Einflüsse der Inversion auf die Verarbeitung mehrstelliger Zahlen bei deutschsprachigen Kindern – ein Überblick
Abstract
Zusammenfassung. Der vorliegende Artikel soll einen Überblick über die aktuelle Literatur zur Verarbeitung von zwei- und mehrstelligen Zahlen bei Kindern liefern. Dabei geht es nicht nur um das Lesen und Schreiben von arabischen Ziffern, das zweifelsfrei die erste Hürde auf dem Weg zum Verständnis des arabischen Stellenwertsystems darstellt. Auch der Umgang mit und das Verständnis von mehrstelligen Zahlen werden detailliert diskutiert. So wird zuerst die Metapher des mentalen Zahlenstrahls, auf dem ein- wie mehrstellige Zahlen von links nach rechts aufsteigend angeordnet sind, erläutert. Anschließend wird diskutiert, wie sich dieser bei Kindern entwickelt und welche besonderen Schwierigkeiten die Inversion in deutschen Zahlwörtern (d. h. die Umkehrung der Reihenfolge von Zehnern und Einern, z. B. «einundzwanzig» vs. «21») bedingen kann. Dazu wird u. a. der Frage nachgegangen, inwieweit zwei- und mehrstellige Zahlen als eine Einheit oder getrennt in Einer, Zehner, etc. verarbeitet werden. Ein weiterer Teil dieser Übersichtsarbeit widmet sich Aspekten des Rechnens mit mehrstelligen Zahlen und den hierfür bereits in der Literatur beschriebenen Effekten. Als zentrale These wird für alle oben genannten Aspekte untersucht, wie die beteiligten Prozesse und Repräsentationen durch die Inversion in deutschen Zahlwörtern negativ beeinflusst werden. Wir kommen zu dem Ergebnis, dass die Inversion ein nicht zu vernachlässigender und erschwerender Faktor für das Lernen von Mathematik in deutschsprachigen Ländern ist. Die Implikationen dieser Befunde für mathematisches Lernen und assoziierte Lernstörungen werden abschließend diskutiert.
Abstract.Background: To date, the majority of research on numerical cognition in general, and of research on numerical development, in particular, is conducted using single-digit numbers as stimuli. However, this only covers a small part of the reality of our everyday experiences with numbers as a lot of numbers we encounter in everyday life are usually multi-digit numbers (e. g., prices, digital clock faces, phone numbers, etc.). Thus, the psychological mechanisms underlying multi-digit number processing should be of specific interest with regard to everyday number processing necessities. In numerical cognition research in adults there is a growing interest into multi-digit numbers. However, this is only partly true for the developmental aspects of multi-digit number processing. Against this background the current article aimed at summarizing and discussing the recent literature on multi-digit number processing and its underlying mechanisms in children. Thereby, particular attention is paid to the dominant organization principle of our Arabic number system – its so-called place-value structure – and how this structural information influences the development of multi-digit number processing in children. Aims: As the majority of research on multi-digit number processing focused on two-digit numbers we will refer to studies investigating two-digit number processing in children most prominently. In particular, we will evaluate empirical studies on the development of children's competencies considering the following aspects: (i) children's reading of two-digit numbers, (ii) children's spatial representation of two-digit numbers (i. e., their mental number line for two-digit numbers), (iii) procession of two-digit numbers as integrated entities or rather decomposed into tens and units, (iv) specificities of arithmetic involving two-digit operands, and finally (v) the issue of numbers exceeding the two-digit number range. Before considering each of these aspects in turn we start with an introduction paragraph on the importance of numerical competencies in modern society and describe briefly what challenges children encounter on their way to numerate adults by referring to the different numerical formats (e. g., Arabic numbers, number words) and the difficulties that arise from transcoding numbers from one format into another. Then we address the transcoding process in greater detail. As regards the reading of two-digit numbers we outline that the number word structure in most languages is not as regular as the Arabic number system and that this results in specific difficulties. Not only that specific number words do not reflect their two-digit nature directly (e. g., “twelve” as a word of its own), these often irregular number words (e. g., “teens”, multiples of ten) also have to be combined following specific composition principles that need to be learned and internalized. For instance, the additive composition principle and the necessary overwriting of zeros have to be understood meaning that to come from »fifty eight” to »58” one has to (i) add the “8” to the “50” and (ii) write the “8” at the position of the “0” within “50” and not add it at the end, which would result in “508”. Moreover, in German the inversion property of number words (i. e., the units are spoken before the tens, e. g., 24 → “vierundzwanzig” → “four and twenty”) makes transcoding even more difficult. In the second section we then describe the metaphor of the mental number line as a proxy for the spatial layout of the mental representation of number magnitude. Furthermore, we then show that