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  • 1.
    Andersson, Ulf
    et al.
    Linköping University, Department of Behavioural Sciences and Learning, Psychology. Linköping University, Faculty of Arts and Sciences.
    Skagerlund, Kenny
    Linköping University, Department of Behavioural Sciences and Learning, Psychology. Linköping University, Faculty of Arts and Sciences.
    Olsson, Linda
    Linköping University, Department of Behavioural Sciences and Learning, Psychology. Linköping University, Faculty of Arts and Sciences.
    Östergren, Rickard
    Linköping University, Department of Behavioural Sciences and Learning, Education, Teaching and Learning. Linköping University, Faculty of Educational Sciences.
    Pathways to arithmetic fact retrieval and percentage calculation in adolescents2017In: British Journal of Educational Psychology, ISSN 0007-0998, E-ISSN 2044-8279, Vol. 87, no 4, p. 647-663Article in journal (Refereed)
    Abstract [en]

    Background

    Developing sufficient mathematical skills is a prerequisite to function adequately in society today. Given this, an important task is to increase our understanding regarding the cognitive mechanisms underlying young people's acquisition of early number skills and formal mathematical knowledge.

    Aims

    The purpose was to examine whether the pathways to mathematics model provides a valid account of the cognitive mechanisms underlying symbolic-number processing and mathematics in adolescents. The pathways model states that the three pathways should provide independent support to symbolic-number skill. Each pathway's unique contribution to formal mathematics varies depending on the complexity and demand of the tasks.

    Sample

    The study used a sample of 114 adolescents (71 girls). Their mean age was 14.60 years (SD = 1.00).

    Methods

    The adolescents were assessed on tests tapping the three pathways and general cognitive abilities (e.g., working memory). A structural equation path analysis was computed.

    Results

    Symbolic-number comparison was predicted by the linguistic pathway, the quantitative pathway, and processing speed. The linguistic pathway, quantitative pathways, and symbolic-number comparison predicted arithmetic fact retrieval. The linguistic pathway, working memory, visual analogies, and symbolic-number comparison predicted percentage calculation.

    Conclusions

    There are both similarities and differences in the cognitive mechanisms underlying arithmetic fact retrieval and percentage calculation in adolescents. Adolescents’ symbolic-number processing, arithmetic fact retrieval, and percentage calculation continue to rely on the linguistic pathways, whereas the reliance upon the spatial pathway has ceased. The reliance upon the quantitative pathway varies depending on the task.

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  • 2.
    Andersson, Ulf
    et al.
    Linköping University, Department of Behavioural Sciences and Learning, Psychology. Linköping University, Faculty of Arts and Sciences.
    Östergren, Rickard
    Linköping University, Department of Behavioural Sciences and Learning, Psychology. Linköping University, Faculty of Arts and Sciences.
    Number magnitude processing and basic cognitive functions in children with mathematical learning disabilities2012In: Learning and individual differences, ISSN 1041-6080, E-ISSN 1873-3425, Vol. 22, no 6, p. 701-714Article in journal (Refereed)
    Abstract [en]

    The study sought out to extend our knowledge regarding the origin ofmathematical learning disabilities (MLD) in children by testing different hypotheses in the same samples of children. Different aspects of cognitive functions and number processing were assessed in fifth- and sixth-graders (1113 years old) withMLD and compared to controls. The MLD group displayed weaknesses withmost aspects of number processing (e.g., subitizing, symbolic number comparison, number-line estimation) and two cognitive functions (e.g., visualspatial working memory). These findings favor the defective approximate number system (ANS) hypothesis, but do not fit well with the access deficit hypothesis. Support is also provided for the defective object-tracking system (OTS) hypothesis, the domain general cognitive deficit hypothesis and to some extent the defective numerosity-coding hypothesis. The study suggests that MLD might be caused by multiple deficits and not a single core deficit.

