Recently there has been a rise in research interests towards early numeracy development and developmental difficulties in it (e.g., Journal of Learning Disabilities vol 38, no.4, July/August 2005) which has been until now a highly neglected topic compared for example the amount of research done in early literacy. One line of the research is trying to find out the best ways to assist children with difficulties in early mathematics learning by seeking more information about the origins of their difficulties. The literature concerning the older children and formal school mathematics suggests that the general intelligence explains only 9-25% of the variance in mathematics performance, and that the potential other explaining factors might be working memory, executive functions and language abilities. The aim of the current symposium is to find out how these cognitive abilities are related to early numeracy, i.e. such mathematical skills which generally develop before the start of the formal teaching of mathematics in school.

The four papers are sound empirical research into the topic. Together there will be results from 483 children aged 4 to 7 years from four different non-English speaking countries. The instruments used to measure different aspects of working memory, executive functions, language abilities and IQ are overlapping between the countries providing interesting possibilities also for cross-sample comparisons. Three of the papers apply one-time-measurement design, and one offers the results from a longitudinal design.

The scientific and educational relevance of the symposium is clear. Firstly, it offers new knowledge about the origins of early numeracy development and its difficulties. Secondly, it gives valuable information for the use of mathematical screening and diagnostic measurements designed to be used with young children. Thirdly, this symposium will give ideas to what to include in the intervention programs planned to support the children’s mathematical development.

The relationship between early numerical abilities and working memory will be investigated. Theoretical background is based on Baddeley´s (1986, 1997, 2000) conceptualisation of working memory as a four-component information processor. The presentation is based on the research project in which we examined how early numeracy in four- to six-year old Finnish children (N=116) is related to working memory (WM), namely central executive (CE), visuo-spatial WM (VSWM) and phonological WM. In addition, we analyzed the impact of language abilities to children’s early numeracy. The general intelligence was a controlled variable. The data was collected during April and May 2006 and will be analysed in autumn 2006. The purpose of the analysis is to investigate whether the performance in different WM tasks (storing, simultaneous storing and processing, visuo-spatial, verbal) explains early numeracy performance (incl. relational and counting skills). We will also examine how much of the early numeracy skills can be explained by language skills including the possible interactions between e.g. language and working memory. Structural regression modelling will be used. The results are relevant especially for the special education field, which designs the early screening measurements and provides support for children with mathematical difficulties.

Background

Several studies have found evidence showing the involvement of working memory (WM) in mathematical performance (for a review, see Ashcraft, 1996). Of the many WM models (Miyake & Shah, 1999) the best-known and empirically thoroughly tested one is Baddeley’s (1997) framework. It includes the three most commonly recognized subcomponents: the central executive (CE), the visuo-spatial WM (VSWM) and the phonological WM. Later, Baddeley (2000) extended the model by suggesting a new component, a domain-free episodic buffer.

In our previous study (Kyttälä, et al., 2003), we observed that early numeracy, more precisely counting skills, were related to VSWM in preschoolers. The results were in line with several other studies (e.g. Gathercole &Pickering, 2000; McLean & Hitch, 1999; Reuhkala, 2001), which, however, are nearly without exception, concentrated on formal school mathematics. We designed this current research to investigate the found relationship between VSWM and early numeracy (children´s abilities to understand and operate with quantities) further. Measures of the other WM components were included as well as the measures of language abilities. The mathematical development in preschool is interesting as then there is a developmental shift from using the biologically primary quantitative abilities to learning the biologically secondary number, counting and arithmetic competencies, the latter being affected more by the language and learning environment than the former (Geary, 2000). It is also a highly relevant timing for screening and providing support for children with mathematical difficulties.

Aims and hypothesis

Gersten, Jordan and Flojo (2005) summarized that, (1) ability to make magnitude comparisons (i.e. knowing which digit in a pair is larger), (2) sophistication of counting strategies, (3) fluent identification of numbers, and (4) well developed numerical related working memory capacity (i.e., reverse digit span) are crucial elements for early numeracy, and can be used as valid and reliable indicators of potential mathematical difficulties in preschoolers. We designed the current study to investigate these suggested factors, and operationalized them as relational (point 1) and counting (points 2 and 3) abilities, and working memory (point  4) in terms of the CE, visuo-spatial component and phonological component. The focus is on four to six years old children.

The purpose of the analysis is to investigate whether performance in different WM tasks (storing, simultaneous storing and processing, visuo-spatial, verbal) explains performance in the early numeracy (incl. relational and counting skills). We will also examine how much of the early numeracy skills can be explained by the language skills. The general intelligence is a controlled variable.  

