Processing of space, time, and number contributes to mathematical abilities above and beyond domain-general cognitive abilities

Journal of Experimental Child Psychology 143 (2016) 85–101

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Processing of space, time, and number contributes to mathematical abilities above and beyond domain-general cognitive abilities Kenny Skagerlund ⇑, Ulf Träff Department of Behavioral Sciences and Learning, Linköping University, SE-581 83 Linköping, Sweden

a r t i c l e

i n f o

Article history: Received 2 June 2015 Revised 9 September 2015

Keywords: Mathematics development Numerical cognition Spatial processing Temporal processing Domain-general abilities Magnitude processing

a b s t r a c t The current study investigated whether processing of number, space, and time contributes to mathematical abilities beyond previously known domain-general cognitive abilities in a sample of 8- to 10-year-old children (N = 133). Multiple regression analyses revealed that executive functions and general intelligence predicted all aspects of mathematics and overall mathematical ability. Working memory capacity did not contribute significantly to our models, whereas spatial ability was a strong predictor of achievement. The study replicates earlier research showing that non-symbolic number processing seems to lose predictive power of mathematical abilities once the symbolic system is acquired. Novel findings include the fact that time discrimination ability was tied to calculation ability. Therefore, a conclusion is that magnitude processing in general contributes to mathematical achievement. Ó 2015 Elsevier Inc. All rights reserved.

Introduction The acquisition of mathematical competency likely depends on several cognitive abilities involved in a distributed neurocognitive network (Fias, Menon, & Szücs, 2013). Identification of these cognitive abilities is vital in order to leverage this information to find children at risk for developing ⇑ Corresponding author. E-mail address: [email protected] (K. Skagerlund). http://dx.doi.org/10.1016/j.jecp.2015.10.016 0022-0965/Ó 2015 Elsevier Inc. All rights reserved.

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mathematical difficulties and also to implement targeted interventions aimed at ameliorating specific weaknesses. Researchers within cognitive psychology and cognitive neuroscience are continuously identifying components pertinent to mathematical achievement. It is increasingly acknowledged that different constellations of cognitive abilities contribute differently to various aspects of mathematics (e.g., Fuchs et al., 2010; Träff, 2013). Therefore, the purpose of the current research was to obtain a better understanding of the relationship among different aspects of mathematical abilities, general cognitive abilities, and more specific abilities pertaining to processing of number, space, and time, all of which have shown some relation to mathematical abilities in independent research (e.g., Dehaene, 2011; Feigenson, Dehaene, & Spelke, 2004; Gunderson, Ramirez, Beilock, & Levine, 2012; Kramer, Bressan, & Grassi, 2011). Here we report a study using hierarchical regression modeling of an extensive battery of tests tapping various cognitive abilities. Even though it is increasingly recognized that mathematical abilities depend on both domaingeneral processes and domain-specific processes, the relative contribution and importance of domain-general abilities vis-à-vis number processing skills remain unresolved. However, few studies have investigated the relative contribution of both domain-general cognitive abilities and number processing skills in the same models and how they relate to different aspects of mathematics (but see Fuchs et al., 2010; Träff, 2013). Studies have found that working memory (WM) capacity is important during multidigit calculation (e.g., Andersson, 2008; Swanson, 2006). However, results are inconclusive. Whereas some researchers have found a link between calculation ability and visuospatial WM capacity (e.g., Swanson, 2006), others have instead found a link between verbal WM and calculation skill (e.g., Andersson, 2008). Adequate mathematical competency also relies on having a network of arithmetical facts stored in long-term memory (Geary, 1993; McCloskey, Caramazza, & Basili, 1985). With experience, children will gradually shift from an overt counting strategy to an automatic retrieval-based strategy. Research indicates that executive functions are important for forming robust and accurate arithmetic facts (Geary, Hoard, & Bailey, 2012; Kaufmann, 2002). Thus, independent efforts have contributed substantially to our understanding of the relationship between cognitive abilities and mathematical abilities, but they have mostly done so by focusing on single variables or in relation to a single composite mathematics outcome (i.e., overall mathematical achievement). In the current study, we investigated the relative contribution of different domain-general cognitive abilities and domain-specific abilities to different aspects of mathematics. Domain-general cognitive abilities and mathematics Pioneering research has found that domain-general cognitive abilities, such as WM (e.g., Bull, Espy, & Wiebe, 2008; Swanson & Beebe-Frankenberger, 2004; Szücs, Devine, Soltesz, Nobes, & Gabriel, 2014) and semantic long-term memory (e.g., Geary, 1993), are important cognitive abilities involved during mathematical reasoning. The widely accepted multicomponent working memory model (Baddeley & Hitch, 1974) comprises three systems: (a) the visuospatial sketchpad, which handles short-term information of visual and spatial character; (b) the phonological loop, which is concerned with acoustic and verbal information; and (c) the central executive, which monitors allocation of attentional resources and execution of sequentially planned tasks. Each of these components has been linked, although inconsistently, to mathematical ability across studies (cf. Andersson, 2008; Passolunghi & Lanfranchi, 2012). The inconsistency may be attributed to the type of mathematical task used as an index of mathematical ability, and different combinations of domain-general abilities may underlie the development of different mathematical competencies (Fuchs et al., 2010). Yet, Fuchs et al. (2010) found that no domain-general cognitive abilities predicted growth in procedural calculation and instead argued that general cognitive abilities support the integration of the symbolic number system and non-symbolic number representations. The central executive has also been linked to mathematical achievement (LeFevre et al., 2013; Meyer, Salimpoor, Wu, Geary, & Menon, 2010) and is thought to be involved in facilitating selection of appropriate strategies and allocation of attentional resources during strategy execution (Geary, 2004). The central executive has been argued to support children’s acquisition of novel procedures and to establish automatic access to facts (Kaufmann, 2002; LeFevre et al., 2013). Recent research has found that executive processes, such as shifting between tasks and inhibiting distracting elements,

