TY - JOUR A1 - Herzog, Moritz A1 - Ehlert, Antje A1 - Fritz, Annemarie T1 - A Competency Model of Place Value Understanding in South African Primary School Pupils JF - African Journal of Research in Mathematics, Science and Technology Education N2 - 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. KW - Place value KW - Rasch test modelling KW - mathematical concepts Y1 - 2017 U6 - https://doi.org/10.1080/18117295.2017.1279453 SN - 1811-7295 SN - 2469-7656 VL - 21 SP - 37 EP - 48 PB - Routledge, Taylor & Francis Group CY - Abingdon ER - TY - JOUR A1 - Fritz, Annemarie A1 - Ehlert, Antje A1 - Leutner, Detlev T1 - Arithmetische Konzepte aus kognitiv-entwicklungspsychologischer Sicht T1 - Artihmetic Concepts From a Cognitive Developmental-psychology Perspective JF - Journal für Mathematik-Didaktik N2 - 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. KW - Arithmetic KW - Development of concepts KW - Developmental model KW - Acquisition of arithmetical concepts KW - Mathematic in the primary school Y1 - 2018 U6 - https://doi.org/10.1007/s13138-018-0131-6 SN - 0173-5322 SN - 1869-2699 VL - 39 IS - 1 SP - 7 EP - 41 PB - Springer CY - Heidelberg ER - TY - JOUR A1 - Herzog, Moritz A1 - Ehlert, Antje A1 - Fritz, Annemarie T1 - Kombinatorikaufgaben in der dritten Grundschulklasse BT - Darstellung, Abstraktionsgrad und Strategieeinsatz als Einflussfaktoren auf die Lösungsgüte JF - Journal für Mathematik-Didaktik N2 - 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. N2 - For several years now, combinatorics has been taught as an independent mathematics topic in primary school that provides various learning opportunities. This raises questions about how learners actually solve and represent combinatorial problems. On the basis of six different combinatorics tasks, the performance and approach of pupils were assessed in an empirical study (N = 548). Particular attention was paid to the representation of the approach the children used, the degree of abstraction as well as the application of overarching (macro) strategies. The influence of each of these three factors on the quality of the solution was investigated. The results show that not all students were able to find suitable solutions for combinatorial problems without guidance. The representations the learners used had relatively little influence on the task performance. In contrast, influences on performance of levels of abstraction and use of strategies were found to varying degree. From the results of the study, conclusions can be drawn about the teaching methods used with regards to the treatment of combinatorics in primary school mathematics. T2 - Combinatorial Problems in Grade 3 Influences of Representation, Level of Abstraction and use of Strategies on Solving Success KW - Primary school KW - Combinatorics KW - Representation KW - Strategies KW - Grundschule KW - Kombinatorik KW - Darstellen KW - Makrostrategien Y1 - 2017 U6 - https://doi.org/10.1007/s13138-017-0118-8 SN - 0173-5322 SN - 1869-2699 VL - 38 SP - 263 EP - 289 PB - Springer CY - Heidelberg ER - TY - JOUR A1 - Fritz, Annemarie A1 - Long, Caroline A1 - Herzog, Moritz A1 - Balzer, Lars A1 - Ehlert, Antje A1 - Henning, Elizabeth T1 - Mismatch of the South African foundation phase curriculum demands and learners’ current knowledge JF - African journal of research in mathematics, science and technology education N2 - Against the background of the low mathematical performance of South African learners in international panel studies, there is an urgent need to improve mathematical education. In particular, the curriculum and its structure raise questions. It is logical that the prescribed curricula should align with learners' developmental trajectories. Given the hierarchical structure of mathematics, the curricular requirements should pay attention to learners' current knowledge of mathematical concepts. The aim of this study was to compare the curricular requirements as defined by the CAPS with the conceptual current knowledge of South African learners. South African Grade 1 learners (N = 602) were assessed on a test of numeracy concepts, based on a theoretically informed and empirically validated model of developing mathematical proficiency. The content of the CAPS for Grade 1 was aligned to the model levels by two experts (Cohen's kappa = .753, p < 0.001). Results show that the curricular requirements go far beyond the current knowledge required to engage with these new concepts of the vast majority of South African Grade 1 learners. The mismatch may to some extent be responsible for the unsatisfactory results in international comparison studies. These results show that the intended curriculum is beyond the grasp of most South African Grade 1 learners. These children are unlikely to develop new arithmetic concepts based on their lack of required foundation knowledge. We therefore argue that the intended curriculum for Grade 1 should focus more on counting skills, ordinal relations between numbers and-most importantly-set-based number representations and part-part-whole relations. KW - Arithmetic development KW - number concepts KW - curriculum development KW - mathematical learning difficulties Y1 - 2020 U6 - https://doi.org/10.1080/18117295.2020.1724466 SN - 1811-7295 SN - 2469-7656 VL - 24 IS - 1 SP - 10 EP - 20 PB - Routledge, Taylor & Francis Group CY - Abingdon ER - TY - JOUR A1 - Hartmann, Julia A1 - Ehlert, Antje A1 - Fritz, Annemarie T1 - Welche Rolle spielen sprachliche Parameter für die Entwicklung integrierter verbal-numerischer Konzepte im vierten Lebensjahr? JF - Frühe Bildung : interdisziplinäre Zeitschrift für Forschung, Ausbildung und Praxis N2 - 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. N2 - This article investigates how and to what extent early verbal competencies predict numerical competencies. In this study, we tested 72 toddlers at the age of 3 years two times with a 3-month period in-between. The results acquired by means of structural equation models reveal that early numerical competencies and early verbal competencies are closely related to each other and are less distinct. Numerical competencies are best predicted by numerical tasks, and at this age verbal competencies do not seem to play a significant role in predicting numerical competencies. We discuss our findings in the context of the current thesis of how speech and numeracy are aligned. T2 - Which Role Do Verbal Parameters Play in the Development of Integrated Verbal-Numerical Concepts at the Age of Four? KW - number KW - verbal competencies KW - numerical competencies KW - prediction KW - early childhood KW - numerik KW - sprachlische Kompetenzen KW - numerische Kompetenzen KW - Vorhersage KW - Kindheit Y1 - 2019 U6 - https://doi.org/10.1026/2191-9186/a000410 SN - 2191-9186 SN - 2191-9194 VL - 8 IS - 1 SP - 44 EP - 52 PB - Hogrefe CY - Göttingen ER -