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Development of a Sustainable Place Value Understanding

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Abstract

A resilient understanding of the decimal place value system is a crucial element of primary mathematical education that has a deep impact on further arithmetical development. Place value understanding has been identified as a good predictor of math performances as well as math difficulties. Difficulties in understanding the place value system affect children in different grades in countries all over the world. Prior and recent research has been focused on ordinal counting schemes and computing strategies. Much attention has been given to transcoding procedures that transfer numerals, number words, and magnitude representations to each other. However, the cardinal aspect of bundling and unbundling has not been taken into account. In this chapter we aim to present a sequence of levels of concepts that describe children’s development of place value understanding. The presented sequence integrates aspects of counting and bundling—in particular, regarding nonstandard partitions. Bundling tasks seem to be more crucial for place value understanding than transcoding tasks. Our own recent empirical studies underpin the hierarchy of the sequence levels.

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... Studies show that many children have difficulties in understanding the base-ten place value system (e.g., Herzog et al. 2019;Scherer 2014) and because of that there is a need for support. Several studies deal with the development of an un-derstanding of the base-ten place value system and of multi-digit numbers (Cobb and Wheatley 1988;Fuson et al. 1997;Herzog et al. 2017aHerzog et al. , b, 2019Ross 1986Ross , 1989Sinclair et al. 1992), focussing on different aspects of the components of place value understanding as shown in Fig. 1. ...
... Types a and b showed difficulties in both principles and therefore the most problems in the tasks. Only writing down the given digits without regrouping like the children of type a.1 is a solution behaviour that Herzog et al. (2019) also describe in a lower level of their model, where the children cannot deal with non-canonical representations and "apply the routine of canonical representations" (p. 575). ...
... A diagnostic conclusion here might be, that some type-e-children should engage in regrouping activities as type a. Type f mainly made regrouping errors when it was necessary to regroup tens into hundreds. The children therefore seem to be able to regroup ten ones into one ten but cannot transfer this idea to regrouping ten tens into one hundred, like also Herzog et al. (2019) describe. Children of this type should work mainly on tasks where tens have to be regrouped in hundreds using ten blocks. ...
Article
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The understanding of place value systems, especially the base-ten place value system, is one of the most important prerequisites to develop numeracy. The understanding of place value systems can be ascribed to two concepts, which in the tradition of the German subject-matter didactics are called regrouping principle and place value principle. Our study aims at clarifying whether these two principles can be used systematically for an effective identification of the gaps in students’ understanding to give a basis for individual support. We therefore conducted a study with N = 100 third graders (8 to 10 years old). We asked the students to work on 7 tasks on translating named units into written numbers using the place value system, in which the place value principle, the regrouping principle or both principles had to be considered. We analysed the errors qualitatively regarding which principle was violated and developed a typification of solution behaviour. The identified individual difficulties by taking the perspective of the two principles can be a starting point for individual support. Our tasks are shown to be a quick and easy way to diagnose students’ individual problems in understanding the base-ten place value system. Online: https://rdcu.be/dMr7B
... Le système de numération indo-arabe est le fruit de longues évolutions culturelles (Dehaene, 2010). Son efficacité réside dans sa capacité à coder des nombres infiniment grands avec un ensemble fini de symboles (c'est-à-dire, dix chiffres), grâce à son système de position (Herzog et al., 2019). La compréhension et l'utilisation de ce système de numération est centrale à toute activité numérique (Nuerk et al., 2015). ...
... La compréhension et l'utilisation du système de numération nécessitent un apprentissage formel et explicite. Herzog et al. (2019) décrit quatre stades dans un modèle de développement conceptuel. ...
... Enfin, les enfants de CE1 semblent avoir atteint le stade 4 car ils ont acquis la lecture et l'écriture des nombres jusqu'à 4 chiffres, qu'ils sont capables de produire un nombre avec objets virtuels ou de comprendre le lien entre un nombre en code indo-arabe et une représentation visuelle (support visuel) et qu'enfin, ils sont capables de décomposer un nombre sans support visuel. Ces données expérimentales soutiennent le modèle d 'Herzog et al. (2019) : l'aspect procédural du système de numération serait acquis avant l'acquisition conceptuelle : il s'agit pour la petite proportion de GSM et la moitié des CP observés d'une utilisation uniquement fonctionnelle avec support, sans compréhension approfondie du concept. Ces résultats suggèrent une bonne validité de contenu. ...
Article
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Résumé : La compréhension du système de numération décimale à valeur positionnelle est essentielle au traitement des nombres. Sa maîtrise est un prédicteur des compétences arithmétiques futures. C'est une compétence difficile à acquérir, également pour les enfants au développement typique. Évaluer la compréhension approfondie du système de numération décimale à valeur positionnelle est primordial ; pourtant, peu d'outils à la disposition des orthophonistes permettent d'évaluer cette compétence pour les enfants en apprentissage, c'est-à-dire entre 5 et 8 ans. Les objectifs de cette recherche étaient de vérifier la cohérence interne des épreuves et du module Numération de la nouvelle batterie Examath 5-8, sa validité concomitante, sa validité de construit en lien avec les caractéristiques des individus (la classe) et l'établissement de normes critériées. L'échantillon était composé de 15 enfants de GSM, 12 enfants de CP et 14 enfants de CE1, sans suivi orthophonique pour des difficultés mathématiques, dans deux écoles différentes. Le matériel utilisé était composé des épreuves du module Numération d'Examath 5-8 et du Tedi-math ainsi que du PicPVT. Les résultats ont montré que le module Numération d'Examath 5-8 possède une bonne cohérence interne, une bonne validité concomitante et une bonne validité de construit en lien avec les caractéristiques des individus (la classe). L'établissement d'une norme critériée a permis de montrer que les nouvelles tâches d'évaluation du système de numération décimale à valeur positionnelle sont adaptées et permettent d'observer les compétences en numération des enfants en apprentissage. D'autres propriétés psychométriques doivent être évaluées pour permettre la validation de cette nouvelle batterie, comme la validité de surface, la validité prédictive ou la validité discriminante (sensibilité). Summary: Understanding the place-value number system is essential to number processing. It predicts the later arithmetic skills. Even typically developing children struggle to acquire it. Assessing the place-value number system understanding is crucial. However, there are few tools available to speech-language therapists to assess this skill in learners between five and eight years old. The aim of this research was to verify the internal consistency of the Number system module of Examath 5-8, its concurrent validity, its construct validity in relation to individual characteristics (grade), and to develop criterized norms. The sample was composed of 15 children in kindergarten, 12 children in first grade, and 14 children in second grade. These children were from two different schools and had no intervention from a speech-language therapist for mathematics difficulties. Number system tasks from Examath 5-8, Tedi-Math, and PicPVT were used. The results showed that the Number system module of Examath 5-8 demonstrated good internal consistency, good concurrent validity, and good construct validity in relation to individual characteristics (grade). The creation of criterized norms allowed to demonstrate that the new place-value number system assessment tasks are adapted and allow to observe children’s place-value skills. Other psychometric properties need to be investigated, such as surface validity, predictive validity, or discriminant validity (sensibility).
