## 1. Introduction

“Number is the ruler of forms and ideas,and the cause of gods and daemons”

(Pythagoras as quoted by Iamblichus of Chalcis in Thomas Taylor, 1986)

Nowadays, these metaphoric words are illustrated empirically in numeracy being necessary for later life achievement (Duncan et al., 2007; Finnie & Meng, 2001; Reyna & Brainerd, 2011). Given the significance of the domain, research has been flourishing around the question of how children's early ability to learn mathematics develops; specifically around the math-specific and nonspecific cognitive precursors of mathematical achievement (Bull, Espy, & Wiebe, 2008; De Smedt, Verschaffel, & Ghesquière, 2009; Geary et al., 2009; Gullick, Sprute, & Temple, 2011; Holloway & Ansari, 2009; Krajewski & Schneider, 2009; Mundy & Gilmore, 2009) and their interrelations (e.g., Logie & Baddeley, 1987; Noël, 2009; Rasmussen & Bisanz, 2005).

Working memory (WM) emerges as a well-established domain-general cognitive predictor of math performance (Raghubar, Barnes, & Hecht, 2010), playing an important role in early mental arithmetic (DeStefano & LeFevre, 2004). At the same time, studies have highlighted a math-specific precursor of mathematical achievement: the very early ability to conduct basic arithmetic operations with large nonsymbolic numerosities, known as nonsymbolic approximate arithmetic skills (Barth et al., 2006; Gilmore, McCarthy, & Spelke, 2010). Some assume that these skills comprise the architectural foundation upon which symbolic math skills are built (e.g., Mundy & Gilmore, 2009), underscoring the importance of a better understanding of their underlying cognitive structure. In this study, we sought to test the assumption that nonsymbolic approximate processing necessitates WM. More specifically, we focused on uncovering which specific WM component is involved in this kind of processing in preschool age.

### 1.1. Nonsymbolic approximate arithmetic

A variety of empirical evidence suggests the existence of a cognitive ability that runs across species and improves with development, an inherent precursor of mathematical skills, known as nonsymbolic approximate arithmetic. Animals (Flombaum, Junge, & Hauser, 2005), human infants (McCrink & Wynn, 2004; Xu & Spelke, 2000), and adults who have received no formal instruction or schooling (Pica, Lemer, Izard, & Dehaene, 2004) can conduct basic mathematic operations with approximate numerical magnitudes. Preschool children who have not yet received any formal math instruction can compare, add, and subtract nonsymbolic approximate numerical magnitudes (Barth et al., 2006; Gilmore et al., 2010; Gilmore & Spelke, 2008) even when the elements presented are different in format and modality (Barth, Beckmann, & Spelke, 2008; Barth, La Mont, Lipton, & Spelke, 2005). Often, these skills are referred to as reflecting an innate “Approximate Number System” (ANS). In the literature, evidence for the ANS comes from several different types of tasks, which, however, have been recently found to be uncorrelated (Gilmore, Attridge, & Inglis, 2011). It should be noted that in this study, we place focus only on the nonsymbolic approximate skills of the ANS, skills that involve the addition and comparison of large numerosities.

It is theorized that exact symbolic verbal mathematic skills—that is, as taught in school—develop on top of and are fostered by approximate nonsymbolic arithmetic skills (Mundy & Gilmore, 2009; see Noël & Rouselle, 2011 for an alternative view). For example, Gilmore et al. (2010) showed that preschoolers' nonsymbolic approximate addition skills were associated with their formal symbolic mathematical performance, even when controlling for intelligence and literacy skills. Thus, it is imperative to understand and uncover the cognitive processes underlying children's nonsymbolic approximate arithmetic skills.

