@article{FritzLongHerzogetal.2020,
  author    = {Fritz, Annemarie and Long, Caroline and Herzog, Moritz and Balzer, Lars and Ehlert, Antje and Henning, Elizabeth},
  title     = {Mismatch of the South African foundation phase curriculum demands and learners' current knowledge},
  series = {African journal of research in mathematics, science and technology education},
  volume    = {24},
  journal   = {African journal of research in mathematics, science and technology education},
  number    = {1},
  publisher = {Routledge, Taylor \& Francis Group},
  address   = {Abingdon},
  issn      = {1811-7295},
  doi       = {10.1080/18117295.2020.1724466},
  pages     = {10 -- 20},
  year      = {2020},
  abstract  = {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.},
  language  = {en}
}
@article{HartmannEhlertFritz2019,
  author    = {Hartmann, Julia and Ehlert, Antje and Fritz, Annemarie},
  title     = {Welche Rolle spielen sprachliche Parameter f{\"u}r die Entwicklung integrierter verbal-numerischer Konzepte im vierten Lebensjahr?},
  series = {Fr{\"u}he Bildung : interdisziplin{\"a}re Zeitschrift f{\"u}r Forschung, Ausbildung und Praxis},
  volume    = {8},
  journal   = {Fr{\"u}he Bildung : interdisziplin{\"a}re Zeitschrift f{\"u}r Forschung, Ausbildung und Praxis},
  number    = {1},
  publisher = {Hogrefe},
  address   = {G{\"o}ttingen},
  issn      = {2191-9186},
  doi       = {10.1026/2191-9186/a000410},
  pages     = {44 -- 52},
  year      = {2019},
  abstract  = {Der Beitrag untersucht, ob und zu welchen Anteilen fr{\"u}he sprachliche Kompetenzen numerische Kompetenzen vorhersagen. An 72 dreij{\"a}hrigen 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. F{\"u}r die numerischen Kompetenzen findet sich bereits in diesem Alter eine hohe interindividuelle Entwicklungsstabilit{\"a}t. 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.},
  language  = {de}
}
@article{FritzEhlertLeutner2018,
  author    = {Fritz, Annemarie and Ehlert, Antje and Leutner, Detlev},
  title     = {Arithmetische Konzepte aus kognitiv-entwicklungspsychologischer Sicht},
  series = {Journal f{\"u}r Mathematik-Didaktik},
  volume    = {39},
  journal   = {Journal f{\"u}r Mathematik-Didaktik},
  number    = {1},
  publisher = {Springer},
  address   = {Heidelberg},
  issn      = {0173-5322},
  doi       = {10.1007/s13138-018-0131-6},
  pages     = {7 -- 41},
  year      = {2018},
  abstract  = {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.},
  language  = {de}
}
@article{HerzogEhlertFritz2017,
  author    = {Herzog, Moritz and Ehlert, Antje and Fritz, Annemarie},
  title     = {A Competency Model of Place Value Understanding in South African Primary School Pupils},
  series = {African Journal of Research in Mathematics, Science and Technology Education},
  volume    = {21},
  journal   = {African Journal of Research in Mathematics, Science and Technology Education},
  publisher = {Routledge, Taylor \& Francis Group},
  address   = {Abingdon},
  issn      = {1811-7295},
  doi       = {10.1080/18117295.2017.1279453},
  pages     = {37 -- 48},
  year      = {2017},
  abstract  = {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.},
  language  = {en}
}
@article{HerzogEhlertFritz2017,
  author    = {Herzog, Moritz and Ehlert, Antje and Fritz, Annemarie},
  title     = {Kombinatorikaufgaben in der dritten Grundschulklasse},
  series = {Journal f{\"u}r Mathematik-Didaktik},
  volume    = {38},
  journal   = {Journal f{\"u}r Mathematik-Didaktik},
  publisher = {Springer},
  address   = {Heidelberg},
  issn      = {0173-5322},
  doi       = {10.1007/s13138-017-0118-8},
  pages     = {263 -- 289},
  year      = {2017},
  abstract  = {Seit einigen Jahren werden kombinatorische Aufgaben als eigenst{\"a}ndiges Thema im Mathematikunterricht der Grundschule behandelt, das diverse Lerngelegenheiten bereith{\"a}lt. Dies wirft Fragen zur Bearbeitungsweise sowie zu den Repr{\"a}sentationen der Sch{\"u}lerinnen und Sch{\"u}ler 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 L{\"o}sungsg{\"u}te untersucht. Die Ergebnisse zeigen, dass nicht alle Sch{\"u}lerinnen und Sch{\"u}ler ohne Anleitung geeignete L{\"o}sungsans{\"a}tze zu kombinatorischen Problemen fanden. Die Darstellungsweise hatte einen vergleichsweise geringen Einfluss auf die L{\"o}sungsquote. Im Gegensatz dazu konnten unterschiedlich starke positive Zusammenh{\"a}nge zwischen Abstraktionsgrad bzw. Strategieeinsatz und L{\"o}sungserfolg nachgewiesen werden. Aus den Ergebnissen der Studie lassen sich Schl{\"u}sse {\"u}ber die Vermittlung und didaktische Aufbereitung des Themenfeldes Kombinatorik in der Grundschulmathematik ziehen.},
  language  = {de}
}