Making sense of numbers

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Early mathematics achievement and working memory in primary

school children

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Cover design: Michel Friso

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ISBN: 978-90-393-6197-9

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Making sense of numbers

Early mathematics achievement and working memory in primary

school children

Inzicht in getallen

Vroege rekenvaardigheden en werkgeheugen bij basisschoolkinderen (met een samenvatting in het Nederlands)

Proefschrift

ter verkrijging van de graad van doctor aan de Universiteit Utrecht op gezag van de rector magnificus, prof.dr. G.J. van der Zwaan, ingevolge het besluit van het college voor promoties in het openbaar te verdedigen op vrijdag 12 september 2014 des middags te 4.15 uur

door

Ilona Friso-van den Bos

geboren op 3 januari 1985 te Sliedrecht

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Copromotor: Dr. E.H. Kroesbergen

Studies reported in this dissertation were funded by the Netherlands Organisation for Scientific Research (NWO).

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Chapter 1

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General introduction

“You see, if there are four apples, and only three oranges – four is a bigger number than three, so that means there are more apples on the table than there are oranges.”

~ Benedict Cumberbatch on Sesame Street, in response to the mystery of apples and oranges.

Understanding numbers and quantities, interpreting them correctly, and using them for making informed decisions is an important task in the lives of young children. Children are confronted with numbers and quantities at school, but also during play, and while doing chores. For example, when distributing marbles for a game, children would want to be able to determine the number of marbles that they have, the number of marbles their friends have, and whether a fair distribution has been made. To do so, they would need to take the same steps as Benedict took in the example above: they would need to count accurately and keep track of counted items, understand what the result meant, remember the result whilst counting the next set, compare both sets, and have sufficient knowledge of numbers to be able to determine that one number is indeed bigger than the other. Later in life, children start using money, comparing prices, and making calculations with time, numbers, or any other quantities, and they continue using number facts for all kinds of decision making. It is thought that a sound grasp on basic number skills, such as counting the number of apples and oranges and comparing between them, constitutes the foundation of more advanced number skills needed for personal and professional goals (e.g., Dehaene, 2001; Jordan, Kaplan, Oláh, & Locuniak, 2006).

This dissertation focuses on these basic number skills, which are often joined under the term number sense (Berch, 2005; Dehaene, 1992, 2001; Gersten & Chard, 1999), as well as the ability to remember and process pieces of information (such as the quantity of a previously counted set) for short amounts of time, also known as working memory (Alloway, Gathercole, Willis, & Adams, 2004; Baddeley, 2007; Baddeley & Hitch, 1974). Number sense and working memory capacity are both thought to be of vital importance for mathematics achievement (Cowan & Powell, 2014; Geary, 2010, 2011; Jordan et al., 2013). In this dissertation, associations between number sense, working memory, and mathematics achievement are investigated using various approaches.

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9 1 | Gen eral In tr o d u cti o n

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Mathematics and number sense

Number sense components and their relation with mathematics

The term number sense refers to a general understanding of number, quantity, and magnitude. The term is difficult to define, and different authors have given different definitions, each highlighting different aspects, and each employing a different delineation of the skills involved in number sense. Broadly speaking, some authors have highlighted the conceptual, abstract part of number processing to define number sense, such as Dehaene (1992, 2001) who stated that number sense is the ability to mentally represent and manipulate number and quantity, or Gersten and Chard (1999), who highlighted fluidity and flexibility with numbers, and a general sense of meaning of numbers. Other researchers, by contrast, have employed operational definitions, focusing on performance that is facilitated by this conceptual understanding of number, such as counting, number identification, number knowledge, estimation, measurement concepts, the ability to perform mental operations with numbers, and to compare between them (e.g., Berch, 2005; Gersten & Chard, 1999; Jordan, Glutting, & Ramineni, 2008; Lago & DiPerna, 2010; Passolunghi, Vercelloni, & Schadee, 2007). Other terms have been used as well when referring to these operational definitions, such as number knowledge, early numeracy, numerical competence, and other terms (Lago & DiPerna, 2010). In this dissertation, the term number sense is used to refer to both the abstract understanding of numbers, mentioned in the conceptual definitions of the construct, and performance on various tasks in which this conceptual understanding is implicit, which is the focus of the operational definitions of number sense.

