- Top of page
- 1. Introduction
- 2. Method
- 3. Results
- 4. Discussion
Preschool children have been proven to possess nonsymbolic approximate arithmetic skills before learning how to manipulate symbolic math and thus before any formal math instruction. It has been assumed that nonsymbolic approximate math tasks necessitate the allocation of Working Memory (WM) resources. WM has been consistently shown to be an important predictor of children's math development and achievement. The aim of our study was to uncover the specific role of WM in nonsymbolic approximate math. For this purpose, we conducted a dual-task study with preschoolers with active phonological, visual, spatial, and central executive interference during the completion of a nonsymbolic approximate addition dot task. With regard to the role of WM, we found a clear performance breakdown in the central executive interference condition. Our findings provide insight into the underlying cognitive processes involved in storing and manipulating nonsymbolic approximate numerosities during early arithmetic.
- Top of page
- 1. Introduction
- 2. Method
- 3. Results
- 4. Discussion
“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.
- Top of page
- 1. Introduction
- 2. Method
- 3. Results
- 4. Discussion
Our aim was to uncover the relationship between nonsymbolic approximate arithmetic and WM. For this purpose, we conducted for the first time a dual-task study with preschool children in which we actively interfered with the phonological, visual, spatial, and central executive WM components while implementing a nonsymbolic approximate addition task, that is, the dot task. At baseline, our results replicated previous findings that show the ability of preschool children to perform above chance level in the dot task. The characteristic ratio effect in accuracy was also replicated. With RT data, however, we did not find a similar effect. Regarding the role of WM in nonsymbolic approximate addition processing, results confirmed our hypothesis showing a predominant clear-cut effect of the CE component. Visual and spatial WM were not confirmed as important factors for nonsymbolic approximate processing. Surprisingly though, we found a PL effect on the secondary task performance; we argue that this effect reflected the role of this component in action control.
A precondition for examining the role of WM in nonsymbolic approximate addition was to show that we indeed provoked approximate addition with our task. Nonsymbolic approximate addition results replicated previous findings that prove preschool children to be able to successfully add large nonsymbolic quantities prior to having received any formal arithmetic instruction in school (Barth et al., 2005, 2006, 2008; Gilmore et al., 2010). Likewise, they did so without resorting to any strategies other than addition per se—for example, using systematic response preferences, such as choosing only the red quantity as being larger—or by basing their responses on perceptual characteristics other than the numerosity of the nonsymbolic stimuli. Children performed above chance level and the characteristic ratio effect was shown for accuracy, which supports the assumption of the existence of a mental number line system underlying approximate quantity estimation abilities (Izard & Dehaene, 2008).
Our study is the first to examine the ratio effect with RTs in nonsymbolic approximate addition. Iuculano, Moro, and Butterworth (2011) used RT data on a similar nonsymbolic approximate addition task but did not examine the corresponding ratio effect. To our surprise, the ratio effect was not evident with the RT data. This effect has been consistently demonstrated in previous research (e.g., Holloway & Ansari, 2009; Rouselle & Noël, 2007; Soltész et al., 2010), which, however, made use of nonsymbolic numerical magnitude comparison tasks. These tasks differ from our nonsymbolic approximate addition task in three main aspects; they (a) do not entail the element of addition, (b) deal with much smaller numerosities, ranging from 1 to 9, and (c) call for a response to simultaneously presented stimuli. This last element suggests that perhaps with the current animated dot-task design it was not possible to collect accurate RTs. The RT interference effect, which was consistent with the corresponding accuracy result, however, contradicts this interpretation. Perhaps, our RT measurement was reliably sensitive to the interference effect, but not sensitive enough to capture the ratio effect due to the inherent design of this task. On the other hand, the remaining two elements of differentiation between the dot task and the previous comparison tasks suggest possible differences in the skills that these tasks actually attempt to measure. Our dot task is a far more complex cognitive task, where children were asked to add large quantities that ranged from 6 to 70.