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are strong predictors of mathematical achievement (Szücs et al., 2014). Researchers have tried to reconcile the discrepant results by highlighting a developmental shift in reliance on WM capacities (cf. Meyer et al., 2010). General intelligence has been tied to mathematical achievement (e.g., de Jong & van der Leij, 1999). Math achievement scores in a sample of ninth graders were predicted by scores on Raven’s Standard Progressive Matrices (Raven, 1976) beyond WM ability (Kyttälä & Lehto, 2008). Geary (2013) suggested that intelligence is important for understanding systematic relations among numerals, which in turn would facilitate the learning of the mental number line. Number processing abilities and mathematics Research has identified that humans are endowed with an innate domain-specific ability to represent and manipulate quantities (Dehaene, 2011; Gelman & Butterworth, 2005), which is an ability phylogenetically shared with other primates and animals (Dehaene, 2011; Feigenson et al., 2004). It has been suggested that this representational system provides the foundation onto which the culturally acquired symbolic system is mapped (Feigenson et al., 2004). The basic ability to apprehend and manipulate quantities, both symbolic and non-symbolic, has been suggested to play a crucial role in mathematical achievement (Libertus, Feigenson, & Halberda, 2011; Mazzocco, Feigenson, & Halberda, 2011; Rousselle & Noël, 2007). Halberda, Mazzocco, and Feigenson (2008) found that the ability to discriminate between sets of objects visible only very briefly on a computer screen predicted mathematical achievement. The ratio between the two sets of objects with which one could reliably discriminate yielded an index of approximate number system (ANS) acuity. A key characteristic of the ANS is its imprecision, where the numerical representations grow increasingly imprecise as a function of quantity (Halberda et al., 2008). This imprecision is sometimes expressed by calculating a Weber fraction, indicating how much the ratio of two magnitudes must increase in order for an individual to perceive a difference. The importance of the innate number sense is further corroborated by studies on developmental dyscalculia (DD), which is a learning disability characterized by impairments in learning and remembering arithmetic facts and in executing calculation procedures (Gelman & Butterworth, 2005). A prominent hypothesis is that DD is caused by a deficit in this innate number ability (Dehaene, 2011; Gelman & Butterworth, 2005). Indeed, findings indicate that children with DD show impaired ANS acuity compared with age-matched peers (Mazzocco et al., 2011; Skagerlund & Träff, 2014), but the relationship between DD and ANS acuity remains debated (cf. De Smedt & Gilmore, 2011; Rousselle & Noël, 2007; Szücs et al., 2014). For example, Halberda et al. (2008) found that ANS acuity retrospectively predicted mathematical achievement in Grade 3, but the direction of causality cannot be determined given that better mathematics skills may have sharpened ANS acuity through experience. Knowledge of Arabic numerals and the magnitudes they represent is highly important for mathematical competency when entering the elementary school system (LeFevre et al., 2010). Basic affinity with the number system has consistently been associated with mathematical achievement (Rousselle & Noël, 2007). It is, however, unclear to what extent knowledge of the number system is dependent on the non-symbolic ANS. A recent study by vanMarle, Chu, Li, and Geary (2014) addressed the question of whether the ANS plays a role in acquiring basic symbolic competencies, and these authors found that ANS acuity facilitates early learning of symbolic knowledge and indirectly influences mathematics achievement. Thus, mounting evidence suggests that the ANS plays a role in achieving mathematical competency. However, the extent of the relative contribution of the ANS and the symbolic skills to mathematical development and DD remains to be determined. Spatial and temporal abilities and mathematics Recent work also indicates that mathematical abilities partly rely on spatial processes (e.g., Gunderson et al., 2012). Zhang et al. (2014) found that spatial visualization skills predicted arithmetical achievement, and Szücs et al. (2014) downplayed the importance of number sense variables and instead emphasized the role of spatial skills. This is further corroborated by a study on children with DD who showed impaired mental rotation skills compared with typically achieving peers (Skagerlund