One of the topical issues in published literature, and one of the themes under investigation in this dissertation, is whether number sense should be seen as a unitary construct, or whether it consists of different components, and if so, which components should be included in a theoretical model. Previous reports include a number of subdivisions of the construct, most of them based on an influential theoretical model known as the ‘triple code model for numerical cognition’ (Dehaene, 1992; Dehaene & Cohen, 1995). This is an information processing model based on the premise that numbers are mentally represented using three different codes. First, there is the auditory verbal word frame, which is the representation of a number word (for example: the number 64 consists of the elements ‘sixty’ and ‘four’, and is processed with the use of these elements). Second, there is the visual Arabic number form, in which numbers are processed in their Arabic form, and recognised as sets of digits. Third, there is the analogue magnitude code, which is the representation of

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quantities with no exact notation. This last numerical code is the most hypothetical, and assumed to be associated with numerical estimation and subitising processes (Dehaene, 1992). The triple code model has formed the foundation for a pertinent number of studies targeting number processing (e.g., De Smedt & Boets, 2010; Moll, Göbel, & Snowling, 2014).

By far the most common distinction between components of number sense is a distinction between symbolic number sense and non-symbolic number sense (e.g., Cirino, 2011; Desoete, Ceulemans, De Weerdt, & Pieters, 2012; Göbel, Watson, Lervåg, & Hulme, 2014; Noël & Rousselle, 2011; Sasanguie, Göbel, Moll, Smets, & Reynvoet, 2013). Non-symbolic number sense, in this framework, refers to the ability to evaluate quantities when these are represented in a non-symbolic way, such as a collection of dots or series of beeps. No formal education is needed to acquire the ability to discriminate between non-symbolic quantities, and it can be compared to the analogue magnitude code (Dehaene, 1992). It is often associated with the approximate number system, which is thought to be an innate capacity to process quantity (Dehaene, 2001; Lipton & Spelke, 2003), although the idea of innateness has been disputed as well (Verguts & Fias, 2004). Symbolic number sense, on the other hand, refers to the ability to interpret and manipulate quantities with the use of culturally taught symbols, such as words and digits. Symbolic number sense manifests itself in tasks in which number words and symbols are actively used to represent quantity, such as counting tasks, number recognition tasks, or number writing tasks. This component can be seen as a combination of the auditory verbal word frame and the visual Arabic code (Dehaene, 1992). Number words and symbols are thought to acquire meaning through their connection with their non-symbolic counterparts, also referred to as mapping (Geary, 2013; Holloway & Ansari, 2009).

A recent debate has emerged concerning the respective roles of symbolic and non-symbolic number sense in the development of mathematical skill. Number sense has been argued to be foundational to the development of mathematical cognition (Dehaene, 2001; Jordan et al., 2006; Kroesbergen, Van Luit, Van Lieshout, Van Loosbroek, & Van de Rijt, 2009; Starr, Libertus, & Brannon, 2013). Traditional theoretical assumptions included non-symbolic number sense as a core system, responsible for successful acquisition of non-symbolic skills as well as mathematical proficiency (e.g., Lipton & Spelke, 2003; Starr et al., 2013). It was argued that symbolic quantities were only meaningful to a person through association with a pre-existing non-symbolic quantity representation (Mundy & Gilmore, 2009; Xenidou-Dervou, De Smedt, Van der Schoot, & Van Lieshout, 2013) and that individual differences in

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non-symbolic number sense could therefore account for individual differences in mathematics achievement. Alternatively, it has been argued that non-symbolic number processing becomes more precise as a result of a connection with symbolic number sense (Kolkman, Kroesbergen, & Leseman, 2013; Noël & Rousselle, 2011; Verguts & Fias, 2004), which would imply that symbolic processing is a catalyser for more advanced non-symbolic number sense as well as mathematics achievement. Either of these views can account for the finding that it is the symbolic system that is directly associated with mathematical achievement (e.g., Göbel et al., 2014; Kolkman, Kroesbergen et al., 2013; Sasanguie et al., 2013).

Measuring number sense

As a result of the variety in definitions of number sense, tasks employed to index the construct have also been diverse. Three types of tasks are especially prevalent in the literature: counting and number recognition tasks, comparison tasks, and number line estimation tasks. All three types of tasks are used to measure number sense in the various chapters of this dissertation, with an emphasis on number line estimation in some chapters.