  • 3.
    Engvall, Margareta
    et al.
    Linköping University, Department of Behavioural Sciences and Learning, Education, Teaching and Learning. Linköping University, Faculty of Educational Sciences.
    Samuelsson, Joakim
    Linköping University, Department of Behavioural Sciences and Learning, Education, Teaching and Learning. Linköping University, Faculty of Educational Sciences.
    Östergren, Rickard
    Linköping University, Department of Behavioural Sciences and Learning, Education, Teaching and Learning. Linköping University, Faculty of Educational Sciences.
    THE EFFECT ON STUDENTS ARITHMETIC SKILLS OF TEACHING TWO DIFFERENTLY STRUCTURED CALCULATION METHODS2020In: Problems of Education in the 21st Century, ISSN 1822-7864, E-ISSN 2538-7111, Vol. 78, no 2, p. 167-195Article in journal (Refereed)
    Abstract [en]

    Mastering traditional algorithms has formed mathematics teaching in primary education. Educational reforms have emphasized variation and creativity in teaching and using computational strategies. These changes have recently been criticized for lack of empirical support. This research examines the effect of teaching two differently structured written calculation methods on teaching arithmetic skills (addition) in grade 2 in Sweden with respect to students procedural, conceptual and factual knowledge. A total of 390 students (188 females, 179 males, gender not indicated for 23) were included. The students attended 20 classes in grade 2 and were randomly assigned to one of two methods. During the intervention, students who were taught and had practiced traditional algorithms developed their arithmetic skills significantly more than students who worked with the decomposition method with respect to procedural knowledge and factual knowledge. These results provided no evidence that the development of students conceptual knowledge would benefit more from learning the decomposition method compared to traditional algorithm.

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  • 4.
    Lindqvist, Henrik
    et al.
    Linköping University, Department of Behavioural Sciences and Learning, Education, Teaching and Learning. Linköping University, Faculty of Educational Sciences.
    Östergren, Rickard
    Linköping University, Department of Behavioural Sciences and Learning, Education, Teaching and Learning. Linköping University, Faculty of Educational Sciences.
    Holme, Lotta
    Linköping University, Department of Behavioural Sciences and Learning, Education, Teaching and Learning. Linköping University, Faculty of Educational Sciences.
    Elevassistenter i skolan: En forskningsöversikt2020In: Pedagogisk forskning i Sverige, ISSN 1401-6788, E-ISSN 2001-3345, Vol. 25, no 2-3, p. 114-137Article in journal (Refereed)
    Abstract [sv]

    Vi har studerat forskning som handlar om elevassistenters arbete för elever i behov av särskilt stöd. Syftet med studien är att undersöka elevassistenters arbete i en allmän och friare roll som resurs i skolan och om det finns forskningsstöd för den typen av insats. Genom sökningar identifierades 16 artiklar och resultaten i artiklarna användes för att genomföra en tematisk analys. Resultaten visar att elevassistenter i en fri roll agerar som (1) Undervisare av elever i behov av särskilt stöd, (2) Experter på elever i behov av särskilt stöd, (3)Stödfunktion till lärare och (4) Upprätthållare av en särskild elevsuppmärksamhet. Temana diskuteras, i ett relationellt perspektiv, utifrån riskeroch möjligheter med elevassistenters arbete med elever i behov av särskiltstöd. En slutsats vi drar av den granskade forskningen är att det är tveksamtom elevassistenter ska användas som primära undervisare för dessa elever.

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    fulltext
  • 5.
    Lindqvist, Henrik
    et al.
    Linköping University, Department of Behavioural Sciences and Learning, Education, Teaching and Learning. Linköping University, Faculty of Educational Sciences.
    Östergren, Rickard
    Linköping University, Department of Behavioural Sciences and Learning, Education, Teaching and Learning. Linköping University, Faculty of Educational Sciences.
    Holme, Lotta
    Linköping University, Department of Behavioural Sciences and Learning, Education, Teaching and Learning. Linköping University, Faculty of Educational Sciences.
    Resurspersonal i skolan: vad visar forskningen?2020In: Venue, ISSN 2001-788X, no 18Article in journal (Other academic)
    Abstract [sv]

    I skolan ökar användningen av resurspersonal för elever i behov av särskilt stöd. ”Resurserna” innefattar främst elevassistenter, lärarassistenter och fritidshemspersonal, men även andra personalkategorier. Det gemensamma för personalgruppen är att den generellt sett har liten eller ingen utbildning för att möta elever i svårigheter. Trots det är det vanligt att resurspersonal, särskilt elevassistenter, anställs för att arbeta med elever i behov av särskilt stöd, men arbetsuppgifterna är sällan tydligt formulerade.