... Based on the notion that transcoding is mostly based on a conceptual understanding of the place value system, procedural and asemantic models such as the ADPAT models have been criticized (e.g., Geary, 2004;Desoete and Grégoire, 2006). However, a profound understanding of the decimal place value system covers both procedural and conceptual aspects such as writing and reading numbers and insight in the iterative relation of the bundling units (Fuson et al., 1997a;Van de Walle et al., 2016;Herzog et al., 2019;Houdement and Tempier, 2019). ...
... To structure the development of place value understanding, Herzog et al. (2019) proposed a developmental model of place value understanding that distinguishes five levels. The levels build up on each other hierarchically. ...
... The influence of the earlier models gets visible in the description of the levels below. However, the earlier models cover only two-digit numbers, in contrast to the model by Herzog et al. (2019). The significance of the relation between bigger bundling units such as hundreds and thousands is highlighted in the literature (Scherer and Moser Opitz, 2010;Houdement and Tempier, 2019). ...
Article
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Recent studies have shown that children’s proficiency in writing numbers as part of the so-called transcoding correlates with math skills. Typically, children learn to write numbers up to 10,000 between Grade 1 and 4. Transcoding errors can be categorized in lexical and syntactical errors. Number writing is thus considered a central aspect of place value understanding. Children’s place value understanding can be structured by a hierarchical model that distinguishes five levels. The current study investigates to what extent a profound understanding of the place value system can explain individual differences in number writing. N = 266 s and third graders (126 girls) participated in the study. The children wrote down 28 verbal given numbers up to 10,000 and completed a place value test based on a hierarchical model to assess number writing skills and place value understanding. Second graders made more number writing errors than third graders and transcoding errors were mostly syntactical errors. In both grades, transcoding performance and place value understanding correlated substantially. In particular complex numbers were more often solved correctly by children with a more elaborated place value understanding. The effect of place value understanding on error rate was smaller regarding lexical errors than syntactical errors. This effect was also comparably small regarding inversion-related errors. The results underpin that writing numbers is an integral part of early place value understanding. Writing numbers can be assumed to be mostly based on the identification of the place values. However, variance in transcoding skills cannot totally be explained by place value understanding, because children with an elaborated place value understanding differed in transcoding performance, too. The differences between the grades indicate that children’s development of writing numbers is also driven by instruction in school. Thus, writing numbers and place value understanding overlap but exceed each other. We discuss how an understanding of the place value relations can be integrated in existing frameworks of place value processing. Since writing numbers is a basic skill in place value understanding, it might serve as an efficient screening method for children, who struggle severely with understanding the decimal place value system.
... A lack of place value understanding adversely affects the ability to succeed in other mathematical concepts and operations (Moeller et al., 2011). If students do not understand place value concepts, they cannot perform mathematical skills such as rounding off to the nearest tens and hundreds (Van de Walle et al., 2016), making arithmetic calculations, comparing multi-digit numbers (Dietrich et al., 2016), solving problems, understanding divisibility laws, prime numbers and exponential numbers (Sharma, 1993), making operations about decimals and fractions (Herzog et al., 2019), and preventing misconceptions in numerical calculations (Kamii & Joseph, 1988;Sar & Olkun, 2019) revious research suggests that the place value concept cannot be effectively understood by primary school students with the current instructional methods used in classrooms today (Dinç-Artut & Tar m, 2006;Herzog et al , 2019;Sar & Olkun, 2019;Thouless, 2014). ...
... A lack of place value understanding adversely affects the ability to succeed in other mathematical concepts and operations (Moeller et al., 2011). If students do not understand place value concepts, they cannot perform mathematical skills such as rounding off to the nearest tens and hundreds (Van de Walle et al., 2016), making arithmetic calculations, comparing multi-digit numbers (Dietrich et al., 2016), solving problems, understanding divisibility laws, prime numbers and exponential numbers (Sharma, 1993), making operations about decimals and fractions (Herzog et al., 2019), and preventing misconceptions in numerical calculations (Kamii & Joseph, 1988;Sar & Olkun, 2019) revious research suggests that the place value concept cannot be effectively understood by primary school students with the current instructional methods used in classrooms today (Dinç-Artut & Tar m, 2006;Herzog et al , 2019;Sar & Olkun, 2019;Thouless, 2014). ...
... Although the term "place value" has been conceptualized in a number of different ways (e.g., Fuson et al., 1997;Herzog et al., 2019;Ross, 1989), two key principles are at the core of the base-10 number system. First, the position of each digit in a numeral is associated with a specific denomination determined by a power of 10 (e.g., 10 0 , or ones; 10 1 , or tens; 10 2 , or hundreds). ...
... Although in line with previous work (Chan et al., 2017;Lambert & Moeller, 2019;Landerl & Kölle, 2009;Moura et al., 2013), our study is one of the first to describe the types of place-value difficulties that are closely aligned with the school curriculum and, therefore, particularly valuable in the context of instruction (Newcombe et al., 2009). Given the complex nature of children's numeration knowledge (Fuson et al., 1997;Herzog et al., 2019), further investigation on other facets of the place-value understanding in this population is warranted. ...
Article
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We investigated the effect of conceptual transparency in the physical structure of manipulatives on place-value understanding in typically-developing children and those at risk for mathematics learning disabilities. Second graders were randomly assigned to one of three manipulatives conditions: (a) attachable beads that did not make the denominations or ones in the denominations transparent, (b) pipe cleaners that made only the denominations transparent, and (c) string beads that made both the denominations and the ones in the denominations transparent. Participants used the manipulatives to represent double- and triple-digit numerals. Statistical analyses indicated that the transparency of the denominations, but not the transparency of the ones in the denominations, is responsible for children’s number representation and place-value understanding. Descriptive analyses of their responses revealed that the at-risk children were at a greater disadvantage than their typically-developing peers with the attachable beads, failing to use place-value concepts to interpret their representations.
... z.B. Cervasoni, 2011;Herzog et al., 2019). -Auf den Punkt gebracht lauten die Ergebnisse, dass sich das Verständnis des dezimalen Stellenwertsystems über mehrere Jahre entwickelt und erst in der 5. Primarstufe von den meisten Kindern verstanden wird. ...
... -Auf den Punkt gebracht lauten die Ergebnisse, dass sich das Verständnis des dezimalen Stellenwertsystems über mehrere Jahre entwickelt und erst in der 5. Primarstufe von den meisten Kindern verstanden wird. Herzog et al. (2019) weisen darauf hin, dass das Verständnis auch bei grossen Zahlen untersucht und auf den höheren Schulstufen geübt werden muss. ...