Counting is an activity that is considered fundamental to learning about relations between numbers, and difficulties with mathematics have been linked to delays in counting skills (Geary, 2003; Geary, Hoard, Byrd-Craven, & DeSoto, 2004). Counting skills can be measured using a number of tasks: 1) enumeration – a task in which a child is asked to determine the number of a set of objects through counting, 2) counting sequence or rote counting – a task in which a child is asked to count as far as they can, or up to a certain number, and 3) the counting principles task – a task in which children detect violations in counting principles as defined by Gelman and Gallistel (1978), usually with the use of a puppet (Chu, VanMarle, & Geary, 2013; Jordan et al., 2006). Counting is considered a basic number skill, together with number recognition, which is the ability to verbalise written numbers (Jordan et al., 2006) and draws upon symbolic number sense (Kolkman, Kroesbergen et al., 2013).

Comparison tasks are found in two versions: non-symbolic versions and symbolic versions. In non-symbolic versions, participants are presented with two arrays of dots, squares, or other shapes, and asked to decide which of the arrays contains a larger number of shapes. Dot size and array diameter are typically controlled for, to prevent participants from basing their answers on visual properties instead of an evaluation of quantity (see Gebuis &

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Reynvoet, 2011). Performance on these non-symbolic comparison tasks has been found to be correlated with mathematics achievement in children (e.g., Desoete et al., 2012; Libertus, Feigenson, & Halberda, 2011; Mazzocco, Feigenson, & Halberda, 2011), although claims have been made that this association is moderated by domain-general cognitive skills (Gilmore et al., 2013). In symbolic versions of the comparison task, participants are asked which of two symbolically presented numbers is the larger of the two. Performance on this task has also been found to predict mathematics achievement at a later age (De Smedt, Verschaffel, & Ghesquière, 2009; Holloway & Ansari, 2009; Mundy & Gilmore, 2009; Sasanguie et al., 2013).

Finally, number line estimation is usually measured with the Number-to-Position task (Siegler & Opfer, 2003). In this task, children are shown an empty number line with a marked beginning- and endpoint, and are asked to indicate the position of a certain number on this line by drawing a hatch mark on the location or pointing to the intended location. As children get older, their number line placements become increasingly accurate, as indexed by a linear association between presented and estimated number (e.g., Ebersbach, Luwel, Frick, Onghena, & Verschaffel, 2008; Holloway & Ansari, 2008; Laski & Siegler, 2008). Before placements show a linear association with their actual values, they have been argued to be distributed logarithmically, with the lower numbers spaced far apart and the higher numbers compressed at the end of the line (Ashcraft & Moore, 2012; Dehaene, 2003; Opfer, Siegler, & Young, 2011; Siegler & Booth, 2004), or according to a power function, with both the lower and higher numbers spaced far apart and small spaces between numbers around the midpoint of the scale (Barth & Paladino, 2011; Hollands & Dyre, 2000; Slusser, Santiago, & Barth, 2013). No consensus with regard to pre-linear models has yet been reached. Linear and accurate placements of numbers on a number line have been shown to be related to higher mathematics achievement (Geary, 2011; Halberda, Mazzocco, & Feigenson, 2008; Sasanguie, De Smedt, Defever, & Reynvoet, 2012; Sasanguie et al., 2013; Sasanguie, Van den Bussche, & Reynvoet, 2012; Siegler & Booth, 2004) and to be impaired in children with mathematical learning disorder (Geary, Hoard, Nugent, & Byrd-Craven, 2008; Landerl, 2013). The relationship between number line estimation and mathematics achievement, however, has been suggested to be bidirectional (LeFevre et al., 2013).

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Training of number sense

It is generally accepted that delays in number sense early in life can have a negative impact on mathematics achievement at a later age. As a result, a variety of intervention programmes has been developed and examined, both for children with low performance on number sense tasks (e.g., Aunio, Hautamäki, & Van Luit, 2005; Toll & Van Luit, 2012; Van Luit & Schopman, 2000) and for children with other risk-factors that are generally associated with mathematical performance, such as working memory deficits (Toll & Van Luit, 2013b), specific language impairment (Mononen, Aunio, & Koponen, 2014), and low socio-economic status (e.g., Baroody, Eiland, & Thompson, 2009; Jordan, Glutting, Dyson, Hassinger-Das, & Irwin, 2012). Moreover, effects of early training on number sense achievements of typically performing children have been investigated (Krajewski, Nieding, & Schneider, 2008).