  • 6.
    Träff, Ulf
    et al.
    Linköping University, Department of Behavioural Sciences and Learning, Psychology. Linköping University, Faculty of Arts and Sciences.
    Olsson, Linda
    Linköping University, Department of Behavioural Sciences and Learning, Psychology. Linköping University, Faculty of Arts and Sciences.
    Skagerlund, Kenny
    Linköping University, Department of Behavioural Sciences and Learning, Psychology. Linköping University, Faculty of Arts and Sciences.
    Östergren, Rickard
    Linköping University, Department of Behavioural Sciences and Learning, Education, Teaching and Learning. Linköping University, Faculty of Educational Sciences.
    Cognitive mechanisms underlying third graders' arithmetic skills: Expanding the pathways to mathematics model.2018In: Journal of experimental child psychology (Print), ISSN 0022-0965, E-ISSN 1096-0457, Vol. 167, p. 369-387Article in journal (Refereed)
    Abstract [en]

    A modified pathways to mathematics model was used to examine the cognitive mechanisms underlying arithmetic skills in third graders. A total of 269 children were assessed on tasks tapping the four pathways and arithmetic skills. A path analysis showed that symbolic number processing was directly supported by the linguistic and approximate quantitative pathways. The direct contribution from the four pathways to arithmetic proficiency varied; the linguistic pathway supported single-digit arithmetic and word problem solving, whereas the approximate quantitative pathway supported only multi-digit calculation. The spatial processing and verbal working memory pathways supported only arithmetic word problem solving. The notion of hierarchical levels of arithmetic was supported by the results, and the different levels were supported by different constellations of pathways. However, the strongest support to the hierarchical levels of arithmetic were provided by the proximal arithmetic skills.

  • 7.
    Träff, Ulf
    et al.
    Linköping University, Department of Behavioural Sciences and Learning, Psychology. Linköping University, Faculty of Arts and Sciences.
    Olsson, Linda
    Linköping University, Department of Behavioural Sciences and Learning, Psychology. Linköping University, Faculty of Arts and Sciences.
    Östergren, Rickard
    Linköping University, Department of Behavioural Sciences and Learning, Education, Teaching and Learning. Linköping University, Faculty of Educational Sciences.
    Skagerlund, Kenny
    Linköping University, Department of Behavioural Sciences and Learning, Psychology. Linköping University, Faculty of Arts and Sciences.
    Heterogeneity of developmental dyscalculia: Cases with different deficit profiles2017In: Frontiers in Psychology, E-ISSN 1664-1078Article in journal (Refereed)
    Abstract [en]

    Context: The aim was to further understand the heterogeneity of  developmental dyscalculia (DD). Utilizing four children (8-9 year-old) performance was contrasted against predominant hypotheses of DD.

    Case report: Despite showing similar mathematical deficits, these children showed remarkable interindividual variability regarding cognitive profile and deficits. Two cases were consistent with the approximate number system deficit account, and the general magnitude-processing deficit account. One case had an access deficit in combination with a general cognitive deficit. One cases suffered from general cognitive deficits only.

    Conclusions: The results showed that DD cannot be attributed to a single explanatory factor. These findings support a multiple deficits account of DD and suggest that some cases have multiple deficits, whereas other cases have a single deficit. We discuss a previously proposed distinction between primary DD and secondary DD, and suggest hypotheses of dysfunctional neurocognitive correlates responsible for the displayed deficits.

  • 8.
    Träff, Ulf
    et al.
    Linköping University, Department of Behavioural Sciences and Learning, Psychology. Linköping University, Faculty of Arts and Sciences.
    Östergren, Rickard
    Linköping University, Department of Behavioural Sciences and Learning, Psychology. Linköping University, Faculty of Arts and Sciences.
    Carlsson, Jessica
    Linköping University, Department of Behavioural Sciences and Learning, Education, Teaching and Learning. Linköping University, Faculty of Arts and Sciences.
    Do number games improve six-year-old childrens number skills and arithmetical skills?2012In: INTERNATIONAL JOURNAL OF PSYCHOLOGY: vol 47, Special Issue: XXX International Congress of Psychology   Supplement: 1, Taylor and Francis (Psychology Press): STM, Behavioural Science and Public Health Titles / Taylor and Francis (Psychology Press) , 2012, Vol. 47, no SI, p. 317-317Conference paper (Refereed)
  • 9. Order onlineBuy this publication >>
    Östergren, Rickard
    Linköping University, Department of Behavioural Sciences and Learning, Psychology. Linköping University, Faculty of Arts and Sciences.
    Mathematical Learning Disability: Cognitive Conditions, Development and Predictions2013Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    The purpose of the present thesis was to test and contrast hypotheses about the cognitive conditions that support the development of mathematical learning disability (MLD). Following hypotheses were tested in the thesis: a) domain general deficit, the deficit is primarily located in the domain general systems such as the working memory, b) number sense deficit, the deficit is located in the innate approximate number system (ANS), c) numerosity coding deficit, the deficit is located to a exact number representation system, d) access deficit, the deficit is in the mapping between symbols and the innate number representational system (e.g., ANS), e) multiple deficit hypothesis states that MLD could be related to more than one deficit.