Methods

Participants

Participants are 116 children (56 girls and 60 boys) from Finnish preschools in capital area. The children have heterogeneous performance profile, including children for instance, with learning difficulties, with multi-language background and average performance. The mean age was 5 years 11 months (in moths: M 70.7, SD 9.01). The volunteer preschools were found via previous research contacts. The research permits were applied from the local authority for early childhood education, the particular preschools’ educators and the parents.

Each participant was tested by the trained research assistant individually in a quiet room. All eight tests were administered in three sessions, one session lasting about 30 minutes. The tests were given in a different random order for each participant.

Measures

We applied one-time measurement design. The scales are given here by the cognitive abilities they measure.

1) Early numeracy

The Early Numeracy Test by Van Luit, Van de Rijt and Aunio (2006) including the scales for relational and counting abilities.

2) WM

VSWM: Matrix pattern task based on the task created by Wilson, Scott and Power (1987)

VSWM: Corsi Blocks (Milner, 1971)

Phonological WM: Non-word repetition task

Non-verbal CE: Odd-one-out

Verbal CE: Backwards word recall

3) Language abilities

Naming abilities: The Boston Naming Test (BNT) by Laine et al.  (1997)

Verbal comprehension of commands: The Token test for children by  DiSimoni (1978)

4) General intelligence

Raven progressive matrices (Raven, 1992)

Outcomes

The data has been collected during April and May 2006. The outcomes from the structural regression model analysis will be ready in January 2007. Based on the previous research we expect that after controlling for the general intelligence there will be specific effects of different cognitive components to the early numeracy. Firstly, the VSWM capacity will explain more the counting than the relational abilities. Secondly, the language abilities and phonological VM capacity will be more important in explaining the relational than the counting scale performance. Thirdly, the executive functions will affect the performance in both early numeracy scales.

The theoretical and educational significance

The outcomes report the relevance of the WM capacity to the early numeracy development. The research will demonstrate the importance of language abilities, in terms of naming and receptive skills, in children’s early numeracy including the possible interactions between for instance language and working memory. In addition, we will know more about why some children have problems in their early numeracy development. The results are important for the professionals working in the field of learning difficulties and especially for those designing early screening measurements and interventions for the children with mathematical difficulties.

The Utrecht Early Numeracy Test (ENT) assesses young children’s number sense. Four subscales refer to the logical principles identified as the key factors underlying children’s understanding of quantities and relations, and four subscales focus more explicitly on the use and understanding of numbers. The purpose of this study was to examine the cognitive correlates in the ENT. Inhibitory processes, working memory, and naming speed were evaluated. Inhibitory processes refer to the central, active suppression of information that is irrelevant to the task at hand. The Working Memory (WM) tasks require children to hold increasingly complex information in memory while responding to the questions about a task. Naming Speed refers to rapidly respond on a variety of the most familiar visual symbols and stimuli in the language. A total of 50 children, aged between 4 and 7 from Cadiz (Spain) schools district participated in this study. The mean age of the participants was 5 y.1 m. Intelligence was evaluated by WISC-IV. The naming speed was assessed by The Rapid Automatized Naming Test. A version of the Stroop task yielded measures of effortful inhibition, and susceptibility to interference. Finally, WM was assessed by the Children Working Memory Test. Students were tested in mobile research laboratories on school grounds during their second- or third preschool years or first primary school year. A variance analysis of results and predictor cognitive variables (WM, inhibitory processes and naming speed) for The ENT performer will be analyzed. Hypothesis prediction suggests that the higher scores in the working memory, and the inhibitory processes, rather than naming speed will results in better ENT performing.

Introduction

The Utrecht Early Numeracy Test (ENT; Van de Rijt, Van Luit & Pennings, 1999) assesses young children’s number sense. Four subscales refer to the logical principles identified as the key factors underlying children’s understanding of quantities and relations (Piaget, 1966): comparison, classification, one-to-one correspondence, and seriation, and the four other subscales focus more explicitly on the use and understanding of numbers (Fuson, 1988; Gelman & Gallistel, 1978): the use of number words, structured counting, resultative counting, and general understanding of numbers.

Mathematics is one of the skills that children learn in school. Large differences between children’s math performance can be detected (Van de Rijt & Van Luit, 1998). Some research does not provide a satisfactory rationalization for this variability (Geary, 2004). Although differences are attributed to intelligence, it can explain only 9-25% of the variance in children’s math achievement (Resing, Ruijssenaars, & Bosma, 2002). Recent studies indicate that other domain-general cognitive abilities, more specifically executive functions and working memory, may provide better explanations for the variability in early math learning (Bull & Scerif, 2001; Kroesbergen & Van Luit, 2005).