For long, many assumed that the skills assessed with the nonsymbolic magnitude comparison and the nonsymbolic approximate addition tasks could be placed under the same “theoretical umbrella”; that of the so-called ANS. This is mainly because of the consistent and common underlying signature effects such as that of the ratio and distance effect (Gilmore et al., 2011). Gilmore et al. (2011), however, provided evidence for the lack of correlation between participants' performances in these tasks. As one of their explanations, they suggest the possibility that these tasks may draw on different domain-general abilities, such as WM. In accordance to that argument, we postulate that the nonsymbolic comparison tasks and nonsymbolic approximate addition tasks, such as our dot task, may call upon different underlying cognitive processes. Of course, the lack of an RT ratio effect must be replicated and further research is needed for its elaborate explanation. Moreover, future research should determine the different mechanisms underlying nonsymbolic magnitude comparison and nonsymbolic approximate addition.
With regard to the role of WM, our findings confirmed our main expectation. WM underlies nonsymbolic approximate addition processing. For interference during the primary task (dot task), our results on both the accuracy and RT data revealed a clear-cut interference effect. Specifically, as expected, preschoolers' performance was hindered in the CE interference condition. There is of course also the matter of the strategic tradeoff between the primary and the secondary tasks. Comparisons of performance between the secondary tasks conducted in the stand-alone and in the dual-task condition indexed once again a breakdown on the CE secondary task. Our findings, therefore, demonstrated a coherent picture for the necessity of CE WM demands. This result is consistent with previous research demonstrating the importance of executive resources in children's mathematical cognition (Noël, 2009; Raghubar et al., 2010). The exact role of the CE demands further elucidation. The CE task that we used, namely the CRT-R, is a task widely utilized to tap the CE (Imbo & Vandierendonck, 2007a; Tronsky, McManus, & Anderson, 2008) as a homunculus subcomponent of WM. The functions of the CE, however, can be further fractionated (Repovš & Baddeley, 2006). During the dot task, a participant must mentally update the mental representation of the first blue array with the second to form a summed set, which can then be compared with the red array. Future research should determine whether it is specifically the executive process of updating that is required during nonsymbolic approximate addition.
On the basis of Rasmussen and Bisanz's (2005) findings on preschoolers' nonsymbolic arithmetic, we had initially also hypothesized a predominant role for the visuo-spatial component of WM in nonsymbolic approximate arithmetic. We, therefore, explored the effect of the Visual and Spatial WM subcomponents. Surprisingly, our results revealed no significant effect for the Visual and hardly any for the Spatial WM component, since the only effect found for the latter was limited to its easiest span. In a dual-task design, if a WM interference effect is to be assumed, it must be evidenced at least in the hardest ratios. Taking a closer look at Fig. 7, which depicts children's performance in the different span levels of the spatial secondary task in the dual and the stand-alone conditions, one can notice that performance drops close to chance (50%) after the easiest span in all conditions. It appears that there was a floor effect. We believe that this task was too hard for our children, resulting in a limited variability in performance, which in turn did not allow for any interference effects to be visible. We advise future studies to make use of easier visuo-spatial interference tasks to illuminate the role of the visuo-spatial sketchpad in nonsymbolic approximate arithmetic. An alternative explanation for this surprising result may be derived from the early studies examining the cognitive processing of expert versus novice chess players (Baddeley, 1996b, 2002; Gobet, 1997). Expert chess playing has been found to not be a result of higher visuo-spatial WM processing, but rather due to the advanced pattern recognition level of the player. Similarly, it is possible that in this assumed innate skill of nonsymbolic approximate arithmetic, some sort of visuo-spatial mental operation does take place, which is not, however, adjunct to visuo-spatial WM.
However, apart from the preceding arguments, why did Rasmussen and Bisanz (2005) find the VSSP playing a predominant role in nonsymbolic processing and we did not? In their nonsymbolic addition task, an experimenter would show a number of chips to the child, cover them up with a box, and then he or she would add more under this box. Children were asked to replicate the amount of chips they saw with their own collection of chips. Operands in this task ranged from 1 to 5, answers from 3 to 8, and it necessitated an exact response. Thus, our differentiated findings imply also differences in the underlying cognitive processing between the two tasks. To our knowledge, research, thus far, has examined the differentiation between exact and approximate symbolic arithmetic processing (Kucian, von Aster, Loenneker, Dietrich, & Martin, 2008), but not between exact and approximate nonsymbolic arithmetic processing. What arises from our pattern of findings is that in nonsymbolic approximate processing, preschool children ultimately rely on their CE for successful implementation. In our dot task, children could not represent each object/dot separately, as in the case of Rasmussen and Bisanz's (2005) task and, thus, due to the large amount of dots, the CE component takes over and compensates by processing condensed whole arrays of dots and updating them within WM. It would be interesting for future studies to examine this assumption by specifically examining the differences in cognitive resources allocated for the processing of nonsymbolic exact and approximate arithmetic.