Although training programmes targeting number sense generally show positive effects on achievement, the trained tasks incorporated in these programmes often target a variety of skills, such as counting and number line estimation (Jordan et al., 2012; Toll & Van Luit, 2013b). It is therefore unclear which aspects of training programmes can be associated with gains in various aspects of number sense. A notable exception are the number line trainings that use linear numerical board games to advance number line placements of young children (Ramani & Siegler, 2011; Siegler & Ramani, 2008, 2009), but these programmes have not yet been compared to other types of interventions. Also, it is debated whether training programmes targeting domain-general skills such as working memory could lead to the same results (Kroesbergen, Van ‘t Noordende, & Kolkman, 2014; Toll & Van Luit, 2012). These issues are addressed in quasi-experimental studies in this dissertation.

Relations between working memory and number-skills

Another factor that is thought to play a central role in the development of number skills (mathematical skill as well as number sense) is the capacity and functioning of working memory and its executive functions: inhibition, shifting, and updating (Bull & Lee, 2014; Bull & Scerif, 2001; Geary et al., 2004; Kolkman, Hoijtink, Kroesbergen, & Leseman, 2013; Passolunghi, Mammarella, & Altoè, 2008). Working memory functioning has been found to predict mathematical achievement across time (Gathercole, Tiffany, Briscoe, & Thorn, 2005; Mazzocco & Kover, 2007; Toll, Van der Ven, Kroesbergen, & Van Luit, 2011) over and above the relation with intelligence (Alloway & Alloway, 2010).

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Definitions of working memory

The most frequently used model of working memory is the tripartite model of Baddeley and Hitch (1974). This model comprises three subcomponents, each having a distinct function in information storage and processing. Two storage systems, the visuospatial sketch pad and the phonological loop, form temporary storage units for visual and spatial information, and phonological and auditory information, respectively. These storage systems are also known as slave systems. Their capacity is typically measured through simple span tasks, in which strings of information, increasing in length, must be replicated (e.g., a dot appearing in different consecutive locations for the visuospatial sketch pad and word lists for the phonological loop). A third component, the central executive, is responsible for the coordination of information stored within the slave systems. Central executive capacity is usually measured using complex span tasks, in which information must be replicated as well as manipulated. Thus, the slave systems are responsible for information storage, while the central executive is responsible for both storage and processing of information (Oberauer, Süß, Wilhelm, & Wittman, 2003).

The coordinating role of the central executive can be further differentiated into a number of sub-processes (Baddeley, 1996). A framework based on both Baddeley (1996) and the executive function literature in which the central executive is subdivided into inhibition, shifting and updating (Miyake et al., 2000) is commonly used. Inhibition is the ability to suppress a dominant response in favour of another or no response, shifting is the ability to switch between response sets, and updating is the ability to monitor information in working memory. Executive functions (inhibition, shifting and updating) are seen as functionally detailed components of the central executive in this dissertation. The distinction between these three executive functions has been confirmed in some factor analytical studies targeting children (e.g., Hughes, 1998, Lehto, Juujärvi, Kooistra, & Pulkkinen, 2003; Rose, Feldman, & Jankowski, 2011) but not in all (e.g., Van der Sluis, De Jong, & Van der Leij, 2007; Van der Ven, Kroesbergen, Boom, & Leseman, 2012). Nevertheless, many studies have employed this distinction in their research on the relations between executive functioning and mathematical performance, and this distinction is maintained in this dissertation.

Relations between working memory and mathematics

The relations between working memory and various measures of mathematical skill have been studied extensively, and there is a general consensus that mathematical achievement is

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associated with working memory performance (Bull & Scerif, 2001; Geary et al., 2004; Passolunghi et al., 2008; St Clair-Thompson & Gathercole, 2006). Working memory has been hypothesised to be involved in the execution of mathematical problems, for example, when children need to remember partial answers to a problem, and in growth in mathematical skill, for example, when instructions need to be remembered to practice sets of problems later on. Yet, it is unclear which components of working memory can predict mathematical performance, and which cannot, because results of various studies contradict one another. For example, relations between mathematical skill and the executive function inhibition have been found in some studies (Bull & Scerif, 2001; Espy et al., 2004; St Clair-Thompson & Gathercole, 2006), but not in others (Andersson, 2008; Monette, Bigras, & Guay, 2011). Similarly, studies regarding associations between phonological loop performance and mathematics achievement have reported both positive relations (Panaoura & Philippou, 2007; Passolunghi et al., 2008) and null-results (Jarvis & Gathercole, 2003; Passolunghi et al., 2007). Such contradictions in outcomes can be found for all other components of working memory.