This study focuses on the children’s knowledge of preparatory math skills that usually receive little attention in schools. Inhibitory processes, working memory, and naming speed were evaluated. Inhibitory processes is considered to be an important component of executive functioning (Carlson & Moses, 2001; Lehto, Juujarvi, Kooistra, & Pulkkinen, 2003). As currently used in the cognitive literature, inhibition refers to the central, active suppression of information that is irrelevant to the task at hand. The Working Memory (WM) has been described as an active information processor responsible for storing and processing information for a short time (Baddeley, 1986, 1997). It includes three components: a central executive controlling system, which is considered primarily responsible for coordinating the activities of the phonological component and the visuo-spatial component, but it also describes resources from long-term memory (Baddeley & Logie, 1999). The WM task requires children hold increasingly complex information in the memory while responding to a questions about a task. Naming Speed refers to rapidly respond on a variety of the most familiar visual symbols and stimuli in the language: letters, numbers, colours, and simple objects.

Method

The purpose of this study was to examine the cognitive correlates (WM, inhibitory processes and naming speed) in the Utrecht Early Numeracy Test. A total of 50 middle-class children, aged between 4 and 7 from Cadiz (Spain) schools district participated in this study. The mean age of the participants was 5.1 (sd. = 1.2); 25 were male and 25 female. Students were assessed with the Utrecht Early Numeracy Test. Intelligence (IQ), working memory, naming speed and inhibitory tests were also used. Intelligence was evaluated by the Wechsler Intelligence Scale for Children (WISC-IV) (Wechsler, 2005). The naming speed was assessed by the Rapid Automatized Naming Test (RAN) (Wolf & Denckla, 2003). A version of the Stroop task (Stroop, 1935) yielded measures of effortful inhibition, and susceptibility to interference. Finally, the working memory was assessed by the Children Working Memory Test (Pickering & Gathercole, 2001). Children were tested in mobile research laboratories on school grounds during the winter of their second- or third preschool years or first primary school year. Tasks were administered over three to four sessions within two weeks.

A variance analysis of results and predictor cognitive variables (WM, inhibitory processes and naming speed) for the Utrecht Early Numeracy Test performer will be analyzed. Hypothesis prediction suggests that higher scores in the working memory, and the inhibitory processes, rather than the naming speed, will results better Utrecht Early Numeracy Test performing.

Scientific and educational relevance

This research brings together knowledge from science education, cognitive psychology and early mathematic learning, implementing programs with children with mathematics learning disabilities. It also has an added value validating the measurement of cognitive skills involved in mathematical learning, and the efficiency for designing mathematical program. The results will have impact in the early diagnosis and identification of children with mathematics learning disabilities and a subsequent improvement in their school performance.

Numerical learning is an essential component in education and deficit in mathematical understanding strongly impair functioning, at school but also in everyday life. Research shows a high incidence of difficulties in mathematics learning in the population; about 7% of school children suffer from a cognitive or neuropsychological deficit that interferes with the acquisition of normal competence in mathematics. This research aims at identifying the precursors of mathematics learning at the beginning of primary school. There are few experimental studies on this topic and existing ones use between-subjects designs and correlation analysis. This paper analyses longitudinal data to investigate whether the relationship between basic abilities and mathematics learning is causally interpretable, rather than one where cognitive abilities are correlated to early mathematics learning in a cross-section design.

The present study tested 170 children at the beginning and the end of first year of primary school. From recent literature, we selected basic cognitive abilities highly likely to predict future mathematics learning. A battery of tests relative to these abilities (working memory resources; phonological ability; numerical competence, i.e. production and understanding of numbers, counting ability), measured pupils’ capacities when first starting primary school. We then looked at the relationships between these test scores and a test of mathematics ability at the end of the first school year.

Linear structural relations, causally interpreted, were used. The model showed that working memory tasks, that in particular tap executive functions, and counting ability tasks are the most discriminating and efficient as precursors of early mathematics learning. In our data, phonological ability is not involved in mathematics learning ability, and in the presence of the cognitive measures included in the model, intelligence level does not directly influence mathematics capacity.

Introduction

This study focuses on early mathematics learning at the beginning of primary school, and aims to identify precursors of mathematics learning. From recent literature, we selected basic cognitive abilities highly likely to predict future mathematics learning. A battery of tests relative to these abilities (working memory, phonological ability; numerical competence), measured pupils’ capacities when first starting primary school. We then looked at the relationships between these test scores and a test of mathematics ability at the end of the first school year.