Experiment Findings
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Im Aufsatz wird ein flexibles Interview von Sharon Ross (1986; zit. nach Kamii, 1989) über das dezimale Stellenwertsystem vorgestellt. Seit 1991 habe ich es in Workshops und in der schulpsychologischen Diagnostik im Kanton Solothurn eingesetzt. Seit 2001 wurden an der HfH verschiedene Studien und Tausende von Fallstudien durchgeführt, in denen diese Aufgabe repliziert und weiterentwickelt wurde (Brugger et al., 2007). Die Aufgabe wurde 2013 im MKT-2 einer repräsentativen Stichprobe von 267 Kindern aus der deutschsprachigen Schweiz und dem Fürstentum Liechtenstein vorgelegt. // The essay presents a flexible interview from Sharon Ross (1986) about the decimal place value, which was published by Constance Kamii (1989). Since 1991 I used it in workshops and diagnostics in the School Psychology of the Canton of Solothurn. Since 2001, various studies and thousands of case studies have been carried out at the HfH, in which this task has been replicated and further developed (Brugger et al., 2007). The task was presented in the MKT-2 to a representative sample of 267 children from German-speaking Switzerland and the Principality of Liechtenstein. 44% of second graders were able to complete it at the end of the school year (see Meyer & Wyder, 2017). The essay explains how to conduct a flexible interview in a pedagogical situation. Subsequently, metacognitive questions are suggested, which are carried out in free conversation with the children or with groups. Thus, the child's maximally possible thinking operations are explored and fostered in a school of thinking cf. Adey, 2008). This procedure expresses the dynamic character of the flexible interview or, as it is also called, the method of critical exploration.
... Das Verständnis des dezimalen Stellenwertsystems heranzubilden, ist eine komplexe und langwierige Aufgabe. Die Forschung bestätigt seit mehreren Jahrzehnten, dass sich die Einsicht bei der Mehrzahl der Schülerinnen und Schüler (abgekürzt: SuS) zwischen der zweiten und fünften Klassenstufe einstellt, bei einigen Kindern erst in der Sekundarstufe 1 (Ross, 1986;Schuler, 2004;Brugger, Sidler & Meyer, 2007;Moser Opitz, 2007;Ruflin, 2008;Herzog, Fritz & Ehlert, 2017;Herzog, Ehlert & Fritz, 2019; siehe Tabelle 1). ...
... Tabelle 1 illustriert die Zusammenhänge zwischen den Denkniveaus nach Piaget (1977aPiaget ( , 1977b, Piaget et al. (1990) sowie zwischen den entwicklungspsychologischen Niveaus der Einsicht in das dezimale Stellenwertsystems nach Ross (1986), Herzog et al. (2019) sowie den Handlungsaspekten der unterrichtlichen Episoden, den Darstellungsformen und den Zyklen der Bildung gemäss Lehrplan 21. Der Stil der Tabelle 1 orientiert sich am Konzept der kognitiven Akzeleration (Adey, 2008). ...
Article
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Der Essay beschreibt, wie das logisch-mathematische Denken und Operieren im Umgang mit dem Stellenwertsystem kreativ, operativ und exemplarisch gefördert werden kann. Theoretisch steht der Essay der kognitiven Akzeleration nahe (vgl. Adey, 2008). Darin wird die Denkschulung mit der genetischen Entwicklungspsychologie (Piaget 1977a, 1977b; Piaget & Voelin, 1980), mit dem Curriculum, mit der Zone der nächsten Entwicklung (Vygotskij, 1986) und der Metakognition vereinigt. Der Essay integriert das Rollenspiel im Sinn eines soziometrischen Experiments (Moreno, 1996, 2007), die Methode der kritischen Exploration (Piaget in Inhelder, Sinclair & Bovet, 1974) und den Schulabakus als Darstellungssystem (Johann, 2002; Johann & Matros, 2003). Diese nonkonformistischen Methoden sind notwendig, um die Orientierung am Verstehen in der mathematischen Bildung für alle zu radikalisieren. Schlüsselwörter: Arithmetik; das dezimale Stellenwertsystem; Abakus; Darstellungsmittel; Rollenspiel. // The paper describes how it is possible to develop logical-mathematical thinking and operational skills regarding the positional decimal numeral system in a creative, operational and illustrative way. Theoretically the paper is close to the construct of «cognitive acceleration» (Adey, 2008) that connects in a “school of thought” the genetic theory of cognitive development (Piaget, 1977a, 1977b; Piaget & Voelin, 1980), the current curriculum (Lehrplan21), the proximal development zone (Vygotskij, 1986) and metacognition. Moreover, the article integrates the role play in the sense of sociometric experiment (Moreno, 1996; 2007), the method of critical exploration (Piaget, in Inhelder, Sinclair & Bovet, 1974) and the abacus as an operational tool of representation of the positional decimal numeral system (Johann, 2002; Johann & Matros, 2003). These methods can guide the understanding process during the learning of mathematics. Keywords: arithmetic; positional decimal numeral system; abacus; tool of representation; role play.
... Secondo Reiss e Schmieder (2005) è un sistema semplice, ma allo stesso tempo anche impegnativo; in effetti giungere alla comprensione del nostro sistema di numerazione è un compito complesso che richiede molto tempo. I risultati della ricerca degli ultimi decenni confermano che la concezione di numerosità, normalmente acquisita da alunni e alunne tra il secondo e il quinto anno della scuola elementare, in alcuni casi si protrae fino alla secondaria e oltre (Ross, 1986;Schuler, 2004;Brugger, Sidler & Meyer, 2007;Moser Opitz, 2007;Ruflin, 2008;Herzog, Fritz & Ehlert, 2017;Herzog, Ehlert & Fritz, 2019; vedi Tabella 1). Dai dati delle ricerche cliniche relative allo sviluppo psicologico del pensiero è difficile dedurre informazioni sullo sviluppo dei livelli cognitivi raggiunti dagli allievi, perché di solito le ricerche sono effettuate in ambiti creati ad hoc. ...
... Psicologia dello sviluppo Ciclo Età (a.) Numero e variabili Livello (Ross, 1986;Herzog et al., 2019) Teoria del morfismo (Piaget et al., 1990) 1 da 4 a 8 anni MA.1.A1 Operare e nominare c) capire e usare i concetti: per, maggiore di, uguale a, minore di, pari, dispari, completare, dimezzare, raddoppiare, decine, unità e i simboli, <, >. Conoscere i numeri naturali fino a 100, leggerli e scriverli. ...