This variety in outcomes, which is addressed in the current dissertation, may be due to the complexity and variety of the tasks employed to measure working memory and the range of aspects that comprise the working memory model (Baddeley, 2007). Furthermore, inconsistencies may be due to characteristics of tasks and constructs specific to the working memory component (for an overview, see Raghubar, Barnes, & Hecht, 2010). Best, Miller, and Naglieri (2011), on the other hand, proposed that varieties in the mathematical task caused the inconsistencies in outcomes: they postulated that mathematical problem solving is strongly dependent on strategy formulation and implementation, and self-monitoring, requiring high working memory involvement. Calculation, on the other hand, was suggested to be more related to fact retrieval, which requires less executive control (Best et al., 2011). Findings reported by Fuchs et al. (2005) suggested the same (see also: Raghubar et al., 2010). Furthermore, it has been proposed that mathematics performance may be related to executive functioning to a different extent in various developmental stages (Best et al., 2011). Age has been reported to moderate the relationship in several studies (Imbo & Vandierendonck, 2007; McKenzie, Bull, & Gray, 2003; Rasmussen & Bisanz, 2005; Van der Ven, Van der Maas, Straatemeier, & Jansen, 2013; Van de Weijer-Bergsma, Kroesbergen, Prast, & Van Luit, 2014), yielding the primary conclusion that older children rely less on working memory in mathematical problem solving than younger children do, suggesting that part of the inconsistencies between studies may be explained by the age of the participants.

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Investigating the relations between mathematics achievement and working memory, as is one of the aims of the current dissertation, is of both theoretical and practical relevance. Research findings concerning this relation may contribute to our understanding of the specific cognitive abilities involved in solving mathematical problems, and how these cognitive abilities play a role in both proficiency and deficiency in mathematical skill. On a more practical level, such findings may guide decisions in development of interventions, but knowledge of working memory involvement in mathematical problem solving may also help practitioners monitor the extent to which various mathematics practice activities impose on relevant working memory components, and help them formulate curriculum plans in which working memory imposition is adequately managed. Taking into account working memory involvement in mathematical problem solving may, through these procedures, both benefit children with a delay in mathematics achievement, and pose challenges to children with high proficiency in the domain of mathematics problem solving.

Relations between working memory and number sense

The relations between mathematics achievement and working memory functioning have been hotly debated and extensively investigated. More recently, research trends started to centre around number sense, generally considered to be a precursor of mathematical skill, and therefore hypothesised to be associated with working memory as well. Children are expected to employ working memory capacity when practising number sense tasks, for example when evaluating which items in a set have already been counted in order to reach a total number of items. There is a limited body of literature available targeting the associations between working memory and its executive functions and number sense (e.g., Bull, Espy, & Wiebe, 2008; Bull & Scerif, 2001; Clark, Pritchard, & Woodward, 2010; Kroesbergen et al., 2009). Differences in numerical skill can be explained using children’s executive functions (Bull & Scerif, 2001; Kroesbergen et al., 2009; Navarro et al., 2011). However, there is no consensus regarding the role that each specific executive function plays in number sense. Several authors have reported that inhibition plays a role in numerical cognition (Kolkman, Hoijtink et al., 2013; Kroesbergen et al., 2009), but this predictive role could not be confirmed in other studies (e.g., Lee et al., 2012; Navarro et al., 2011). Updating is usually seen as the most important predictor of number sense (Kroesbergen et al., 2009; Lee et al., 2012), but despite these claims, research targeting the relations between number sense and executive functions is limited. More information with regard to the associations between working memory and its

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executive functions on the one hand, and number sense on the other hand, is needed. This need is addressed in the current dissertation. Findings regarding the association between number sense and working memory functioning may provide insight into the cognitive abilities needed to support domain-specific knowledge acquisition, which in turn may help formulate curriculum plans and intervention plans. Children with working memory deficits have been found to be able to improve in number sense after a number sense directed intervention, but not to reach the same level of performance as their peers with typical working memory functioning after intervention (Toll & Van Luit, 2013a). Insight into working memory involvement in various number sense activities may guide intervention strategies that bring children with working memory deficiencies to the same level as their typicall developing peers.

