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The acquisition of basic arithmetic concepts of children at pre-school and primary-school age (about 4 to 8 years of age) can be described by a cognitive developmental model with 6 levels: (1) count number, (2) mental number line, (3) cardinality and decomposability, (4) class inclusion and embeddedness, (5) relationality, and (6) units in numbers (bundling and unbundling). In this paper, 3 studies for longitudinally testing the model are presented. In Studies 1 (N = 26; heterogeneous age) and 2 (N = 62; homogeneous age) it shows that the individual development of arithmetic concepts across 18 months (Study 1:4 points of measurement) respectively 17 months (Study 2:3 points of measurement before, at, and after entering primary school) follows the levels of the model. In Study 3 (N = 243) it shows that the acquisition of curricular mathematical competencies at the end of Grade 2 is better predicted by conceptual arithmetic understanding at the end of Grade 1 than by intelligence. The results substantiate the validity of the model und confirm the relevance of basic arithmetic concepts for mathematical learning at school with respective consequences for the remedial training of children with math learning difficulties and dyscalculia.
Teachers, as role models, are crucial in promoting inclusion in society through their actions. Being perceived as fair by their students is linked to students' feelings of belonging in school. In addition, their decisions of resource allocations also affect students' academic success. Both aspects underpin the importance of teachers' views on justice. This article aims to investigate what teachers consider to be just and how teacher characteristics and situational factors affect justice ratings of hypothetical student-teacher-interactions. In an experimental design, we randomly varied the description of the interacting student in text vignettes regarding his/her special educational need (SEN) (situational factor). We also collected data on teachers' attitudes toward inclusion and experiences with persons with disabilities (individual factors). A sample of in-service teachers in Germany (N = 2,254) rated randomized versions of two text vignettes. To also consider the effect of professional status, a sample of pre-service teachers (N = 275) did the same. Linear mixed effect models point to a negative effect of the SEN on justice ratings, meaning situations in which the interacting student is described with a SEN were rated less just compared to the control condition. As the interacting student in the situations was treated worse than the rest, this was indicative for the application of the need principle. Teachers with more positive attitudes toward inclusion rated the vignettes as significantly less just. Professional status also had a negative effect on justice ratings, with in-service teachers rating the interactions significantly lower than the pre-service teachers. Our results suggest that the teachers applied the principle of need in their ratings. Implications for inclusive teaching practices and future research are discussed further.
Although the general development of mathematical abilities in primary school has been the focus of many researchers, the development of place value understanding has rarely been investigated to date. This is possibly due to the lack of conceptual approaches and empirical studies related to this topic. To fill this gap, a theory-driven and empirically validated model was developed that describes five sequential conceptual levels of place value understanding. The level sequence model gives us the ability to estimate general abilities and difficulties in primary school pupils in the development of a conceptual place value understanding. The level sequence model was tried and tested in Germany, and given that number words are very differently constructed in German and in the languages used in South African classrooms, this study aims to investigate whether this level sequence model can be transferred to South Africa. The findings based on the responses of 198 Grade 2-4 learners show that the English translation of the test items results in the same item level allocation as the original German test items, especially for the three basic levels. Educational implications are provided, in particular concrete suggestions on how place value might be taught according to the model and how to collect specific empirical data related to place value understanding.
Seit einigen Jahren werden kombinatorische Aufgaben als eigenständiges Thema im Mathematikunterricht der Grundschule behandelt, das diverse Lerngelegenheiten bereithält. Dies wirft Fragen zur Bearbeitungsweise sowie zu den Repräsentationen der Schülerinnen und Schüler auf. In einer empirischen Studie (N = 548) wurden anhand von sechs Kombinatorikaufgaben Leistungen und Bearbeitungsweisen von Lernenden in der dritten Klasse erhoben. Besonderes Augenmerk wurde auf die verwendeten Darstellungen und den Abstraktionsgrad sowie den Einsatz von Makrostrategien gerichtet. Diese drei Faktoren wurden hinsichtlich ihres Einflusses auf die Lösungsgüte untersucht.
Die Ergebnisse zeigen, dass nicht alle Schülerinnen und Schüler ohne Anleitung geeignete Lösungsansätze zu kombinatorischen Problemen fanden. Die Darstellungsweise hatte einen vergleichsweise geringen Einfluss auf die Lösungsquote. Im Gegensatz dazu konnten unterschiedlich starke positive Zusammenhänge zwischen Abstraktionsgrad bzw. Strategieeinsatz und Lösungserfolg nachgewiesen werden. Aus den Ergebnissen der Studie lassen sich Schlüsse über die Vermittlung und didaktische Aufbereitung des Themenfeldes Kombinatorik in der Grundschulmathematik ziehen.