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& Träff, 2014). It has also been argued that numbers are mentally represented along a horizontal leftto-right number line. This notion is derived from the fact that individuals tend to make decisions faster with smaller numbers with a button located to the left and with higher numbers with a button located to the right (Moyer & Landauer, 1967). This effect has been named the spatial–numerical association of response codes (SNARC; Dehaene, Bossini, & Giraux, 1993). Gunderson et al. (2012) found that mental rotation ability predicted the linearity of number line knowledge. These authors hypothesized that spatial ability plays an important role in mathematics by helping children to develop a meaningful linear mental number line. In addition, they found that spatial skills at age 5 years predicted approximate calculation skill at age 8 years, which provides a direct link between spatial skill and arithmetic (see also Hegarty & Kozhevnikov, 1999). A longitudinal study found spatial attention to be strongly related to number naming and processing of numerical magnitude (LeFevre et al., 2010). Training effects on a mental number line task were observed in both healthy children and children with DD when participating in a training regime over 5 weeks (Kucian et al., 2011). These children demonstrated improvements on the training task but also showed improved mathematical skills. Prior to the training period, children with DD showed abnormal activation patterns in the parietal lobules. After the training period, children demonstrated increased activation patterns in and around the intraparietal sulcus, suggesting that training potentially ameliorated the previously dysfunctional hypoactive region (Kucian et al., 2011). These neuroscientific data show promising results and also highlight the overlapping brain regions pertaining to space, number, and arithmetic. Corroborating evidence shows that mental rotation tasks elicit the same cortical substrates as number processing tasks (Kucian et al., 2007), which further points to the intricate relationship between space and number. These findings firmly suggest that spatial ability is indeed tied to mathematical competency. Time is another dimension that has also been suggested to play a role in mathematics (Kramer et al., 2011; Skagerlund & Träff, 2014; Vicario, Rappo, Pepi, Pavan, & Martino, 2012). Temporal processing has mostly been indirectly tied to mathematics, where individuals with DD often complain about poor perception of time (Cappelletti, Freeman, & Butterworth, 2011). The link between time and mathematics can be traced to Walsh (2003), who proposed that humans are equipped with a general magnitude system (a theory of magnitude, ATOM) responsible for handling amodal mental representations of ‘‘more than” or ‘‘less than.” This includes dimensions such as time, space, number, size, and luminance. The neural underpinnings of this magnitude system can be traced to the parietal cortex, where this system integrates analogue magnitude information of various modalities to encode for sensorimotor action (Bueti & Walsh, 2009). Data from Kramer et al. (2011) support this view, where time estimation ability predicted mathematical intelligence. Vicario et al. (2012) and Skagerlund and Träff (2014) investigated temporal processing in children with DD, and both studies found abnormal temporal skills. Skagerlund and Träff suggested that the concomitant temporal, spatial, and numerical impairments indicate that children with DD suffer from a general magnitude processing deficit rather than a deficit limited to quantity processing. In addition, Brannon, Suanda, and Libertus (2007) found that temporal and number discrimination ability follows the same developmental trajectory in infants, further highlighting the close link between time and number. The exact mechanism by which temporal processing would support mathematical computations is not clear, but Kramer et al. (2011) proposed that temporal processing is inherently spatial in nature, thereby supporting numerical representations on the mental number line. In sum, processing of space and time has been tied to mathematical development and linked to DD. However, it is not currently known whether these dimensions predict different mathematical abilities. That is the topic of the current study. The current study Our objective was to gain a better understanding of the cognitive underpinnings supporting mathematical development during childhood. To date, no study has investigated whether processing of time, space, and number predicts mathematical skill when including previously known predictors. An additional objective was to explore whether time, space, and number all predict the same mathematical abilities or whether they differ.