Article
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L’articolo descrive come è possibile sviluppare il pensiero logico-matematico e le competenze operative concernenti il sistema di numerazione decimale posizionale in modo creativo, operativo ed esemplificativo. Teoricamente l’articolo è vicino al costrutto di «accelerazione cognitiva» (Adey, 2008) che collega in una “scuola del pensiero” la teoria genetica dello sviluppo cognitivo (Piaget, 1977a, 1977b; Piaget & Voelin, 1980), il Piano di studio in vigore (Lehrplan21), la zona di sviluppo prossimale (Vygotskij, 1986) e la metacognizione. Inoltre, l’articolo integra il gioco di ruolo nel senso di esperimento sociometrico (Moreno, 1996; 2007), il metodo dell’esplorazione critica (Piaget, in Inhelder, Sinclair & Bovet, 1974) e l’abaco scolastico come mezzo operativo di rappresentazione del sistema di numerazione decimale posizionale (Johann, 2002; Johann & Matros, 2003). Questi metodi possono orientare il processo di comprensione durante l’apprendimento della matematica.
... Moreover, children begin comprehending basic arithmetic operations with tangible objects (Barth et al., 2005). Recent studies accentuate the capacity of preschoolers to perceive object sets as units (bundling), albeit with counting challenges, fostering multiplicative thinking and a preliminary grasp of the place value system (Herzog et al., 2019;Van de Walle et al., 2013;Wege et al., 2023). Another critical skill is understanding the concept of zero (Krajcsi et al., 2021;Lee & Md-Yunus, 2016). ...
Conference Paper
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This study explores the potential of Gebeta, a traditional Ethiopian board game, as a pedagogical tool to foster early number sense and mathematical engagement among Norwegian preschool children. Leveraging Bishop's six core mathematical activities as a theoretical framework, we conducted an observational case study involving four children from two kindergartens in Norway. Through video analysis, we scrutinized children's verbal and non-verbal interactions while engaging with the game, both according to the game's prescribed rules and in free play. Our findings show a prominent engagement in counting and locating activities, with notable instances of one-to-one correspondence, bundling, and conceptual subitizing. The study offers promising insights into play-based mathematical learning and highlights the Gebeta game's rich affordances in fostering mathematical exploration and creativity among young learners, suggesting a promising avenue for further research in early childhood mathematics education.
... Place value understanding has been identified as a good predictor of mathematics performances as well as of mathematical difficulties [for example, see [6]]. Research shows that especially low achievers, even in higher grades, have great difficulties with this mathematical topic [7,8]. ...
Chapter
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Focusing on diagnosis and support of basic mathematical competencies is not only important for learning mathematics in general but also especially relevant for students with mathematical learning difficulties. Fostering basic mathematical competencies, like understanding numbers and operations as well as place value, is a central objective of the project “MaCo” (Catching up in Mathematics after Covid), which will be presented in this chapter. The project itself addresses both, teachers and students, and, not least, tries to meet central challenges after the Covid pandemic. The developmental research and research activities in this project were guided by a design-based research approach. Exemplary for the project, developed concepts and materials for understanding place value in inclusive settings on primary level are illustrated, completed by first evaluations. On the one hand, these concepts and materials comprise concrete activities and tasks, explanatory videos, and web applications designed for students. On the other hand, the professionalization of teachers and facilitators is pursued by massive open online courses, materials for professional development programs, and accompanying didactic manuals for students’ materials.
... Place value is a fundamental building block of primary mathematics education. The place value system relates to arithmetic capabilities (Herzog et al., 2019). Experts believe that the understanding of the place value system predicts math performance and math difficulties. ...
... Addressing these gaps, Herzog and Fritz-Stratmann link their recently proposed hierarchical model of place-value learning (Herzog et al., 2019) to number transcoding (i.e., writing numbers to dictation or reading digital-Arabic numbers aloud)-a commonly employed task used to index place-value understanding. The authors found that transcoding may indeed be a valid index for place-value understanding because 2nd and 3rd graders demonstrating more advanced levels of place-value understanding also performed better in writing Arabic numbers to dictation, especially for syntactically more complex numbers (e.g., including zeros). ...
... In ähnlicher Weise unterstützt die Relationalität die Entwicklung tragfähiger Stellenwertvorstellungen. Der "Zehner" als wesentlicher Bestandteil früher Stellenwertvorstellungen wird erst durch seine regelmäßige Wiederholbarkeit zur Bündeleinheit (Cobb & Wheatley, 1988;Herzog, Ehlert & Fritz, 2019). ...
Article
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Zusammenfassung. Hintergrund: Die Subjektwissenschaftliche Lerntheorie betrachtet Lernprozesse vom Standpunkt des Subjektes. Dabei wird ein besonderer Fokus auf Begründungsstrukturen für gelingende oder nicht gelingende Lernhandlungen gelegt. Die subjektwissenschaftliche Perspektive scheint gerade für die Betrachtung von Lernprozessen in der Alphabetisierung und Grundbildung gewinnbringend, da bisherige (negative) Lernerfahrungen, unterschiedlich zusammengesetzte Kompetenzen sowie spezifische Bildungsziele (wie z.B. gesellschaftliche Teilhabe) auf besondere Weise das Lernen beeinflussen können. Grundbildungszentren (GBZen) können als regionale Anlaufstellen für unterschiedliche Fragestellungen bezüglich der Grundbildung und Alphabetisierung Erwachsener Lernenden gelten. Dabei halten GBZen sowohl niedrigschwellige Lernangebote für Menschen mit Alphabetisierungs- und Grundbildungsbedarfen vor, als auch regionale Netzwerk- und Kooperationsstrukturen, um das Umfeld der Lernenden zu sensibilisieren. Methoden: Anhand von zwölf Interviews und vier Selbstberichten, die im Rahmen einer Evaluationsstudie generiert wurden, wurde folgende Fragestellung sekundäranalytisch beantwortet: Welches Potential haben die GBZen Lernprozesse subjektorientiert zu gestalten? Ergebnisse und Diskussion: Die Ergebnisdarstellung erfolgt anhand einer institutionellen, interaktionellen sowie individuellen Ebene. Die Ergebnisse zeigen, dass die Struktur der GBZen eine wesentliche Vorbedingung darstellt, Lernenden subjektorientiert zu unterstützen, sich das Potenzial in vollem Umfang aber nur durch das Zusammenwirken aller drei Ebenen entfalten kann.
... But, their study finds no support in the majority of studies that hold this view (Moeller, Pixner, Zuber, Kaufmann & Nuerk, 2011;MacDonald, Westenskow, Moyer-Packenham, & Child, 2018). Based on these results, it can be concluded that understanding place value is an important milestone in learning arithmetic and mathematics in general (Herzog, Ehlert & Fritz, 2019). ...
Preprint
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Children’s learning of place value is critical to their future study of mathematics. However, its learning presents a huge challenge for many learners. As result they develop misconceptions and thencommit errors that become an encumbrance to their understanding of place value. It is therefore important to identify these misconceptions and their associated errors early and help children to overcome them. This article reports the findings from literature on what misconceptions learners display in the learning of place value and how they can best be supported to overcome them. The results show that despite the importance of this topic there are limited studies devoted to its research. The dearth of these studies creates a gap in the study of place value.
... The high interest in learning mathematics students will support increased student learning achievement. The same thing was expressed by Herzog, Ehlert, & Fritz (2019) that interest has a strong positive relationship with success related to mathematics. The correlation between interests and achievements is quite high because the more people who are attracted to an object the more knowledge they obtain. ...