Training of working memory

It has been suggested that working memory capacity can be trained in a similar way to the training of domain-specific skills: through ability-guided practice and repetition (Klingberg et al., 2005; Klingberg, Forrsberg, & Westerberg, 2002; Thorell, Lindqvist, Bergman Nutley, Bohlin, & Klingberg, 2009). Improvements in working memory capacity after working memory training have been found in children as young as four or five years old (Diamond, Barnett, Thomas, & Munro, 2007; Röthlisberger, Neuenschwander, Cimeli, Michel, & Roebers, 2012). Especially working memory trainings that are adaptive to the level of the child can facilitate working memory, and mathematical achievement, associated with working memory performance, has been found to improve after working memory training as well (Holmes & Gathercole, 2013; Holmes, Gathercole, & Dunning, 2009), suggesting that gains in working memory can not only lead to better information processing, but also that there is a causal relationship between mathematics performance and working memory functioning. It must be noted though that the general effectiveness of working memory training is disputed, and that transfer to other skills such as numerical skills is, on the average, weak (Melby-Lervåg & Hulme, 2012; Rapport, Orban, Kofler, & Friedman, 2013).

No study to date has yet compared the gains after a number sense training directly with the gains after a working memory without a numerical component, in number sense skills or in working memory performance. A training study in which number sense and working memory trainings were compared to one another contained a strong number component in the working memory training (Kroesbergen et al., 2014). This issue, therefore,

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is addressed in this dissertation. Also, although effects of working memory training in kindergartners have been investigated, and transfer towards academic domains has been studied previously (Diamond et al., 2007; Holmes & Gathercole, 2013; Holmes et al., 2009; Röthlisberger et al., 2012), only a limited number of studies has yet investigated transfer of working memory training in children as young as kindergartners. Addressing training possibilities for working memory is not only of practical relevance, providing evidence for the intervention strategies applicable in scholastic settings, but can also contribute to our theoretical understanding of the relation between working memory and number sense: if a training targeting working memory indeed shows the capacity to improve working memory functioning, and if these gains in working memory transfer to number sense, this can serve as experimental support for the causal relationship between working memory and number sense. However. This can only be interpreted as such if variance in the gains in working memory can be related to variance in gains in number sense, and if gains in number sense do not exceed the gains in working memory performance. Mean gains in a trained group in number sense that do not relate directly to gains in working memory, or exceed them, would be indicative of the involvement of other processes that are both facilitated by working memory training and related to number sense.

Outline of this dissertation

In summary, current research trends centre around critical investigation of number sense, its components, and relation to working memory. Also, number sense and working memory are considered important predictors of mathematics achievement, both concurrently and longitudinally, but which specific assets of these predictors make up key-determinants of success or failure in mathematics remains unclear. Predictive associations hypothesised in current literature between mathematics achievement, number sense, and working memory are summarised in Figure 1.1: working memory is considered a predictor of both mathematical skill and number sense, and number sense is considered a predictor of mathematics achievement. These are the relations under investigation in the current dissertation. This dissertation contributes to the growing body of knowledge regarding the relations between these constructs by summarising, replicating, reformulating, and expanding upon the model presented in Figure 1.1.

Relations between number sense, working memory, and mathematics achievement are investigated using various approaches: 1) meta-analytical studies targeting working memory

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as a predictor of mathematics achievement and number sense, 2) relations between working memory and number sense, measured concurrently, 3) predictive value of working memory for number sense, and of number sense for mathematics achievement, and 4) facilitation of number sense and working memory through training studies. These approaches are embedded in the longitudinal MathChild study; a three-year longitudinal research project consisting of three part-projects, and with the general aim to investigate the predictive relationships between non-symbolic and symbolic number sense, and mathematics achievement in children from kindergarten to the early primary school years. This dissertation is a product of one of these part-projects, originally aiming only to investigate facilitation of number sense and working memory through training, but which has expanded over the course of the study to address the relations between these constructs in a more complete fashion.

Figure 10.1. Hypothesised relations between working memory performance, number sense, and mathematics achievement.

Section 1: Meta-analyses

Chapters 2 and 3 focus on working memory as a predictor of mathematics achievement (chapter 2) and number sense (chapter 3). In these chapters, reported associations are aggregated using meta-analysis, and variance around the mean effect size is explained using predictor variables that described differences between measures, samples, designs, and studies.

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In chapters 4 and 5, relations between number sense and working memory are further investigated using original data. In chapter 4, the predictive role of executive functions for a number-to-position task is examined in children from kindergarten to first year of primary school. In chapter 5, the factor structure of number sense is being investigated with a diverse battery of number sense tasks, and resulting factors are predicted using working memory variables measured concurrently.