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We assessed the relative contributions of WM, executive functions, processing speed, and reasoning on different aspects of mathematics in a sample of second and fourth graders. These aspects consisted of written multidigit calculation, equations with an unknown operand (e.g., ‘‘Solve for x”), and arithmetic fact retrieval (i.e., calculation fluency). We assessed the relative contribution of magnitude processing abilities pertaining to space, time, and number to investigate whether these skills contributed above and beyond domain-general skills. The importance of basic number processing skills, such as digit comparison and non-symbolic number comparison, for mathematical development is disputed; by incorporating predictors of other dimensions of magnitude into our model, we may deepen our knowledge of how the ANS contributes to mathematics. Therefore, we used hierarchical multiple regression modeling to investigate the relative importance of these cognitive abilities. Although the current approach was mainly explorative, findings from previous research justified the following hypotheses: 1. In line with previous research, executive processing should predict performance on arithmetic fact retrieval (Kaufmann, 2002; LeFevre et al., 2013) and possibly on written calculation and equations as well (LeFevre et al., 2013). Visuospatial WM (e.g., Swanson, 2006) and verbal WM ability (e.g., Andersson, 2008) will predict performance on written calculation. 2. Number processing skills should predict multidigit calculation and arithmetic fact retrieval (Träff, 2013). Findings by Kibbe and Feigenson (2015) showed that preschoolers can solve these types of problems intuitively using the ANS; thus, we hypothesized that ANS acuity would predict performance on this task. 3. Spatial ability was predicted to contribute to performance on multidigit calculation and equations (Hegarty & Kozhevnikov, 1999). Method Participants The participants were recruited from 15 different schools and consisted of 133 Swedish schoolchildren (53 boys and 80 girls, mean age = 9.69 years, SD = 0.90) enrolled in the fourth grade (n = 67, mean age = 10.59 years, SD = 0.38) and second grade (n = 66, mean age = 8.78 years, SD = 0.32). All participants had Swedish as their primary language and had normal or corrected-tonormal vision and normal color vision. Parents gave informed consent for their children to participate. The study was approved by the local ethics committee. We excluded children with a history of neurologically based impairments such as attention deficit/hyperactivity disorder (ADHD) and other known learning disabilities (e.g., dyslexia). Materials Assessment of mathematical ability Multidigit calculation. This test consisted of eight addition and eight subtraction problems written in Arabic numerals that became progressively more difficult (e.g., 56 + 47, 545 + 96). Children needed to solve as many problems as possible within 10 min, and they had only paper and pencil at their disposal. The total number of problems to solve was 16, and each correctly solved problem yielded 1 point. Arithmetic equations. This test contained 12 equations to be solved. The equations were written as arithmetic problems in Arabic notation. The problems involved subtraction, addition, and multiplication where one of the operands was missing (e.g., 41 + ___ = 53). Children needed to solve as many of these equations as possible within the allotted time (7 min). Each correctly solved problem yielded 1 point for a maximum score of 12. Arithmetic fact retrieval. The ability to quickly retrieve arithmetical facts from long-term memory was measured using a computer-based task consisting of 36 problems. The problems in the task were easy