Article
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Student interest in learning is a very important factor in determining student success in learning mathematics. Various attempts were made by educators and educational researchers to increase student interest in learning. This research is a classroom action research model by Kemmis and Mc Taggart that aims to describe the application of Guided Discovery learning in optimizing students' interest in learning mathematics. The increase in students' interest in learning mathematics is also supported by the results of student achievement. The research data consisted of students' interest in learning mathematics, learning achievement data, and observations of learning outcomes. Data on learning interest in mathematics is obtained through a questionnaire, data on learning achievement is obtained through tests and data on the results of observations of learning achievement are obtained through observation sheets during learning. In general, the results of the study showed that the average student interest in learning mathematics at 83.93 reached the good category. The completeness of student achievement test results reached 83.87% of students achieving the minimum completeness criteria with an average student score of 85.61. The percentage of teacher and student learning outcomes respectively at 83.80% and 76.91% reached the good category. Therefore it can be concluded that the Guided Discovery learning model can be applied to optimize students' interest in mathematics learning especially by paying attention to the results of reflections from this study.
... Einigkeit bei allen Erklärungsmodellen besteht jedoch darüber, wie sich Schwierigkeiten beim Mathematiklernen äußern: im Vergleich zu ihren Peers stark unterdurchschnittlichen Mathematikleistungen in spezifischen Bereichen.Betroffene Schülerinnen und Schüler haben zentrale Inhalte der Grundschulmathematik -den sogenannten mathematischen Basisstoff -nicht oder nur unzureichend erworben(Freesemann, 2014, S. 31) . Probleme zeigen sich insbesondere beim verbalen Zählen(Desoete, Ceulemans, Roeyers, & Huylebroeck, 2009;Moser Opitz, 2013a) , beim Verständnis des dezimalen Stellenwertsystems(Herzog, Ehlert, & Fritz, 2019; Vukovic & Siegel, 2010) und beim Bearbeiten von Text-und Sachaufgaben(Kingsdorf & Krawec, 2014;Zhang & Xin, 2012) . Zudem gelingt es Schülerinnen und Schülern mit Schwierigkeiten beim Mathematiklernen nur in eingeschränktem Maße, Aufgaben des Einspluseins oder des kleinen Einmaleins zu automatisieren; dies wird vor allem auf Probleme mit dem Arbeitsgedächtnis zurückgeführt (Geary, Hoard, & Bailey, 2012) . ...
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Inklusiver Mathematikunterricht muss sich einerseits der Herausforderung stellen, gemeinsame Lernsituationen für Schülerinnen und Schüler mit sehr unterschiedlichen Lernvoraussetzungen anzubieten, um Separation möglichst zu vermeiden·und sozial- interaktive Lernprozesse zu ermöglichen. Andererseits muss die gezielte Förderung aller Lernenden gewährleistet sein; das heißt, es müssen auch spezifische Maßnahmen ergriffen werden, um Schülerinnen und Schüler mit mathematischen Lernschwierigkeiten gezielt zu unterstützen. Der Beitrag macht auf der Grundlage empirischer Studien und konzeptio- neller Überlegungen Vorschläge, wie beides gleichermaßen gelingen kann. Dabei werden sonderpädagogische und mathematikdidaktische Konzepte zu fünf Leitideen zusammen- geführt und anhand eines Unterrichtsbeispiels konkretisiert.
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The challenges inherent in assessing mathematical proficiency depend on a number of factors, amongst which are an explicit view of what constitutes mathematical proficiency, an understanding of how children learn and the purpose and function of teaching. All of these factors impact on the choice of approach to assessment. In this article we distinguish between two broad types of assessment, classroom-based and systemic assessment. We argue that the process of assessment informed by Rasch measurement theory (RMT) can potentially support the demands of both classroom-based and systemic assessment, particularly if a developmental approach to learning is adopted, and an underlying model of developing mathematical proficiency is explicit in the assessment instruments and their supporting material. An example of a mathematics instrument and its analysis which illustrates this approach, is presented. We note that the role of assessment in the 21st century is potentially powerful. This influential role can only be justified if the assessments are of high quality and can be selected to match suitable moments in learning progress and the teaching process. Users of assessment data must have sufficient knowledge and insight to interpret the resulting numbers validly, and have sufficient discernment to make considered educational inferences from the data for teaching and learning responses.
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The Welsh language uses a regular counting system, whereas English uses an irregular counting system, and schools within Wales teach either through the medium of Welsh or English. This provides the opportunity to compare linguistic effects on arithmetical skills in the absence of many other confounding factors that arise in international comparisons. This study investigated the hypothesis that language properties influence children's performance in certain numerical tasks by comparing the performance of 20 Welsh- and 20 English-medium Year Two pupils in non-verbal line estimations and transcoding. Groups did not differ on global arithmetic abilities, but the pupils taught through the medium of Welsh on average performed better in the non-verbal line estimation tasks than the English-medium group. This superiority was most apparent in comparisons involving numbers over 20: a result which was complicated by the fact that Welsh-medium pupils showed a lower range of error scores than the English-medium pupils. These results were thought to be related to the increased transparency of the Welsh counting system.
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The relative linguistic transparency of the Asian counting system has been used to explain Asian students’ relative superiority in cross-cultural comparisons of mathematics achievement. To test the validity and extent of linguistic transparency in accounting for mathematical abilities, this study tested Chinese and British primary school children. Children in Hong Kong can learn mathematics using languages with both regular (Chinese) and irregular (English) counting systems, depending on their schools’ medium of instruction. This makes it possible to compare groups with varying levels of exposure to the regular and irregular number systems within the same educational system, curriculum, and cultural environment. The study included three groups of first/second graders and third/fourth graders with varying degrees of experience to the Chinese language and counting systems: no experience (UK; n = 49); spoke Chinese at home and learnt to count in English at school (HK-E; n = 43); spoke Chinese at home and learnt to count in Chinese at school (HK-C; n = 47). They were compared on counting, numerical abilities and place value representation. The present study also measured nonverbal reasoning, attitude toward mathematics, involvement of parents, and extra-curricular mathematics lessons to explore alternative explanations of children’s numeric ability. Results indicated that students in HK-C were better at counting backward and on the numeric skills test than those in HK-E, who were in turn better than the UK students. However, there was no statistical difference in counting forward, place value understanding, and a measure of arithmetic. Our findings add to existent literature suggesting that linguistic transparency does not have an all-pervasive influence on cross-national differences in arithmetic performance.