Most of the literature shows that working memory is related to a variety of numerical and mathematical abilities (Adams & Hitch, 1997; Logie, Gilhooly, & Wynn, 1994; Passolunghi, Cornoldi, & Di Liberto, 1999; Passolunghi & Pazzaglia, 2004). On the other hand, the research on working memory as precursor to mathematics learning at early age is still very limited. One aim of this paper is to analyse the causal role of working memory as an actual precursor to the acquisition of numerical and mathematical skills in children starting primary school. Our hypothesis is that working memory, and in particular the executive functions, is causally related to early mathematics learning.

Very few studies take account of the relationship between level of phonological ability and performance in mathematics and arithmetic tasks; moreover, these studies show conflicting results (see Leather & Henry, 1994; Bryant et al., 1990). On this background, our paper aims to disentangle the contrasting evidence from the literature, and investigate whether there is a causal relationship between phonological ability and early mathematics learning.

Finally, on the basis of some previous results (Geary, Bow-Thomas & Yao, 1992; Geary, Hoard & Hamson, 1999) we suggest that counting capacity predicts mathematics learning, but believe that production and comprehension of numbers do not predict later performance since differences in these capacities at the outset of the school year tend to cancel out with the natural progression of learning.

Method

Participants

170 first grade children (72 girls and 98 boys), with average age of 6 years 4 months.

Procedure

The children were tested in two successive phases, the first in October and November, the second in the following May. In the first phase they were given tests to evaluate their basic cognitive abilities. These were: 1) working memory (Listening Span Completion Task, Passolunghi & Siegel, 2004; Word and Digit Span Backwards Tasks) and short-term memory tasks (Word and Digit Span Forwards Tasks); 2) tests of phonological ability (tasks of blending and phonetic analysis); 3) tasks for evaluating numerical competence (number production and comprehension tasks; Counting Knowledge Task; Verbal Counting Task; Counting speed task); and 4) tests measuring IQ (block design, and vocabulary, WISC-R, 1974, Italian standardisation 1987). The two scores can be used to obtain an overall value for IQ (Sattler, 1988). In the second phase a test of achievement in mathematics (subtest: logic, arithmetic, and geometry, Amoretti, Bazzini, Pesci, & Reggiani, 1993) was given.

Results

To test our hypothesis, we constructed structural equation models starting in a systematic way, as a top-down procedure, from the most complex model that includes variables deduced from the literature, simplifying to the final model. The final model has 10 observed variables and is presented in Figure 1.

Figure 1. Final model (for 10 observed variables).

Discussion

The structural linear model confirms the hypothesis that working memory is a distinct and significant predictor of mathematics learning at the beginning of primary school. More specifically, the results suggest that a general executive system underlies development in mathematics achievements at an early age.

Another aspect concerns the arguments about the controversial role of phonology. The results clearly suggest that phonological awareness is not generally relevant in mathematics learning, but that the specific phonological-numerical ability of verbal counting is a good and early predictor of mathematics learning during the first primary school year. We consider these findings might be useful in identifying measures that are more sensitive than standardised mathematics achievement tests (see Geary, 2005) or test of general intelligence for early identification of the general and specific deficits of children with forms of learning disability in mathematics. Moreover, on the basis of this results it is possible to develop appropriate training that can improve children’ early mathematics abilities.

The aim of this study was to investigate the relations between executive functions and preparatory math skills in normally developing kindergartners. The hypothesis was tested that executive functions explain a large part of the variability in early math skills, more than traditional intelligence does. Early math skills are operationalized in terms of counting skills. Executive functions are defined as the higher-order functions that are necessary for the adequate execution of complex goal-directed activities. Different executive functions were measured: planning, updating, and inhibition. 143 children, aged five to six, participated in the study. Next to the measures of  counting skills, executive functions, and intelligence, the language level of the children was measured, because this is an important predictor for the counting skills. The results show that two of the three tests for executive functions show stronger correlations with the counting skills than the intelligence. The test expected to measure inhibition did not correlate with the early math performance. The executive functions planning and updating together explain 41% of the variation in the counting skills, while the intelligence can only account for 21%. The results confirm the hypothesis that the executive functions are more closely related to the early math performance than the intelligence. Although further research is essential, the results are promising. The concept of executive functions should be used for the early identification of children at risk for math learning difficulties and can give direction to the remediation programs.