    Three studies examined the connection between cognitive abilities and arithmetic. Study one and three compared different groups of children with or without MLD (or risk of MLD). Study two investigated the connection between early number knowledge, verbal working memory and the development of arithmetic ability.

    The results favoring the multiple deficit hypothesis, more specifically the result indicate that number sense deficit together with working memory functions constitutes risk-factors to the development of MLD in children. A simple developmental model that is based on von Asters and Shalev´s (2007) model and the present results is suggested, in order to understand the development of MLD in children.

    List of papers
    1. Number magnitude processing and basic cognitive functions in children with mathematical learning disabilities
    Open this publication in new window or tab >>Number magnitude processing and basic cognitive functions in children with mathematical learning disabilities
    2012 (English)In: Learning and individual differences, ISSN 1041-6080, E-ISSN 1873-3425, Vol. 22, no 6, p. 701-714Article in journal (Refereed) Published
    Abstract [en]

    The study sought out to extend our knowledge regarding the origin ofmathematical learning disabilities (MLD) in children by testing different hypotheses in the same samples of children. Different aspects of cognitive functions and number processing were assessed in fifth- and sixth-graders (1113 years old) withMLD and compared to controls. The MLD group displayed weaknesses withmost aspects of number processing (e.g., subitizing, symbolic number comparison, number-line estimation) and two cognitive functions (e.g., visualspatial working memory). These findings favor the defective approximate number system (ANS) hypothesis, but do not fit well with the access deficit hypothesis. Support is also provided for the defective object-tracking system (OTS) hypothesis, the domain general cognitive deficit hypothesis and to some extent the defective numerosity-coding hypothesis. The study suggests that MLD might be caused by multiple deficits and not a single core deficit.

    Place, publisher, year, edition, pages
    Elsevier, 2012
    Keywords
    Mathematical learning disabilities, Cognitive functions, Number processing, Subitizing, Enumeration
    National Category
    Social Sciences
    Identifiers
    urn:nbn:se:liu:diva-77933 (URN)10.1016/j.lindif.2012.05.004 (DOI)000312683900007 ()
    Funder
    Swedish Research Council, 421-2007-1881
    Available from: 2012-05-31 Created: 2012-05-31 Last updated: 2017-12-07Bibliographically approved
    2. Early number knowledge and cognitive ability affect early arithmetic ability
    Open this publication in new window or tab >>Early number knowledge and cognitive ability affect early arithmetic ability
    2013 (English)In: Journal of experimental child psychology (Print), ISSN 0022-0965, E-ISSN 1096-0457, Vol. 115, no 3, p. 405-421Article in journal (Refereed) Published
    Abstract [en]

    Previous literature suggests that early number knowledge is important for the development of arithmetic calculation ability. The domain-general ability of verbal working memory also has an impact on arithmetic ability. This longitudinal study tested the impact of early number knowledge and verbal working memory on the arithmetic calculation ability of children in preschool (N = 315) and then later in Grade 1 using structural equation modeling. Three models were used to test hypotheses drawn from previous literature. The current study demonstrates that both early number knowledge and the domain-general ability of verbal working memory affect preschool and Grade 1 arithmetic ability. Early number knowledge had a direct impact on the growth of arithmetic ability, whereas verbal working memory had only an indirect effect via number knowledge and preschool arithmetic ability. These results fit well with von Aster and Shalevs developmental model of numerical cognition (Developmental Medicine and Child Neurology, 2007, Vol. 49, pp. 868-873) and highlight the importance of considering arithmetic ability as independent from early number knowledge. Results also emphasize the importance of training early number knowledge before school entry to promote the development of arithmetic ability.

