4) numerical facts that sum up to 10 (e. g., 9 + 1, 8 + 2, 7 + 3, …),
(5) addition up to 10 and decomposition of numbers,
(6) subtraction and inversion problems,
(7) base-10 system and transcoding up to 100, counting in steps, multistep calculations,
(8) multiplication and repeated addition,
(9) division and inversion problems.
Mixed-age groups with 2 to 6 children were tutored once per week in block sessions similar in length to two regular school lessons (1.5 hours). To ensure comparable settings for all groups, the procedures and exercises were standardized. Intensity and duration of single modules varied depending on the severity of deficits children of the respective groups suffered from (on average 37 sessions in total).
Results: Comparisons of children's mathematical performance before and after the intervention revealed a significant interaction effect of time (pre, post) and group (MD, MD/RD, CC, RD, CC/MD/RD). While children with MD who took part in the program were able to improve their arithmetic performance significantly (ps ≤ .013), children of the waiting and control groups did not show comparable performance changes (ps ≥ .249). These findings demonstrate that the numerical intervention program was effective for both experimental groups. At the same time, MD and MD/RD did respond differently to the intervention, i. e. the improvement in performance was larger in children with MD (η p2 = .561) compared to MD/RD (η p2= .328). This group difference can be explained mainly by differential performance changes in tasks that assess addition, subtraction and the counting of objects. These tasks specifically load on fact and procedural knowledge. On the other hand, performance increases in tasks loading on conceptual knowledge and visual-spatial abilities were comparable for both experimental groups.
Discussion: The findings of the present study demonstrate that the adopted numerical intervention program (Kaufmann et al., 2003) is effective and, thus, appropriate for the remediation of MD in primary school children. Basic numerical, conceptual, procedural and arithmetic fact knowledge did improve significantly in the course of the intervention period. The overall effects were remarkable considering the fact the intervention program was extensive and implemented in a group setting, i. e. factors which typically result in rather small effect sizes (Baker et al., 2002; Kroesbergen & van Luit, 2003).
Another important result was that MD outperformed MD/RD with respect to performance gains during the intervention. One possible explanation for the specific disadvantage of MD/RD could be their difficulties in establishing fact and procedural knowledge (Geary & Hoard, 2001; Powell et al., 2009). Consequently, future studies should focus on the comparison of intervention strategies for children with isolated mathematical difficulties and children who suffer from additional reading problems. Furthermore, intensities and contents of the program should be manipulated in order to filter out the most effective components of this rather complex intervention program.
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