Das Kardinalzahlkonzept

Abstract

This case-study investigates different aspects of the concept of cardinality of an eighteen-year-old student with mental retardation. At the age of six she could not relate number words, finger and objects in counting. These errors still persist in the classroom situation. This investigation shows that nevertheless her concept of cardinality is fairly highly developed. She knows that in counting she must match number words and objects one to one, the number word sequence she uses is stable, and her insight into the irrelevance of order of enumeration when counting, which she finds by trial, is a sign of the robustness of her cardinal concept. She also understands the relationships of equivalence and order of sets, and she solves arithmetical problems by counting on or down, which means that she understands the number words as cardinal and at the same time as sequence numbers. Errors occur in complex situations, where several components have to be considered. But her concept of cardinality is also incomplete: she has special difficulties concerning counting out objects bundled in tens. The same problems occur when she uses multidigit numbers: she does not see a ten-unit as composed of ten single unit items, that is to say, she replaces the hierarchic structure of the number sequence by a concatenated one. These difficulties must be interpreted as a consequence of her special weakness concerning synthetic thinking and simultaneous performing, as similar patterns can be seen in her spatial perception and in her speech. In the syntactic structure of her utterances, too, the combination of simple entities to complicated units is replaced by a mere concatenation. This means that due to brain dysfunction her behavior is determined by a particular pattern which repeatedly appears intrapersonally, and which is characteristic of some mentally retarded persons though not of all of them. Evidently mathematical thinking is also not a determined system, but a variable one. Mentally retarded students may therefore have great difficulties concerning some areas and at the same time make better progress in others. In particular, difficulties in counting objects are no obstacle to knowledge of cardinality.

Kurzreferat

Die Studie untersucht verschiedene Aspekte des Kardinalzahlkonzeptes bei einer 18jährigen Schülerin. Infolge von Hirnfunktionsstörungen ist ihr intelligentes Verhalten durch ein besonderes Muster bestimmt, das intrapersonell gleichförming auftritt und für manche, aber nicht für alle Menschen mit geistiger Behinderung charakteristisch ist. Die zugrundeliegenden Prozesse, die sich auch im Sprechen und im räumlichen Denken zeigen, erschweren besonders das Abzählen und das Arbeiten mit bestimmten Veranschaulichungsmitteln. Trotzdem konnte im Laufe der Schulzeit ein entwickeltes Kardinalzahlkonzept erworben werden. Das mathematische Denken ist also kein determiniertes System, sondern ein variables. Bei behinderten Schülern können deshalb große Schwierigkeiten in bestimmten Bereichen des Denkens und Lernens und zugleich bessere Fortschritte in anderen auftreten.