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The challenges inherent in assessing mathematical proficiency depend on a number of factors, among which are an explicit view of what constitutes mathematical proficiency, an understanding of how children learn and the purpose and function of teaching. All of these factors impact on an approach to assessment. In this article we distinguish between two broad types of assessment, classroom-based and systemic assessment. We argue that the process of assessment informed by Rasch measurement theory (RMT) can potentially support the demands of both classroom-based and systemic assessment, particularly if a developmental approach to learning is adopted, and an underlying model of developing mathematical proficiency is explicit in the assessment instruments and their supporting material. An example of a mathematics instrument and its analysis which illustrates this approach, is presented. We note that the role of assessment in the 21 st century is potentially powerful. This influential role can only be justified if the assessments are of high quality and can be selected to match suitable moments in learning progress and the teaching process. Users of assessment data must have sufficient knowledge and insight to interpret the resulting numbers validly, and have sufficient discernment to make considered educational inferences from the data for teaching and learning responses. Introduction Assessing mathematical proficiency is a complex task. The challenges inherent in this process
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The mission of German special schools is to enhance the education of students with Special Educational Needs in the area of Learning (SEN-L). However, recent studies indicate that graduate students with SEN-L from special schools show difficulties in basic arithmetical operations, and the development of basic mathematical skills during secondary special school is not warranted. This study presents a newly developed test of basic arithmetical skills, based on already established tests. The test examines the arithmetical skills of students with SEN-L from fifth to ninth grade. The sample consisted of 110 students from three special schools in Munich. Testing took place in January and June 2013. The test shows to be an effective tool that reliably and precisely assesses students’ performance across different grades. The test items can be used without creating floor and ceiling effects among fifth to ninth grade students with SEN-L. The items’ conformity to the dichotomous Rasch model is demonstrated. The students’ skills turn out to be very heterogeneous, both overall and within grades. Many of the students do not even master basic arithmetical skills that are taught in primary school, although achievement improves in higher grades.
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A common theme of models of conceptual growth is to establish the hierarchical structures of abilities that can be interpreted along developmental lines. Integrating the literature on the development of mathematical concepts and skills in children, a comprehensive 6 level model for describing, explaining and predicting the development of key numerical concepts and arithmetic skills from age 4 to 8, is proposed. Two studies will be presented. In the first study, 1095 preschool children completed a mathematics test (MARKO-D0) based on a 5-level model. The test fitted with a one-dimensional Rasch model. The extension of the model to a sixth level was verified in a new study: 312 first-graders took part in a mathematics test based on the six levels (MARKO-D1). In order to check whether the data of both samples adhered to the principle of unidimensionality, the data of MARKO-0 and MARKO-1 were used in a common analysis for comparative purposes. The applicability of these findings for a qualitative diagnostics and an adaptive training will be discussed.
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We compared the cognitive representation of number of American, Chinese, Japanese, and Korean first graders, and Korean kindergartners, to determine if there might be variations in those representations resulting from numerical language characteristics that differentiate Asian and non-Asian language groups. Children were asked to construct various numbers using base 10 blocks. Chinese, Japanese, and Korean children preferred to use a construction of tens and ones to show numbers; place value appeared to be an integral component of their representations. In contrast, English-speaking children preferred to use a collection of units, suggesting that they represent number as a grouping of counted objects. More Asian children than American children were able to construct each number in 2 ways, which suggests greater flexibility of mental number manipulation.
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Ways in which children think of 10 are considered first. Then a study with 14 second graders is reported; students were placed at three levels with respect to their addition and subtraction concepts. Findings are detailed, along with implications for instruction. (MNS)
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The authors propose that conceptual and procedural knowledge develop in an iterative fashion and that improved problem representation is 1 mechanism underlying the relations between them. Two experiments were conducted with 5th- and 6th-grade students learning about decimal fractions. In Experiment 1, children's initial conceptual knowledge predicted gains in procedural knowledge, and gains in procedural knowledge predicted improvements in conceptual knowledge. Correct problem representations mediated the relation between initial conceptual knowledge and improved procedural knowledge. In Experiment 2, amount of support for correct problem representation was experimentally manipulated, and the manipulations led to gains in procedural knowledge. Thus, conceptual and procedural knowledge develop iteratively, and improved problem representation is 1 mechanism in this process. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
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This book provides a comprehensive overview of numerical cognition by bringing together writing by leading researchers in psychology, neuroscience, and education, covering work using different methodological approaches in humans and animals. During the last decade there had been an explosion of studies and new findings with theoretical and translational implications. This progress has been made thanks to technological advances enabling sophisticated human neuroimaging techniques and neurophysiological studies of monkeys, and to advances in more traditional psychological and educational research. This has resulted in an enormous advance in our understanding of the neural and cognitive mechanisms of numerical cognition. In addition, there has recently been increasing interest and concern about pupils' mathematical achievement, resulting in attempts to use research to guide mathematics instruction in schools, and to develop interventions for children with mathematical difficulties. This book aims to provide a broad and extensive review of the field of numerical cognition, bringing together work from varied areas. The book covers research on important aspects of numerical cognition, involving findings from the areas of developmental psychology, cognitive psychology, human and animal neuroscience, computational modeling, neuropsychology and rehabilitation, learning disabilities education and individual differences, cross-cultural and cross-linguistic studies, and philosophy. It also includes an overview 'navigator' chapter for each section to provide a brief up-to-date review of the current literature, and to introduce and integrate the topics of the chapters in the section.
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This book provides a comprehensive overview of numerical cognition by bringing together writing by leading researchers in psychology, neuroscience, and education, covering work using different methodological approaches in humans and animals. During the last decade there had been an explosion of studies and new findings with theoretical and translational implications. This progress has been made thanks to technological advances enabling sophisticated human neuroimaging techniques and neurophysiological studies of monkeys, and to advances in more traditional psychological and educational research. This has resulted in an enormous advance in our understanding of the neural and cognitive mechanisms of numerical cognition. In addition, there has recently been increasing interest and concern about pupils' mathematical achievement, resulting in attempts to use research to guide mathematics instruction in schools, and to develop interventions for children with mathematical difficulties. This book aims to provide a broad and extensive review of the field of numerical cognition, bringing together work from varied areas. The book covers research on important aspects of numerical cognition, involving findings from the areas of developmental psychology, cognitive psychology, human and animal neuroscience, computational modeling, neuropsychology and rehabilitation, learning disabilities education and individual differences, cross-cultural and cross-linguistic studies, and philosophy. It also includes an overview 'navigator' chapter for each section to provide a brief up-to-date review of the current literature, and to introduce and integrate the topics of the chapters in the section.
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Wie begegnet man Schwächen im Rechnen gezielt? Haben Kinder gute mathematische Vorkenntnisse im Vorschulalter erworben, sind meist gute mathematische Leistungen in der Grundschule zu erwarten. Geringe Vorkenntnisse hingegen werden selten kompensiert. Warum fällt vielen Kindern das Rechnen schwer? Wann muss man von einer Dyskalkulie sprechen? Die Autorinnen erklären den Prozess des Rechnenlernens, machen den Leser mit individuellen Strategien von Kindern vertraut und stellen Fördermaßnahmen für effektives Rechnen vor. Dieser Titel ist auf verschiedenen e-Book-Plattformen (Amazon, Libreka, Libri) auch als e-Pub-Version für mobile Lesegeräte verfügbar.