The most common task for assessing nonsymbolic approximate arithmetic skills is a computer-animated task (Barth et al., 2005, 2006; Gilmore et al., 2010), which we will refer to from here on as the dot task. A trial of the addition dot task consists of the following steps: An initial blue dot array appears on the screen and is then covered by a rectangular box, then an additional array of blue dots hides within the box, and lastly a set of red dots appears next to it. At the end of each trial, children have to estimate whether they saw more red dots or more blue dots. In the dot task, a ratio effect on performance arises from the distance of the summed blue set and the red set. As the numerical difference or distance between the two sets becomes smaller, their ratio approaches 1 and performance declines (e.g., Barth et al., 2006). For example, if the two blue dot sets add up to 40, it is easier to estimate the correct response when they are compared to a set of 70 red dots than to a set of 40. The large numerical distance makes their comparison much easier. It is postulated that this occurs because the mental representations of two numerical magnitudes, which are close to each other, overlap and are therefore harder to compare (Izard & Dehaene, 2008). This ratio effect is also presumed to be reflected in the participants' mean reaction response times (Noël, Rouselle, & Mussolini, 2005 as cited in De Smedt & Gilmore, 2011). This assumption, however, has not been tested in the nonsymbolic approximate arithmetic domain because tasks used so far did not allow reaction time (RT) registration. We developed a dot task that permitted the recording of RT data and thus the acquisition of a more fine-grained illustration of children's nonsymbolic approximate arithmetic cognitive processes. Previous research with tasks assessing comparison of small nonsymbolic numerosities has demonstrated the ratio effect in RTs by showing children's performance being slower in harder to compare trials (Holloway & Ansari, 2009; Rouselle & Noël, 2007; Soltész, Szücs, & Szücs, 2010).

So, what underlies the process of nonsymbolic approximate addition? In the dot task, participants must mentally retain and add the two blue dot sets, remember the summed numerosity, and then compare it to the red dot set. This procedure appears to involve working memory. Barth et al. (2006) already assumed working memory load involvement in their nonsymbolic approximate addition and subtraction tasks' implementation. However, to our knowledge, no previous study has examined in detail the role of WM in nonsymbolic approximate arithmetic processing.

### 1.2. Working memory and arithmetic

The most prominent theoretical account of WM is the tripartite WM model, originally conceptualized by Baddeley and Hitch (1974). According to this model, WM is a multicomponent cognitive architectural system that is responsible for the short-term storage and manipulation of a limited amount of elements during the execution of cognitive activities (Baddeley, 1986, 2002, 2003). It entails a master system, *the central executive (CE),* and two slave subsystems, the *phonological loop (PL)* and the *visuo-spatial sketchpad (VSSP)*. The central executive component has a supervising role; it is an executive system which regulates and controls cognitive processes run by the two slave subsystems. The phonological loop is responsible for retaining verbal information, whereas visuo-spatial information is maintained within the visuo-spatial sketchpad. Since its original conceptualization, empirical accounts have led to the development and extension of this multicomponent model (see Baddeley, 1996a, 2000, 2002, 2003). The role of the central executive was for a long time unclear. On the basis of accumulating findings, Repovš and Baddeley (2006; p. 14) proposed that “*in the realm of working memory tasks, executive processes seem to be involved whenever information within the stores needs to be manipulated*.” In other words, the slave subsystems are free of executive processes only when they involve simple representation and maintenance.

The literature distinguishes two kinds of methodological designs utilized for assessing the role of these WM components (Raghubar et al., 2010): experimental dual-task studies and correlational designs. The dual-task methodology is considered the most reliable experimental design since it uncovers the online underlying WM resources allocated in complex task processing. However, it has been predominantly used in studies with adults (e.g., Fürst & Hitch, 2000; Lee & Kang, 2002; Trbovich & LeFevre, 2003). The dual-task design involves the execution of a primary task (e.g., arithmetic task) while simultaneously performing a secondary task which loads—and therefore interferes with—a specific WM component. It is based on the principle that, if a specific WM component is necessary for the cognitive processing of the primary task, one will identify a performance breakdown or reaction time increase on either the primary or the secondary task in the corresponding interference condition compared to the conditions where these tasks were performed in a stand-alone form (baseline).