Section 3: Longitudinal studies

In chapters 6 and 7, longitudinal studies of number sense are reported, with a focus on performance on the number line estimation task. In chapter 6, latent growth modelling is applied, and latent class analysis is used to identify trajectories of development in performance on this task. Growth variables and class membership are predicted using both the factors reported on in chapter 5, and working memory variables. Mathematics performance is used as an outcome variable. In chapter 7, performance on the Number-to-Position task is investigated using various models of number line placement, each referring to the use of certain reference points in making these placements. Moreover, longitudinal associations with mathematics achievement are investigated using cross-lagged panel analyses.

Section 4: Training studies

In chapters 8 and 9, the possibilities to ameliorate number sense are investigated using various training programmes aimed to improve kindergartners’ number sense. In chapter 8, a training in which number sense skills are targeted directly is compared to a training study in which number sense skills are targeted indirectly, through the training of working memory. In chapter 9, two trainings are compared that both focus on number sense: a counting training and a number line training.

Finally, in chapter 10, which consists of a general discussion, the findings of the eight studies included in this dissertation are summarised and discussed, and conclusions regarding the role of number sense and working memory skills in the development of mathematics achievement are drawn.

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Section 1 Meta-analyses

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Chapter 2 Working memory and mathematics in primary school children:

A meta-analysis

I. Friso-van den Bos S. H. G. van der Ven E. H. Kroesbergen J. E. H. van Luit

Printed as: Friso-van den Bos, I., Van der Ven, S. H. G., Kroesbergen, E. H., & Van Luit, J. E. H. (2013). Working memory and mathematics in primary school children: A meta-analysis. Educational

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Abstract

Working memory, including central executive functions (inhibition, shifting and updating) are factors thought to play a central role in mathematical skill development. However, results reported with regard to the associations between mathematics and working memory components are inconsistent. The aim of this meta-analysis is twofold: to investigate the strength of this relation, and to establish whether the variation in the association is caused by tests, sample characteristics and study and other methodological characteristics. Results indicate that all working memory components are associated with mathematical performance, with the highest correlation between mathematics and verbal updating. Variation in the strength of the associations can consistently be explained by the type of mathematics measure used: general tests yield stronger correlations than more specific tests. Furthermore, characteristics of working memory measures, age and sample explain variance in correlations in some analyses. Interpretations of the contribution of moderator variables to various models are discussed.

Introduction

Solving mathematical problems is an important activity in the lives of children. From an early age onwards, learning to count, acquiring number skills, and performing mathematical operations become part of children’s daily activities. These activities remain important throughout a person’s life. Research concerning the underpinnings of success and deficiencies in mathematical skills has expanded during the past two decades. A number of cognitive mechanisms underlying these mathematical skills have been proposed and their contribution to the development of mathematical skill has been investigated. A factor that is thought to play a central role in mathematic skill development is the capacity and the efficiency of working memory and the executive functions: inhibition, shifting and updating (e.g., Bull & Scerif, 2001, Geary et al., 2004; Passolunghi et al., 2008; St Clair-Thompson & Gathercole, 2006). Working memory capacity is frequently used as a predictor of skills in mathematics at a later point in time (see LeFevre, DeStefano, Coleman, & Shanahan, 2005). The number of studies in which the predictive value of working memory and executive functions for mathematical performance is investigated has increased sharply, but the pattern of results is inconsistent: mathematics performance is not consistently predicted by one or all working memory components. Therefore, the present study serves as a meta-analysis of studies in which this relationship was investigated, to investigate whether each working memory component is related to mathematics performance. Moreover, to find an explanation for conflicting results, we investigated the influence of various moderator variables: the type

Making sense of numbers - Early mathematics achievement and working memory in primary school children

Making sense of numbers

Early mathematics achievement and working memory in primary

school children

Making sense of numbers

Early mathematics achievement and working memory in primary

school children

Inzicht in getallen

Proefschrift

Ilona Friso-van den Bos

Table of contents

Chapter 1

General introduction

1

Mathematics and number sense

1

1

Relations between working memory and number-skills

1

1

Outline of this dissertation

1

References

1

1

1

1

2

Section 1

Meta-analyses

2

Chapter 2

Working memory and mathematics in primary school children:

A meta-analysis

Abstract

Introduction