    Place, publisher, year, edition, pages
    Elsevier, 2013
    Keywords
    Early number knowledge; Mental number line; Numerical knowledge; Arithmetic calculation; Verbal working memory; Arithmetic development
    National Category
    Social Sciences
    Identifiers
    urn:nbn:se:liu:diva-96454 (URN)10.1016/j.jecp.2013.03.007 (DOI)000320740400002 ()
    Available from: 2013-08-23 Created: 2013-08-20 Last updated: 2017-12-06
    3. Cognitive conditions of children at risk of developing mathematical learning disabilities
    Open this publication in new window or tab >>Cognitive conditions of children at risk of developing mathematical learning disabilities
    2013 (English)Manuscript (preprint) (Other academic)
    Abstract [en]

    Various hypotheses have been advanced regarding the origin of mathematical learning disabilities (MLD). The present study set out to test a number of hypotheses regarding the underlying condition for the development of MLD, namely: the domain general hypothesis, number sense deficit, numerosity coding deficit, access deficit and multiple deficits. These hypotheses were tested on a sample of eight-year-old children that was divided into three groups: MLD (N = 13), typical achievers (TA, N = 57) and high achievers (HA, N = 25) based on a longitudinal approach of stability in performance on arithmetic tests. The development of the three groups was also assessed using data from preschool to grade two. The results revealed support for the hypothesis of multiple deficits that are primarily located in intraparietal sulci (IPS) and manifest themselves as both number sense deficits and deficits in spatial processing. This type of deficit is supported by less developed general abilities in the domains of both phonological ability and nonverbal intelligence, resulting in poorer number knowledge for those children at risk of MLD. The HA group displayed a superior early number knowledge in combination with superior domain general abilities, which support the development of number knowledge.

    Keywords
    Mathematical learning disability, development of arithmetic skills, number knowledge, spatial ability, number sense deficit
    National Category
    Social Sciences
    Identifiers
    urn:nbn:se:liu:diva-96798 (URN)
    Available from: 2013-08-27 Created: 2013-08-27 Last updated: 2013-08-27Bibliographically approved
    Download full text (pdf)
    Mathematical Learning Disability: Cognitive Conditions, Development and Predictions
    Download (pdf)
    omslag
  • 10.
    Östergren, Rickard
    et al.
    Linköping University, Department of Behavioural Sciences and Learning, Education, Teaching and Learning. Linköping University, Faculty of Educational Sciences.
    Ringborg Lindgren, Marie
    Norrköpings Municipality, Sweden.
    Lindgren, Britt-Marie
    Norrköpings Municipality, Sweden.
    Samuelsson, Joakim
    Linköping University, Department of Behavioural Sciences and Learning, Education, Teaching and Learning. Linköping University, Faculty of Educational Sciences.
    Learning declarative knowledge in special education treatment group2018In: journal of education and training, ISSN 2330-9709, Vol. 5, no 1Article in journal (Refereed)
    Abstract [en]

    An organizing structure that in recent years has had a major impact on how to work with students who don’t respond to regular instruction is Response to Intervention (RTI). Efforts in RTI are divided into three different tiers of instruction: primary, secondary and tertiary. In our study, we investigate the impact of intensive secondary-tier instruction on students’ knowledge of basic combinations of digits in addition. We also focus on how the students develop their use of more advanced calculations in addition during the intervention.

    The results showed that students became faster at performing simple addition tasks, which indicates that their fluency – declarative knowledge – developed during the intervention phase. Our results thereby strengthen suggestions that a secondary-tier intervention level should take place in a small group of students 20-40 minutes four to five times a week. Meanwhile, the students developed their ability to solve two-digit arithmetic tasks in addition and subtraction, which could be explained by the fact that students had automated simple number combinations and thus could focus on the calculation procedure.

