Several cross-sectional and longitudinal studies have looked at numerical magnitude processing (NMP) and working memory (WM) as possible domain-specific and domain-general cognitive precursors of children’s mathematics performance and development (e.g. Alloway & Alloway, 2010; De Smedt et al., 2013; Friso-van den Bos et al., 2013; Kroesbergen & van Dijk, 2015; Li et al., 2018; Toll et al., 2016). Both NMP and WM have been linked with individual differences in mathematics performance, and although weakness in either of them seem to result in inferior performance (e.g. Brankaer et al., 2017; Cañizares et al., 2012), surprisingly few studies have investigated them simultaneously (Chan & Wong, 2019; Kroesbergen & van Dijk, 2015; Passolunghi et al., 2014; Toll et al., 2016). The intriguing findings from these few studies, varying in designs and methods, on one hand suggest WM to be a stronger predictor than NMP of early mathematics performance (Passolunghi et al., 2014), but on the other hand, children having weakness in both NMP and WM to struggle most in mathematics compared to those having no weakness or weakness in either NMP or WM (Kroesbergen & van Dijk, 2015; Toll et al., 2016).
Most of these studies looking at how the patterns of NMP and WM among children predict mathematics performance have used measures of non-symbolic NMP or employed varying operationalisations of NMP, while recent research suggests that symbolic NMP (SNMP) might provide more consistent findings and stronger predictions on mathematics performance among school-aged children (De Smedt et al., 2013). Prior studies have often used visual WM as a measure of WM, but it would seem beneficial to include verbal WM and central executive in these investigations as well, as both are known to be associated with mathematics performance (Friso-van den Bos et al., 2013). Along with NMP and WM, several domain-general factors, such as fluid intelligence, rapid automatized naming and language skills, have also been linked with mathematics performance (Koponen et al., 2018; Purpura & Ganley, 2014), due to which further attention to these individual differences might be informative. Finally, as to the criterion variables, focus in terms of mathematics performance has mainly been on arithmetic facts and word problem–solving, while also some other core skills, such as counting, have shown to be important predictors of later mathematics performance (Aunola et al., 2004). Of particular interest is also the fact that previous studies have typically used a priori defined cut-off scores for defining different performance levels, and, consequently, grouping children accordingly to those levels (Kroesbergen & van Dijk, 2015; Toll et al., 2016). It would therefore seem informative to examine the extent to which an alternative, data-driven classification matched the previous findings.
With this study, we aim to expand on previous research on the joint role NMP and WM play in early mathematics performance. We will focus particularly on SNMP (1- and 2-digit comparison), due to the more robust and stronger predictions obtained with it, and include all three different components of WM for a more inclusive representation of WM. As to classifying children according to their performance profiles, we will extract empirical profiles of SNMP and WM by means of latent class clustering. Finally, we will also use a relatively comprehensive set of measures for mathematics performance and take into account the influence of various domain-general skills, in order to tease out better the unique effects of SNMP and WP on mathematics performance.
The relation between symbolic numerical magnitude processing and mathematics performance
The ability to approximately represent numerical magnitude, often called approximate number system (ANS), is considered to underlie both non-symbolic and symbolic numerical magnitude processing (Feigenson et al., 2013; Goffin & Ansari, 2019). The association between NMP and mathematics performance has been substantially investigated in recent years, and this association seems to partly depend on the number format used in the NMP tasks. Studies using a symbolic format (i.e. Arabic digits) have demonstrated more consistent findings and stronger effects on mathematics performance than studies using non-symbolic format (i.e. dots) (De Smedt et al., 2013; Schneider et al., 2017). The inconsistencies regarding the non-symbolic format might partly be due to methodological issues, such as using many different types of non-symbolic comparison measures, or the measures not being sensitive enough. Alternatively, the non-symbolic magnitude processes measured may simply be less critical in the context of school mathematics (De Smedt et al., 2013). Given that these constraints do not seem to apply when NMP is measured using a symbolic format (De Smedt et al., 2013), we focused only on SNMP in this study.
SNMP predicts mathematics achievement within and across different grades in elementary school (Brankaer et al., 2017; Holloway & Ansari, 2009), also after controlling for age, intellectual ability and speed of number identification (De Smedt et al., 2009). Furthermore, students performing well in SNMP exhibit more effective arithmetic strategy use (i.e. being faster in retrieving facts and using procedural strategies), even when taking into account differences in intellectual ability, digit naming and general mathematics achievement (Vanbinst et al., 2012).