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to solve (e.g., 3 + 4) so that the children would be prompted to use retrieval strategies rather than needing to rely on calculation procedures. Each problem was presented individually on a computer screen, and the participants were instructed to answer verbally as quickly as possible. Response times (in milliseconds) were recorded by the experimenter using the SuperLab environment. The number of problems answered correctly within 3 s was used as a dependent measure (cf. Russell & Ginsburg, 1984). Domain-general cognitive abilities General intelligence. Raven’s Standard Progressive Matrices (RPM; Raven, 1976) was administered as a measure of general intelligence. Because of the extensive test battery administered to the children, only Sets B, C, and D were used to measure general intelligence. The raw scores were used in the regression analyses. Processing speed. This task was administered on two separate sheets of A4 paper (common letter-sized paper), where 30 strings of ‘‘XXX” (Arial 22-point font) were printed in different colors and in two separate columns. Children were instructed to name the color in which the strings were printed as fast as possible without making any errors. A stopwatch was used to measure the total response time used as the performance measure. The combined response times for the two sheets of paper were used as a measure of speed of access to semantic information in long-term memory (Temple & Sherwood, 2002). Central executive. Capacity of the central executive was measured in terms of shifting ability. This was assessed using a paper-and-pencil version of the Trail Making Test (van der Sluis, de Jong, & van der Leij, 2004), which was composed of two conditions. The first condition (A) contained 22 circles, each with a digit, whereas the second condition (B) also contained 22 circles but with either a digit or a letter. In Condition A, the task was simply to draw a line between the circles in ascending order as quickly as possible. In Condition B, the participants were told to draw the line and connect the circles in ascending order once again, but this time in alternating order (1–A–2–B–3–C, etc.) and as fast as possible. Time (in seconds) to complete each condition was used as dependent measure. Shifting ability was assessed by subtracting the completion time of Condition A from that of Condition B. Visuospatial working memory. The computerized visuospatial WM task consisted of a matrix of 2.5  2.5-cm squares. The initial matrix consisted of 3  3 empty squares. Two black dots appeared in one of the squares, and the participants were asked to estimate whether these dots were of equal size and to press the ‘‘*” key if they were equal or the ‘‘A” key if they were not. After 3 s, two additional dots appeared in another square while the former dots were still visible. Once again, the participants needed to decide whether these two dots were of equal size. While carrying out this task, they concurrently needed to remember in which squares these dots appeared. The matrix disappeared after a given sequence of dots had been presented. Participants then needed to mark their answers in corresponding squares on a piece of paper containing an identical but empty matrix. The number of squares in each matrix increased and the number of dots increased, making each subsequent trial more difficult. The initial matrix of 3  3 squares had two squares with black dots and, thus, a trial of span size 2. The subsequent matrix had 3  4 squares and black dots appeared in three squares, giving a span size of 3. The task became progressively more difficult, and the final span level had 7 dot pairs to be remembered. For each span size, two matrices were presented to the participants. The total number of correctly recalled dot pairs was used as the dependent variable. Verbal working memory. Three-word sentences were read out loud to the participants, and the initial task was to make a judgment of whether the sentence made semantic and syntactic sense. Thus, the participants were to respond ‘‘yes” if the sentence made sense or ‘‘no” if the presented sentence was absurd. Participants were instructed to try to remember the first word in each sentence. After they answered ‘‘yes” or ‘‘no,” the next sentence was presented. The first span size level was 2, meaning that the participants were read two sentences, after which the participants needed to recall, in correct serial order, the target words. The span size ranged from 2 to 5, with two trials for each span size,