children's initial mental number line for two-digit numbers is not comparable to that of adults but still has to develop its characteristic equidistant spacing between adjacent numbers. Initially, children overestimate the spatial extent of single-digit as compared to two-digit numbers. This specific differentiation between single- and two-digit numbers is interpreted as an influence of the place-value structure of the Arabic number system. Comparable to detrimental influences of the inversion property on number reading we also describe similar influences of inversion on German-speaking children's number line representation. In the following section we discuss recent evidence questioning the dominant view that multi-digit numbers are represented as integrated entities. In particular, the so-called unit-decade compatibility effect is illustrated. The unit-decade compatibility effect describes the finding that it is more difficult to single out the larger number of a pair when the decision biases induced by separate comparisons of tens and units are incompatible (e. g., 47_62, with 4 < 6 but 7 > 2) than when they are compatible (e. g., 42_57, with 4 < 5 and 2 < 7) even when overall numerical distance is held constant (e. g., 15 in both above examples). The compatibility effect is usually assessed in magnitude comparison tasks presenting a balanced set of unit-decade compatible and incompatible number pairs. Interestingly, the compatibility effect seems to be influenced by the presence or absence of within-decade filler items in the stimulus set (e. g., 42_47). As the unit digit is decision relevant in these number pairs it is not a beneficial strategy to focus on the tens digits as it is for between-decade pairs. Having to consider the unit digits as well in a certain amount of the trials is then assumed to increase the influence of the decision irrelevant unit digits in between-decade comparisons. Taken together, the compatibility effect clearly indicates that tens and units of two-digit numbers are processed separately. We present evidence that children as early as grade one are able to process tens and units in a decomposed manner, thereby, retaining the place-value information of the respective numbers. Interestingly, children from grade one on seem to be able to adapt their processing strategy to task demands. Depending on stimulus properties (i. e., the inclusion of within-decade fillers) they are able to process tens and units sequentially (i. e., comparing the tens first) or in parallel indicating a quite flexible use of different strategies specifically considering place-value information. Additionally, we also present evidence suggesting an influence of the inversion property, as the compatibility effect is more pronounced in a language with inversion as compared to a language without. In the fourth section we then review the existing empirical data on two-digit arithmetic with the focus on addition. Again, it is argued that there are specific influences of the place-value structure of the Arabic number system. Comparable to adult data the eye-movement pattern of third-graders indicated that carry addition problems require specific processing of the unit digits as these determine whether a carry operation is needed (unit sum > 9) or not. As in the previous sections, we discuss first evidence suggesting a detrimental influence of the inversion property of German number words on children's addition performance. In a last section, we extend the scope from two- to multi-digit numbers and discuss first empirical evidence on the processing of three- and higher-digit numbers. In particular, we describe that the compatibility effect can also be observed as interference between the processing of hundreds and tens as well as hundreds and units in three-digit number comparison. Additionally, there is now first evidence from adults (that still awaits replication for children) that processing six-digit numbers is a mixture of sequential and parallel processing of the single digits constituting the respective numbers. Discussion: Taken together, we conclude that there is undisputed evidence for a vital influence of the place-value structure of the Arabic number system on numerical cognition, not only in adults, but particularly so in children. Recent data even indicates that early place-value understanding in first grade is a reliable predictor of arithmetic competencies in third grade: The better children's place-value understanding in first grade, the higher their performance in an addition task two years later. Additionally, inversion-related transcoding errors indicating insufficient place-value understanding even predicted the mathematics grade at the end of grade three. Thereby, not only place-value understanding per se seems to influence numerical development in children but also the inversion property of German (and other languages') number words. It is important to note that these developmental trajectories were observed while controlling for influences of general cognitive functioning and working memory. Against this background, we suggest to consider place-value understanding as an important determinant of numerical competencies. As a consequence we advocate systematic training of place-value understanding during the first years of schooling.
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