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To assess children’s individual stages of mathematical development, as well as their developmental trajectories, different diagnostic approaches have been developed. Based on the test results of such approaches, children with mathematics learning difficulties are identified and programs to enhance mathematical learning can be developed and evaluated. In this chapter, quality characteristics of assessment are described (reliability, objectivity, validity, and provision of norms). Different categories are presented to classify approaches to assessing mathematical competence and performance (norm-referenced versus not-norm-referenced tests, individual versus group testing, paper-and-pencil tests versus interviews versus computer-based tests, chronological versus educational age–oriented tests, speed versus power tests, and principles of task selection). Drawing on selected approaches, different principles of task selection are discussed (curriculum-based, based on neuropsychology theories, or based on developmental psychology theories) together with their consequences for the interpretation of respective test results. Finally, some promising research trends are outlined.
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In this chapter, we highlight the importance not only of an understanding of place value, but the importance of a flexible understanding. We describe the principles of our decimal place value system and the development processes of children. Embedded in artefact-centric activity theory, we present an education-oriented design of a virtual place value chart and its potential to support this development and understanding. We also present results of a qualitative study with second graders as well as results of a quantitative study with third graders that can guide further research in that area.
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When a student mentally computes 32 + 59 by thinking 30 + 50 is 80, 9 + 2 is 11, and 80 + 11 is 91, that student has had to call on some well-developed concepts of numerical partwhole relationships and place value. People with good number sense make frequent and flexible use of these two related concepts to perform mental computations and numeric estimates. Students find these concepts difficult; their understanding develops slowly over a period of several years. To be successful at teaching number sense, we must design instruction that respects students' need to construct their own knowledge.
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Researchers from 4 projects with a problem-solving approach to teaching and learning multidigit number concepts and operations describe (a) a common framework of conceptual structures children construct for multidigit numbers and (b) categories of methods children devise for multidigit addition and subtraction. For each of the quantitative conceptual structures for 2-digit numbers, a somewhat different triad of relations is established between the number words, written 2-digit marks, and quantities. The conceptions are unitary, decade and ones, sequence-tens and ones, separate-tens and ones, and integrated sequence-separate conceptions. Conceptual supports used within each of the 4 projects are described and linked to multidigit addition and subtraction methods used by project children. Typical errors that may arise with each method are identified. We identify as crucial across all projects sustained opportunities for children to (a) construct triad conceptual structures that relate ten-structured quantities to number words and written 2-digit numerals and (b) use these triads in solving multidigit addition and subtraction situations.
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Children's counting strategies reflect how much they understand the place-value structure of numbers. In Study 1, a novel task, namely the strategic counting task, elicited strategies from kindergarteners and first graders that showed a trend of increasing place-value knowledge – from perceiving number as an undivided entity to seeing it as a collection of independent groups of powers of ten. In Study 2, first-graders' strategic counting task scores at the end of fall semester were better predictors of year-end mathematical achievement than the traditional place-value tasks. In Study 3, a five-item subset of strategic counting was the best among 15 various cognitive predictors of end of second-grade mathematical achievement. Growth curve modeling revealed that low-mathematics achievers at the end of second grade had been lagging behind their peers in strategic counting since early first grade. Implications for early support for children with difficulties in place-value knowledge are discussed.
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Place value notation is essential to mathematics learning. This study examined young children's (4- to 6-year-olds, N = 172) understanding of place value prior to explicit schooling by asking them write spoken numbers (e.g., "six hundred and forty-two"). Children's attempts often consisted of "expansions" in which the proper digits were written in order but with 0s or other insertions marking place (e.g., "600402" or "610042"). This partial knowledge increased with age. Gender differences were also observed with older boys more likely than older girls to produce the conventional form (e.g., 642). Potential experiences contributing to expanded number writing and the observed gender differences are discussed.
Article
Arithmetic difficulties have long captured the attention of teachers and researchers, but intervention programs for assisting children are seldom successful for all. Recent Australian research suggests that this is because we have failed to recognise the complexity of arithmetic difficulties. Analysis of some 30,000 one on- one clinical interviews conducted over three years during the Early Numeracy Research Project provided rich data for charting the pathway of young children’s number learning in four domains (Counting, Place Value, Addition and Subtraction Strategies, and Multiplication and Division Strategies), and for identifying children who were having difficulty. We describe such children as being vulnerable or at risk of not being able to take advantage of everyday classroom experiences. The data show that the combinations of domains in which children were vulnerable were diverse, and suggest that there is no single ‘formula’ for describing children who are vulnerable in number learning, or for describing the instructional needs of students. Indeed, children have learning needs that call for teachers to make individual decisions about the instructional approach for each child. Further, the diversity of children’s mathematical knowledge in the four domains suggests that knowledge in any one domain is not necessarily prerequisite for knowledge construction in another domain. This finding has implications for both intervention programs and for the way in which school mathematics is introduced to children. It seems likely that children may benefit from concurrent learning opportunities in all number domains, and that experiences in one domain should not be delayed until a level of mathematical knowledge is constructed in another domain.
Article
Recent research has shown that place value remains difficult in third and fourth grade, in spite of the fact that it is taught repeatedly in every primary grade. This study was conducted to understand the cognitive processes underlying this difficulty. A counting task was devised, based on Piaget's theory of number, to find out if children in grades 1–5 are constructing a system of tens on a system of ones. Only some children in grades 2–5 evidenced this construction. The implications of the findings for place value instruction are discussed, with observations from second grade classrooms in which children are encouraged to invent their own ways of doing double-column addition.
Article
This paper aims to highlight the significance of a particular aspect of magnitude processing, namely counting and subitizing or the rapid enumeration of small sets of items, for learning. Emphasis is laid on the historical roots and the conceptual framework as well as on studies on pre-verbal and school-age children. Evidence of the potential value of this research for the assessment of children at risk of mathematical learning disabilities, is presented. Inherent to its nature, subitizing relies on rapid, preverbal analogue magnitude comparisons being triggered. We will highlight the differences with counting, and the implications of shortcomings in counting and subitizing in children with mathematical learning disabilities for the automaticity of number magnitude processing. Furthermore we especially look in this paper at the varying assessment paradigms which are used in research with different age groups, something which has received insufficient attention in the past. Finally, we outline the challenges for future research on mathematical learning disabilities.
Article
A learning/teaching approach used base-ten blocks to embody the English named-value system of number words and digit cards to embody the positional base-ten system of numeration. Steps in addition and subtraction of four-digit numbers were motivated by the size of the blocks and then were carried out with the blocks; each step was immediately recorded with base-ten numerals. Children practiced multidigit problems of from five to eight places after they could successfully add or subtract smaller problems without using the blocks. In Study 1 six of the eight classes of first and second graders (N=169) demonstrated meaningful multidigit addition and place-value concepts up to at least four-digit numbers; average-achieving first graders showed more limited understanding. Three classes of second graders (N=75) completed the initial subtraction learning and demonstrated meaningful subtraction concepts. In Study 2 most second graders in 42 participating classes (N=783) in a large urban school district learned at least four-digit addition, and many children in the 35 classes (N=707) completing subtraction work learned at least four-digit subtraction.