Background

Mathematical development in primary school reveals already at an early stage large differences between children. Traditionally, intelligence has been viewed as the most important predictor of academic performance, especially in the area of mathematics. However, evidence shows that intelligence, as measured with traditional tests such as WISC-III, can explain only 9-25% of the variance in children’s math achievement (Resing, Ruijssenaars & Bosma, 2002). Recent studies indicate that other domain-general cognitive abilities, more specifically executive functions (Efs), may provide better explanations for variability in early math learning (e.g., Bull & Scerif, 2001; Kroesbergen, Van Luit & Naglieri, 2003). Executive functions are an umbrella term for different higher-order functions such as planning, inhibition, updating and shifting and are necessary for the adequate execution of complex goal-directed activities (e.g., Barry, Lyman & Klinger, 2002). EFs are especially important in novel situations in which one cannot rely on routine, and have in common the regulation of other cognitive skills. This study focuses on EFs in children. EFs develop relatively late, but are already found in children aged two to three years old. However, at this age, EFs are not as differentiated as in older children or adults. In this study, the relations between EFs and mathematics are further investigated. Because EFs are good predictors for math development at later ages, they are expected to also be related to preparatory mathematics.

Aims

The aim of this study was to investigate the relations between EFs and preparatory math skills in normally developing kindergartners. The hypothesis was tested that EFs explain a large part of the variability in early math skills, more than traditional intelligence does. Early math skills are operationalized in terms of counting skills.

Research design

The study was conducted in five regular elementary schools in the Netherlands. A total of 143 children from second year of kindergarten participated. The sample consisted of 58 boys and 85 girls, aged 5;3 till 7;0 years (M = 6.0, sd = 0.4). In two sessions at different days, the children were administered tests for preparatory math skills (Early Numeracy Test; Dutch version, Van Luit et al., 2006); language (Language for Kindergartners; CITO, 2004); IQ (Raven’s Coloured Progressive Matrices; Raven, 1992) and executive functions (Tower of London, expected to measure planning; Digit Span Backwards, expected to measure updating; Expressive Attention, expected to measure inhibition) (Gathercole & Pickering, 2000; Korkman, Kirk & Kemp, 1997). Language level was measured because it is an important control variable in all academic skills.

First, the factor structures and reliabilities of the scales for language, mathematics and executive functions were studied. It appeared that for both language and mathematics, one scale could be calculated (Cronbach’s alpha’s were .87 and .81 respectively). The three executive function tasks could not be combined into one scale. Digit span Backwards correlated significant with both Tower of London (r = .37) and Expressive Attention (r = .27), but the latter two tasks did not correlate at all (r = .01).

Second, the effects of age, sex, and ethnicity were explored. No effects (p > .10) were found of age or sex on any of the other variables. However, children from ethnic minorities (N = 53) were found to have significant lower scores (p < .01) on all tests except Expressive Attention (p > .10). Ethnicity has therefore been controlled for in all analyses.

Third, the relations between IQ, executive functions, language and math were investigated. Pearson’s correlations were calculated for each explaining factor. Furthermore, a stepwise regression analysis was conducted, with (1) language, (2) Raven’s SPM and (3) the three measures of executive functions.

Outcomes

The scores on the early numeracy test correlated significantly with language (r = .61, p < .01), IQ (r = .46, p < .01), Digit span Backwards (r = .59, p < .01), and Tower of London (r = .47, p < .01). However, no correlation was found between the early numeracy and Expressive Attention (r = .06). Digit span Backwards and Tower of London together explain 41% of the variance in the early numeracy. The results confirm the hypothesis that executive functions explain more variance in the early numeracy than IQ.

A stepwise regression analysis was conducted, with language, IQ and the three executive functions. The results show that Executive Functions explain variance in early numeracy scores above language and IQ. Expressive attention was not a significant factor in the model, and was therefore excluded. The results show that there is a strong relation between language and early numeracy. Furthermore, IQ explains a significant part of the variance early numeracy. Executive functions do have an additive value in explaining this variance. The regression coefficients of the final model show that Digit span Backwards, measuring updating, is the most important explaining factor next to language. The difference between the Raven’s CPM and the TOL, measuring planning, is small.

Theoretical and educational significance of the research

This study has some important implications. First, the results show that executive functions can be measured at the age of five or six, but that further research is necessary to specify tasks for different executive functions. When these functions are compared to intelligence, it is also important to differentiate between fluid and crystallized intelligence, because the former is expected to be more closely related to executive functioning than the latter. Furthermore, the results show that executive functions are indeed more closely related to the early math performance than intelligence. Although further research is essential, the results are promising. The concept of executive functions should be used for early identification of children at risk for math learning difficulties and can give direction to remediation programs.

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