Unexpectedly, secondary task performance results also identified PL involvement. According to Krajewski and Schneider's (2009) theoretical model, children from a very young age start utilizing quantity discrimination words such as “much” or “more.” It may be assumed that children made use of such a strategy to solve the nonsymbolic approximate arithmetic problems, that is, by applying phonological tags on the arrays presented. Such an explanation, even though interesting, is also unsafe. Other studies have shown children to start utilizing phonological WM and corresponding strategies at a later age (McKenzie et al., 2003; Rasmussen & Bisanz, 2005). We believe, therefore, that this unexpected result of PL involvement was shown due to the unavoidable instructions that were given to the children during dot-task implementation. During testing, we observed these young children being easily distracted while conducting the given complex tasks. For this reason, it was necessary in some occasions to give them instructions to sustain their attention during dot-task implementation such as “look at the dots,” “pay attention.” It is very plausible that this may be the practical explanation of the PL interference effect evident on the corresponding secondary task when performed under the dual-task condition. Children heard the verbal instructions and at the same time had to remember the series of letters. This explanation is in line with the findings that regard the PL also as playing a role in the control of one's behavior (Baddeley, 2003; Baddeley, Chincotta, & Adlam, 2001). At the same time, this observation constitutes a limitation of our study. Future dual-task research should focus in developing a context where no such verbal instructions are needed, so that clear WM PL storing implications can be concluded.
This study is also limited by the fact that different WM task designs were utilized to load and interfere with the corresponding WM abilities. The CE interference task was a continuous one, whereas the rest of the secondary tasks took place “before” and “after” each primary-task trial and entailed discrete levels of difficulty. Nevertheless, this is common practice within the dual-task literature (e.g., Imbo & Vandierendonck, 2007b) due to the practical restrictions of wanting to load and interfere purely on a specific WM component without also interfering with the actual skill (not related to WM) that the primary task is tapping. We argue, however, that our results cannot be interpreted based on the differences between the designs of the tasks. Performance in each interference condition was only compared with that of baseline. In other words, interference conditions were not compared among each other. Also, the effects of disruption of the CE cannot be attributed to a higher task difficulty, as all other WM tasks included harder levels of difficulty than usual performance in this age and analyses were conducted on the span level. Future research in this field could pursue the design of similar WM interference tasks that would allow also the examination of the difference of effects between each condition. For example, future studies may alternatively use articulatory suppression, where in the dual condition participants repeat an irrelevant word such as “the,” as an alternative to our PL interference condition (see Baddeley, 2002). Furthermore, innovative VSSP interference conditions could be developed, such as those used by Lanfranchi, Baddeley, Gathercole, and Vlanello (2012), that interfere with the primary task during its completion. The important issue in developing and using these secondary tasks for a dual-task study is that they tap the different WM components as purely as possible. Our secondary tasks were developed in that manner. The tasks we used for tapping the slave subsystems of WM were free of executive resources since they necessitated sole representation and maintenance of the corresponding information (Repovš & Baddeley, 2006). On the other hand, the CE secondary task necessitated manipulation within the store. Actually, it called for manipulation within the PL store, as the task had verbal characteristics. Future research should indicate if this CE interference result would also be evident in a condition where the corresponding task needed manipulation within the VSSP store.
The findings of this study generate methodological as well as cognitive, developmental, and applied educational psychology implications. We demonstrated that effective dual-task studies with active WM interference can be conducted with children as young as preschoolers. Nonsymbolic approximate representations have been characterized as being central to human knowledge of mathematics (Gilmore & Spelke, 2008). It is even assumed that nonsymbolic approximate arithmetic skills comprise the building blocks on top of which symbolic exact arithmetic skills are developed and enhanced (Mundy & Gilmore, 2009). We showed that preschoolers' nonsymbolic approximate addition skills necessitate central executive resources. Now, the question is raised of whether it is actually nonsymbolic approximate skills that play a role in later math development or do these skills mediate the effect of WM processing on mathematical achievement? Our findings constitute a stepping stone in the path for uncovering and understanding the underlying cognitive architecture of early arithmetical skills.