Tasks measuring SNMP aim to tap a person’s ability to access the number magnitudes in symbols (Rousselle & Noël, 2007). A vast majority of studies have used 1-digit numbers on symbolic comparison tasks (Brankaer et al., 2017). Some studies show that children solve multi-digit comparison tasks slower than single-digit tasks (Landerl et al., 2009), which may stem from the fact that in such tasks, children process multi-digit numbers differently than single-digit numbers. Two alternative explanations have been proposed; one suggesting that children process the number as a uniform unit (i.e. holistic view) (Reynvoet & Brysbaert, 1999), and another suggesting that children process decade-digit and unit-digit of the number independently (i.e. compositional model) (Nuerk et al., 2001). In support of the compositional model, research has found children to compare compatible number pairs (i.e. when both digits of the number are bigger than in the number to be compared; 25 vs. 68) faster than incompatible number pairs (e.g. 51 vs. 37).
The role of WM in mathematics performance
A significant relation between WM and mathematics performance has been evidenced in several studies (Friso-van den Bos et al., 2013; for a meta-analysis). Previous research has often employed the multicomponent WM model by Baddeley and Hitch (1974), hence referring to the three subcomponents of WM: the two slave systems, phonological loop and visuospatial sketchpad for storing verbal and visuospatial information, respectively, and central executive for coordinating information of the slave systems. This three-component model is still drawing the most attention in educational research, even though a fourth component, episodic buffer, has been included in the model (Baddeley, 2010).
According to the meta-analysis by Friso-van den Bos et al. (2013), all three WM components are linked with children’s mathematics performance. This relation, however, was dependent on the type of mathematics test used. General mathematics tests, such as national curriculum tests, which demand more switching between different operations and updating sets of information, yielded stronger correlations with WM than those focusing only on some specific mathematical skills. When looking at the relation between WM and specific mathematical skills, both verbal and visuospatial WM seem to be important predictors of counting (e.g. tasks involving number sequences or linking quantities to number words) at kindergarten age (Preßler et al., 2013). Kyttälä et al. (2019) showed that verbal, but not visuospatial WM, predicted word problem–solving from kindergarten to second grade, whereas Andersson (2008) found both verbal and central executive functions to predict word problem–solving among second to fourth graders. The use of central executive resources in solving single-digit arithmetic problems has also been highlighted, and verbal WM seems to play a role if such problems are solved using counting strategies (DeStefano & LeFevre, 2004, for a review).
However, the effect of WM seems to diminish when other factors are controlled. For example, basic academic skills (e.g. reading and calculation) and fluid intelligence have been found to account for some of the effects of WM on word problem–solving accuracy in early grades (Fung & Swanson, 2017; Zheng et al., 2011). Also, the effect of verbal WM on counting skills diminished when vocabulary, morphology, phonology, intelligence, task orientation and gender were controlled (Koponen et al., 2018).
SNMP and WM as predictors of mathematical learning difficulties
Mathematical learning difficulties (MLD) and dyscalculia (i.e. severe and persistent learning difficulties in mathematics) are another relevant issue in the present context. Given the impact of SNMP and WM on mathematics performance and development, children with MLD or dyscalculia would then be expected to display inferior SNMP and/or WM skills compared to their peers without such difficulties. Indeed, over the years, different theoretical models have been proposed as a cause of MLD (for an overview, see e.g. Siemann & Petermann, 2018), highlighting a deficit in domain-specific skills, namely in NMP—either non-symbolic NMP (i.e. defective number module, Butterworth, 2005) or SNMP (i.e. access deficit, Rousselle & Noël, 2007), or in domain-general skills, such as WM (i.e. cognitive deficit, Geary, 2004; Karagiannakis et al., 2014) or in both (i.e. double deficit, Kroesbergen & van Dijk, 2015; Wolf & Bowers, 1999). Regarding school beginners, as is the case in the current study, a weak performance in either NMP or WM or both might thus point out to a risk for MLD.
Some studies using non-symbolic NMP measures have shown children with dyscalculia to perform poorly in tasks of comparing two magnitudes (Desoete et al., 2012; Landerl et al., 2009; Mazzocco et al., 2011), or matching them (i.e. deciding whether two magnitudes are same or not) (Lafay et al., 2019), thus supporting the “defective number module” (Butterworth, 2005) as a primary cause of dyscalculia. Other studies, however, have not found such effects (e.g. De Smedt & Gilmore, 2011; Lafay et al., 2019). In contrast, more consistent and robust differences between children with or without MLD or dyscalculia (Cañizares et al., 2012; De Smedt et al., 2013; De Smedt & Gilmore, 2011; Desoete et al., 2012; Landerl et al., 2009) have been found in studies using SNMP measures, thus supporting “access deficit” (Rousselle & Noël, 2007) as an alternative explanation for dyscalculia, according to which the core deficit lies in not being able to access the number magnitude in symbols.