Der Beitrag untersucht, ob und zu welchen Anteilen frühe sprachliche Kompetenzen numerische Kompetenzen vorhersagen. An 72 dreijährigen Kindern wurden numerische, verbal produktive und rezeptive sowie grammatische Leistungen zwei Mal im Abstand von drei Monaten erhoben. Mithilfe von Strukturgleichungsmodellen kann gezeigt werden, dass sprachliche und numerische Leistungen in diesem Alter noch wenig distinkt sind. Für die numerischen Kompetenzen findet sich bereits in diesem Alter eine hohe interindividuelle Entwicklungsstabilität. Ein bedeutsamer Einfluss sprachlicher Kompetenz auf den Zuwachs mathematischer Kompetenz im vierten Lebensjahr konnte nicht nachgewiesen werden. Wir diskutieren die Ergebnisse vor dem Hintergrund der aktuellen Thesen zum Zusammenhang von Sprache und Numerik in der Entwicklung.
Intelligence, as well as working memory and attention, affect the acquisition of mathematical competencies. This paper aimed to examine the influence of working memory and attention when taking different mathematical skills into account as a function of children’s intellectual ability. Overall, intelligence, working memory, attention and numerical skills were assessed twice in 1868 German pre-school children (t1, t2) and again at 2nd grade (t3). We defined three intellectual ability groups based on the results of intellectual assessment at t1 and t2. Group comparisons revealed significant differences between the three intellectual ability groups. Over time, children with low intellectual ability showed the lowest achievement in domain-general and numerical and mathematical skills compared to children of average intellectual ability. The highest achievement on the aforementioned variables was found for children of high intellectual ability. Additionally, path modelling revealed that, depending on the intellectual ability, different models of varying complexity could be generated. These models differed with regard to the relevance of the predictors (t2) and the future mathematical skills (t3). Causes and conclusions of these findings are discussed.
The goal of the present study was to analyze how students' attitudes towards inclusive education develop over the course of a school year and how these attitudes relate to students' peer relations. Sixth- and seventh-graders of 44 inclusive classes filled out a questionnaire at two measurement points within one school year to assess attitudes towards inclusive education and peer relations. Applying multilevel regression analyses it turned out that changes in peer relations over time were positively predicted by students' attitudes towards instructional adaptations for students with behaviour difficulties. Further, students with self-perceived behavior difficulties reported lower scores for peer relations compared to students without self-perceived difficulties.
Results are discussed with respect to structural factors and individual characteristics affecting inclusive education.
Intelligence, as well as working memory and attention, affect the acquisition of mathematical competencies. This paper aimed to examine the influence of working memory and attention when taking different mathematical skills into account as a function of children’s intellectual ability. Overall, intelligence, working memory, attention and numerical skills were assessed twice in 1868 German pre-school children (t1, t2) and again at 2nd grade (t3). We defined three intellectual ability groups based on the results of intellectual assessment at t1 and t2. Group comparisons revealed significant differences between the three intellectual ability groups. Over time, children with low intellectual ability showed the lowest achievement in domain-general and numerical and mathematical skills compared to children of average intellectual ability. The highest achievement on the aforementioned variables was found for children of high intellectual ability. Additionally, path modelling revealed that, depending on the intellectual ability, different models of varying complexity could be generated. These models differed with regard to the relevance of the predictors (t2) and the future mathematical skills (t3). Causes and conclusions of these findings are discussed.
Im inklusiven Unterricht ist es von zentraler Bedeutung, das inhaltliche und pädagogische Handeln auf die individuellen Lernvoraussetzungen der Schüler_innen anzupassen. Der aktuelle Lernstand der Kinder muss daher aus deren unterrichtlichen Handeln heraus festgestellt werden. Dafür stellt die Diagnosekompetenz der Lehrkräfte eine wichtige Grundlage dar. Um diese Kompetenz zu fördern, wurde im Rahmen des Projekts „Qualitätsoffensive Lehrerbildung“ an der Universität Potsdam ein Seminarkonzept im Rahmen des Praxissemesters entwickelt. Die diagnostischen Kompetenzen der Studierenden werden dabei durch das Prinzip des Micro-Teachings anhand von Videovignetten gefördert.
Für den inklusiven Unterricht und die dadurch wachsende Heterogenität im Unterricht sind die Einschätzung der Lernvoraussetzungen und die Anpassung der Unterrichtsinhalte daran von großer Bedeutung. Für die Umsetzung dieser Anforderungen müssen die Lehrkräfte über diagnostische Kompetenzen verfügen. Um diese zu erfassen wurde im Rahmen des Projekts „Qualitätsoffensive Lehrerbildung“ an der Universität Potsdam ein Erhebungsinstrument entwickelt. Dabei werden die diagnostischen Kompetenzen im mathematischen Bereich anhand eines Fragebogens in Verbindung mit Videovignetten erfasst. Das Instrument wurde sowohl mit Lehramtsstudierenden als auch mit aktiven Lehrkräften und Sonderpädagogen evaluiert und erreicht sehr gute Reliabilitäts- und Validitätswerte. Außerdem ergeben sich durch den Einsatz von Videosequenzen authentische Praxisbeispiele und somit kann eine wichtige Kompetenz abgebildet werden.