Dual-task studies with children are very limited. In their review, Raghubar et al. (2010) identify only two with primary school-aged children (Imbo & Vandierendonck, 2007a; McKenzie, Bull, & Gray, 2003). McKenzie et al. (2003) examined the developmental changes in the use of the slave WM components in exact verbal symbolic arithmetic (i.e., with Arabic numbers in the form of a + b = c). Two age groups of children were used: one with mean age 6.91 years and the other 8.94 years. Phonological and visuo-spatial interference occurred with the concurrent presentation of secondary tasks. In the respective interference conditions, children either heard irrelevant speech or looked at dynamic visual noise without needing to react to these secondary tasks. This type of interference is characterized as passive. In an active interference condition participants are asked to also respond to the secondary task (e.g., Imbo & Vandierendonck, 2007a). In this way, the interactive effect of the interference is indexed and thus performance breakdowns due to the load are reflected on either the primary or the secondary task. As highlighted by McKenzie et al. (2003), one reason for them to choose passive secondary tasks was because it was uncertain whether, especially the younger children, could perform active concurrent secondary tasks. Their results showed that younger children relied solely on visuo-spatial strategies when solving verbally presented exact symbolic arithmetic problems, whereas older children used also phonological strategies. Our study takes WM research in mathematical cognition a step further. We introduced for the first time active WM interference to children as young as preschoolers showing both the feasibility and the effectiveness of such an experimental design in this age group.

We know most about the early stages of learning arithmetic and the role of WM from studies using correlational designs. It has been argued that preschoolers' performance in arithmetic is in fact restricted due to their limited WM capacity (Klein & Bisanz, 2000). Specifically, Rasmussen and Bisanz (2005) demonstrated the developmentally differentiated relationship of WM components with distinct arithmetic problem formats. They tested 5- and 6-year-old children's PL, VSSP, and CE skills, and their performance in two different arithmetic problem formats: verbal (using story problems) and nonverbal (using chips). Preschool children's performance on the nonverbal simple addition task was found to be related to their VSSP WM capacity, contrary to older children who relied on their PL capacity. The authors argued that preschool children make use of a mental model to represent objects and conduct arithmetic manipulations, contrary to older children who make use of phonological coding strategies. Their nonverbal task was nonsymbolic in nature. In their study, however, approximate arithmetic was not examined since exact responses were required for the arithmetic problems, with operand set sizes ranging from 1 to 7.

Essentially, Rasmussen and Bisanz's (2005) study revealed the importance of the VSSP WM component for early nonsymbolic arithmetic but was limited by the fact that only one task was utilized to assess it. WM literature has shown evidence for the fractionation of this component into a visual and a spatial subcomponent (Baddeley, 2003; Logie, 1986; Darling, Della Sala, Logie, & Cantagallo, 2006). Notably, Hegarty and Kozhevnikov's (1999) results highlight the importance of this fractionation, since spatial and visual representations were shown to be differentially related to mathematical success. For this reason, we designed both a visual and a spatial interference condition to test whether they play different roles in the process of mentally representing nonsymbolic approximate arithmetic information.

On the other hand, Noël's (2009) research on preschoolers' simple addition skills emphasized the role of the CE component. Children were presented with drawings of objects, such as cows, with which they were asked to conduct basic additions. Contrary to the previously mentioned studies, here, children were free to solve the problems in any way they preferred and could even use their fingers or tokens in the process. Noël's arithmetic problems were presented in a combined visual and verbal manner, and in the given presentation format both symbolic and nonsymbolic information was involved. It was shown that in a free situation, the predictive power of the CE appears stronger and more significant than that of the other components in preschoolers' simple addition. Nevertheless, nonsymbolic approximate processing was not examined. To our knowledge, our study is the first to study the underlying WM processing in nonsymbolic approximate arithmetic.

### 1.3. The present study

Our main aim was to examine the relationship between nonsymbolic approximate arithmetic and WM as conceptualized by Baddeley's multicomponent model. Thus, a dual-task study was conducted with active phonological, visual, spatial, and central executive interference during the completion of a nonsymbolic approximate addition task, that is, the dot task. On the basis of Rasmussen and Bisanz's (2005) findings, we hypothesized that its underlying processing will depend on VSSP WM and not the PL in preschoolers. As indicated earlier, the CE appears to play an important role in children's arithmetic processing (see also Raghubar et al., 2010). During the implementation of the dot task, the mental representation of the first appearing blue quantity set must be updated after the second one is presented to form the basis against which the red set can be compared. On the basis of this updating process, we also hypothesized the CE WM component being involved in nonsymbolic approximate arithmetic processing (Morris & Jones, 1990).

Our secondary aim was to replicate existing findings on preschoolers' ability to successfully conduct addition with large nonsymbolic approximate quantities (Barth et al., 2006). Moreover, with our dot task, we scoped for the use of RT data as an additional source of information, which will facilitate the acquisition of a more coherent picture of the processes underlying children's nonsymbolic approximate arithmetic skills.