  • 11.
    Östergren, Rickard
    et al.
    Linköping University, Department of Behavioural Sciences and Learning, Education, Teaching and Learning. Linköping University, Faculty of Educational Sciences.
    Samuelsson, Joakim
    Linköping University, Department of Behavioural Sciences and Learning, Education, Teaching and Learning. Linköping University, Faculty of Educational Sciences.
    Is repeated testing of declarative knowledge in mathematics moderated by feedback?2018In: journal of education and culture studies, ISSN 2573-0401, Vol. 2, no 3, p. 209-212Article in journal (Refereed)
    Abstract [en]

    This study set out to examine the effects of repeated testing of students’ declarative knowledge in mathematics in grade 7 (13-14 years old) and to what extent feedback moderates the effect of continually testing students’ declarative knowledge. Students who have automated the 400 basic arithmetical combinations (200 addition combinations and 200 subtraction combinations) have gained declarative knowledge. Mastering these combinations gives students an advantage where doing various calculations and performing different mathematical procedures are concerned (Dowker, 2012). If a student has automated the basic combinations, their attention will not be diverted from the procedure when solving calculation tasks, and there is thereby less risk of incorrect answers (Dowker, 2012). Previous studies have also shown that declarative knowledge in mathematics predict future results in more advanced mathematics (Hassel Bring, Goin, & Bransford, 1988; Gersten, Jordan, & Flojo, 2005; Rathmell & Gabriele, 2011).

  • 12.
    Östergren, Rickard
    et al.
    Linköping University, Department of Behavioural Sciences and Learning, Psychology. Linköping University, Faculty of Arts and Sciences.
    Skagerlund, Kenny
    Linköping University, Department of Behavioural Sciences and Learning, Psychology. Linköping University, Faculty of Arts and Sciences.
    Träff, Ulf
    Linköping University, Department of Behavioural Sciences and Learning, Psychology. Linköping University, Faculty of Arts and Sciences.
    Cognitive conditions of children at risk of developing mathematical learning disabilities2013Manuscript (preprint) (Other academic)
    Abstract [en]

    Various hypotheses have been advanced regarding the origin of mathematical learning disabilities (MLD). The present study set out to test a number of hypotheses regarding the underlying condition for the development of MLD, namely: the domain general hypothesis, number sense deficit, numerosity coding deficit, access deficit and multiple deficits. These hypotheses were tested on a sample of eight-year-old children that was divided into three groups: MLD (N = 13), typical achievers (TA, N = 57) and high achievers (HA, N = 25) based on a longitudinal approach of stability in performance on arithmetic tests. The development of the three groups was also assessed using data from preschool to grade two. The results revealed support for the hypothesis of multiple deficits that are primarily located in intraparietal sulci (IPS) and manifest themselves as both number sense deficits and deficits in spatial processing. This type of deficit is supported by less developed general abilities in the domains of both phonological ability and nonverbal intelligence, resulting in poorer number knowledge for those children at risk of MLD. The HA group displayed a superior early number knowledge in combination with superior domain general abilities, which support the development of number knowledge.

  • 13.
    Östergren, Rickard
    et al.
    Linköping University, Department of Behavioural Sciences and Learning, Psychology. Linköping University, Faculty of Arts and Sciences.
    Träff, Ulf
    Linköping University, Department of Behavioural Sciences and Learning, Psychology. Linköping University, Faculty of Arts and Sciences.
    Early number knowledge and cognitive ability affect early arithmetic ability2013In: Journal of experimental child psychology (Print), ISSN 0022-0965, E-ISSN 1096-0457, Vol. 115, no 3, p. 405-421Article in journal (Refereed)
    Abstract [en]

    Previous literature suggests that early number knowledge is important for the development of arithmetic calculation ability. The domain-general ability of verbal working memory also has an impact on arithmetic ability. This longitudinal study tested the impact of early number knowledge and verbal working memory on the arithmetic calculation ability of children in preschool (N = 315) and then later in Grade 1 using structural equation modeling. Three models were used to test hypotheses drawn from previous literature. The current study demonstrates that both early number knowledge and the domain-general ability of verbal working memory affect preschool and Grade 1 arithmetic ability. Early number knowledge had a direct impact on the growth of arithmetic ability, whereas verbal working memory had only an indirect effect via number knowledge and preschool arithmetic ability. These results fit well with von Aster and Shalevs developmental model of numerical cognition (Developmental Medicine and Child Neurology, 2007, Vol. 49, pp. 868-873) and highlight the importance of considering arithmetic ability as independent from early number knowledge. Results also emphasize the importance of training early number knowledge before school entry to promote the development of arithmetic ability.

1 - 13 of 13
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