Article
Differences in mathematical competence between U S and Chinese children first emerge during the preschool years, favor Chinese children, and are limited to specific aspects of mathematical competence The base-10 structure of number names is less obvious in English than in Chinese, differences between these languages are reflected in children's difficulties learning to count Language differences do not affect other aspects of early mathematics, including counting small sets and solving simple numerical problems Because later mathematics increasingly involves manipulation of symbols, this early deficit in apprehending the base-10 structure of number names may provide a basis for previously reported differences in mathematical competence favoring Chinese schoolchildren
Article
Comparing numerical performance between different languages does not only mean comparing different number-word systems, but also implies a comparison of differences regarding culture or educational systems. The Czech language provides the remarkable opportunity to disentangle this confound as there exist two different number-word systems within the same language: for instance, "25" can be either coded in non-inverted order "dvadsetpät" [twenty-five] or in inverted order "pätadvadset" [five-and-twenty]. To investigate the influence of the number-word system on basic numerical processing within one culture, 7-year-old Czech-speaking children had to perform a transcoding task (i.e., writing Arabic numbers to dictation) in both number-word systems. The observed error pattern clearly indicated that the structure of the number-word system determined transcoding performance reliably: In the inverted number-word system about half of all errors were inversion-related. In contrast, hardly any inversion-related errors occurred in the non-inverted number-word system. We conclude that the development of numerical cognition does not only depend on cultural or educational differences, but is indeed related to the structure and transparency of a given number-word system.
Article
It is assumed that basic numerical competencies are important building blocks for more complex arithmetic skills. The current study aimed at evaluating this interrelation in a longitudinal approach. It was investigated whether first graders' performance in basic numerical tasks in general as well as specific processes involved (e.g., place-value understanding) reliably predicted performance in an addition task in third grade. The results indicated that early place-value understanding was a reliable predictor for specific aspects of arithmetic performance. Implications of the role of basic numerical competencies for the acquisition of complex arithmetic are discussed.
Article
This study examined how four domain-specific skills (arithmetic procedural skills, number fact retrieval, place value concept, and number sense) and two domain-general processing skills (working memory and processing speed) may account for Chinese children's mathematics learning difficulties. Children with mathematics difficulties (MD) of two age groups (7-8 and 9-11 years) were compared with age-matched typically achieving children. For both age groups, children with MD performed significantly worse than their age-matched controls on all of the domain-specific and domain-general measures. Further analyses revealed that the MD children with literacy difficulties (MD/RD group) performed the worst on all of the measures, whereas the MD-only group was significantly outperformed by the controls on the four domain-specific measures and verbal working memory. Stepwise discriminant analyses showed that both number fact retrieval and place value concept were significant factors differentiating the MD and non-MD children. To conclude, deficits in domain-specific skills, especially those of number fact retrieval and place value understanding, characterize the profile of Chinese children with MD.
Article
This paper provides a tutorial introduction to numerical cognition, with a review of essential findings and current points of debate. A tacit hypothesis in cognitive arithmetic is that numerical abilities derive from human linguistic competence. One aim of this special issue is to confront this hypothesis with current knowledge of number representations in animals, infants, normal and gifted adults, and brain-lesioned patients. First, the historical evolution of number notations is presented, together with the mental processes for calculating and transcoding from one notation to another. While these domains are well described by formal symbol-processing models, this paper argues that such is not the case for two other domains of numerical competence: quantification and approximation. The evidence for counting, subitizing and numerosity estimation in infants, children, adults and animals is critically examined. Data are also presented which suggest a specialization for processing approximate numerical quantities in animals and humans. A synthesis of these findings is proposed in the form of a triple-code model, which assumes that numbers are mentally manipulated in an arabic, verbal or analogical magnitude code depending on the requested mental operation. Only the analogical magnitude representation seems available to animals and preverbal infants.
Article
The present study examined whether children's variation in arithmetic performance was related to differences in their understanding of place-value. Training in place-value concepts was provided to a group of Chinese children who were poor in arithmetic. Their performance before and after the training was compared to that of the children in two control groups. The results showed that there were reliable connections between place-value understanding and addition and subtraction skills. Furthermore, training in place-value concepts was found to be effective in enhancing the children's place-value understanding and addition skills. Implications for instructions in arithmetic were discussed. Copyright 1997Academic Press
Article
Transcoding Arabic numbers from and into verbal number words is one of the most basic number processing tasks commonly used to index the verbal representation of numbers. The inversion property, which is an important feature of some number word systems (e.g., German einundzwanzig [one and twenty]), might represent a major difficulty in transcoding and a challenge to current transcoding models. The mastery of inversion, and of transcoding in general, might be related to nonnumerical factors such as working memory resources given that different elements and their sequence need to be memorized and manipulated. In this study, transcoding skills and different working memory components in Austrian (German-speaking) 7-year-olds were assessed. We observed that inversion poses a major problem in transcoding for German-speaking children. In addition, different components of working memory skills were differentially correlated with particular transcoding error types. We discuss how current transcoding models could account for these results and how they might need to be adapted to accommodate inversion properties and their relation to different working memory components.
The Routledge international handbook of dyscalculia and mathematical learning difficulties
  • A Dowker
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Dowker, A., & Morris, P. (2015). Interventions for children with difficulties in learning mathematics. In S. Chinn (Ed.), The Routledge international handbook of dyscalculia and mathematical learning difficulties (pp. 256-264). London: Routledge.
Proceedings: The development of mathematical understanding
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Hart, K. (2009). Why do we expect so much? In J. Novotná, & H. Moraova (Eds.), SEMT 2009. International symposium elementary maths teaching. August 23-28, 2009. Proceedings: The development of mathematical understanding (pp. 24-31). Prague, Czech Republic: Charles University.
The development of place value concepts across primary school
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Herzog, M., Ehlert, A. & Fritz, A. (in prep). The development of place value concepts across primary school.
Tools and processes in mathematics teacher education
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Nührenbörger, M., & Steinbring, H. (2008). Manipulatives as tools in teacher education. In D. Tirosh & T. Wood (Eds.), The international handbook of mathematics teacher education Vol. 2. Tools and processes in mathematics teacher education (pp. 157-182). Rotterdam, The Netherlands: Sense Publishers.
The Oxford handbook of mathematical cognition
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Okamoto, Y. (2015). Mathematics learning in the USA and Japan. In R. C. Kadosh & A. Dowker (Eds.), The Oxford handbook of mathematical cognition (pp. 415-429). Oxford: 2 Medicine UK.
Why do we expect so much
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