In support of the cognitive deficit as a cause for MLD, not only has it been shown that children with higher WM capacity outperform their peers with lower WM capacity in different mathematics tasks (e.g. Preßler et al., 2013), but also that children with varying degrees of MLD display weaker WM skills compared to their peers without MLD (e.g. Geary et al., 2004; Menon, 2016; Passolunghi & Siegel, 2004). In a meta-analysis comparing children with MLD to average-achieving age-matched children on measures of WM, large effects were found for central executive and visuospatial sketchpad (d = 0.95 and d = 0.59, respectively) and medium effects for phonological loop (d = 0.36), in favour of the average-achieving children (David, 2012).
Investigating the roles of (S)NMP and WM simultaneously in mathematics performance
Surprisingly few studies have investigated the roles of NMP or SNMP and WM in mathematics performance, or in relation to MLD, simultaneously. Passolunghi et al. (2014) found WM to be a stronger predictor of mathematics performance than non-symbolic NMP in the beginning of the first grade (6-year-olds), after controlling for intelligence (i.e. verbal and fluid intelligence). Furthermore, NMP lost its significance by the end of the first grade when predicting teacher-rated mathematics performance in a similar manner. Note, however, that intelligence turned out to be an even stronger precursor of mathematics performance than either WM or NMP.
Recently, Chan and Wong (2019) tested whether the prediction of visuospatial WM at grade 1 (7-year-olds) on mathematics achievement at Grade 2 was mediated by numerical magnitude representation (i.e. SNMP and computation) and problem representation (i.e. word problems). Both pathways were found to be significant, even after controlling nonverbal intelligence, reading fluency, processing speed, and verbal WM. However, as the direct effect from visuospatial WM to mathematics achievement remained significant as well, the authors concluded that numerical magnitude representation and problem representation failed to fully explain the relation between visuospatial WM and mathematics achievement.
Kroesbergen and van Dijk (2015) examined visuospatial WM and NMP (although they used the term “number sense”, which was represented by a combined score of non-symbolic and symbolic comparison, and number line tasks) in relation to arithmetic fluency and word problem–solving among 6–10 years old, also including a subgroup of children with MLD. Using a cut-off point of scoring below the 25th percentile, the participants were first divided into four groups according to their performance in number sense and WM: weakness in number sense, weakness in WM, weakness in both number sense and WM (i.e. double weakness) and without weakness. When comparing these groups on mathematics performance, they found a deficit in either number sense or WM, or both, to be connected with lower performance. Those identified as having double weakness displayed the most inferior performance of all groups, even after controlling for age, IQ and verbal WM, thus supporting the double-deficit hypothesis of MLD.
Partly replicating the study by Kroesbergen and van Dijk (2015), Toll et al. (2016) also investigated the predictions of NMP (or “number sense”, in their study) and visuospatial WM from the first year of kindergarten (5-year-olds) to grade one (7-year-olds). Non-symbolic NMP was operationalised in terms of dot comparison tasks, and although conceptualised as “symbolic number sense”, which refers to SNMP in some other studies, the authors used counting-based tasks instead of a digit comparison task. The results showed non-symbolic NMP to predict only word problem–solving 2 years later, whereas symbolic number sense predicted both arithmetic facts and word problems. When grouping the participants scoring below the 25th percentiles, weakness in both visual WM and number sense was connected with the lowest performance in arithmetic facts and word problems, even when controlling for fluid intelligence, thus replicating the results of Kroesbergen and van Dijk (2015).
By expanding on previous captivating findings on the roles of SNMP and WM in mathematics performance, we will in this study first explore the patterning of SNMP and WM through children’s performance profiles, and then link these profiles with several aspects of children’s mathematics performance (i.e. counting, arithmetic facts and word problem–solving). In contrast to previous studies using a priori defined cut-off points in performance (i.e. below or above the 25th percentile) to classify children into different groups, we examined the relative strength in children’s SNPM and WM skills by extracting empirical profiles through latent class clustering. Thus, instead of forming fixed categories of specific combinations (e.g. high/low, high/high), we relied on more naturally occurring data-driven patterns representing groups of children similar to each other, but different from the others. In a sense, then, this approach might result in a somewhat less artificial account of the patterning of SNMP and WM among children. Prior studies have mainly used visuospatial WM as an indicator of WM, whereas we included three WM components—verbal, visual and central executive—in the classification, thus seeking to have a more comprehensive empirical representation of individual differences in WM. As to the outcome measures, we included verbal counting skills in addition to arithmetic facts (addition and subtraction) and word problem–solving used in previous studies (Kroesbergen & van Dijk, 2015; Toll et al., 2016), as it is one of the core skills developing in this age group, and a significant predictor of later mathematics performance (Aunola et al., 2004). Finally, we controlled for several demographics (i.e. age, gender, parental educational level, status of second language learner) and domain-general cognitive skills (i.e. fluid intelligence, rapid automatized naming and word comprehension) shown to be associated with children’s mathematics performance (e.g. Alloway & Alloway, 2010; Koponen et al., 2018; Purpura & Ganley, 2014), to capture the unique contribution of children’s skill profiles on mathematics performance.