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and all span sizes were presented to the participants. The total number of correctly recalled words was the dependent measure in this task. Half of the sentences made sense and half were absurd. Number processing tasks Symbolic number comparison. This task consisted of two subtasks: a one-digit task and a two-digit task. In the one-digit task, two numerals (1–9) were simultaneously displayed on the computer screen. The task was to decide which of the two numerals was the numerically larger one and to respond with either ‘‘A” (corresponding to the left numeral) or ‘‘*” (corresponding to the right-most numeral). Before each trial, a fixation cross was displayed for 1000 ms, after which two digits were presented and remained exposed to the children until they pressed a button. Two numerical distances were used, 1 (e.g., 2–3, 5–6) and 4–5 (e.g., 1–6, 4–9, 3–7), and each pair was presented twice, resulting in a total of 32 trials. Response times for correct responses and response accuracy were used as dependent measures. The two-digit task was essentially identical to the one-digit condition except that this time the stimuli consisted of two-digit pairs rather than one-digit pairs. The numerical distance between the pairs was either 1 or 5 (e.g., 21–22, 46–47, 31–36, 54–59). Non-symbolic number discrimination. Individual ANS acuity was measured using a number discrimination task. Participants saw two intermixed arrays of blue and yellow dots for 800 ms, after which they were asked to determine which color was more numerous and subsequently to press a color-coded key on the computer keyboard. This task was identical to the one used in Halberda et al. (2008). The task contained five practice trials followed by 75 test trials. The arrays contained between 5 and 16 dots, and the ratio of colors varied among four ratio bins (1:2, 3:4, 5:6, and 7:8). Half of the trials contained more blue dots, and the other half contained more yellow dots. Half of the trials were controlled for surface area so that the total blue and yellow surface area was equal, and the dots varied in size to ensure that numerosity was the critical aspect of the task. We determined each participant’s Weber fraction (w) by fitting psychophysical model to the data as a measure of ANS acuity. Spatial and temporal processing Spatial transformation. A mental rotation task was used to assess spatial transformation ability. This paper-and-pencil test involved two sets of stimuli with two subtests. The first subtest contained alphabetic letters as stimuli, and the second subtest consisted of cube figures. Each subtest contained 16 items, where the reference was located on the left side accompanied by four comparison stimuli located on the right side adjacent to the target. The comparison stimuli always consisted of two ‘‘correct” and two ‘‘incorrect” items. The primary task was to identify the two matching items and to respond by marking them with a pencil. The incorrect items were visually mirrored instances of the correct target. All comparison stimuli were rotated in the picture plane and in one of six rotation angles: 45°, 90°, 135°, 225°, 270°, or 315°. Participants needed to mark both correct comparison stimuli to obtain 1 point for each item, yielding a maximum score of 16 for each subtest and 32 for the entire test. The time limit was 2 min for the letter condition and 4 min for the figures condition. Spatial visualization. Spatial visualization was measured using a paper-folding task containing 20 items. Each item involved the visual presentation of a square piece of paper being folded a given number of times followed by a hole being punched through the paper, thereby piercing all the layers of the paper. The task was to imagine how this piece of paper would look when unfolded again. Beneath the folded paper, the participants were given five alternatives, one of which was correct. Participants provided answers by marking an alternative using a pencil. The test items became progressively more difficult, and difficulty was manipulated by increasing the number of times the paper was folded. The simplest items were folded once, and the hardest items were folded twice. Participants had 10 min to complete the test. Time processing. The processing of time was measured using a time discrimination task. The task was to estimate which of two visually presented stimuli was presented the longest. Participants were presented with a reference stimulus (a red ball) centered on the screen on a white background. The reference stimulus presentation lasted for 3000 ms, followed by a blank screen for 500 ms, after which a

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target stimulus (a blue ball) appeared centered on the screen. After the target stimulus disappeared, a response screen followed prompting a response. The reference was always presented for 3000 ms and before the target, whereas the target stimuli duration ranged from 1500 to 6000 ms. Participants pressed either the ‘‘a” key (marked with red) or the ‘‘*” key (marked with blue). The ratios between the reference and the targets were such that they corresponded to the Weber fractions found in other magnitude dimensions (e.g., Halberda et al., 2008). Thus, all trials belonged to four different bins corresponding to specific ratios. The four ratio bins were 1:2, 3:4, 4:5, and 5:6 across 60 test trials, where each participant’s w could be determined by fitting a psychophysical model to the estimation ability. The test included four practice trials, and the participants were asked not to use any counting strategies, such as sub-vocal counting, during the task. Procedure The study was conducted over two sessions, one group session and one individual session, each lasting approximately 120 min, which included a mid-session break. All testing was completed within 1 month. Instructions were read aloud from a printed manuscript, and all tests were administered in the same order for all study participants. Computer-based tasks were run on a laptop using SuperLab PRO 4.5. During the group session, the following tasks were administered: RPM, assessment of mathematical ability, mental rotation task, and the paper-folding task. All of the other tests were performed individually. Results Prior to data analysis, intra-individual trials were examined to remove outliers; response times (RTs) < 200 ms were removed, as were RTs > 2.5 SD of the individual within a test. When analyzing the multiple regression models, we ensured that, for each model, multicollinearity was not an issue. For all models, the variance inflation factor (VIF) was always below 1.78 and tolerance was always above 0.56. Means and standard deviations for all tasks and the correlations between them can be found in Table 1. To investigate whether the relative contribution of spatial processing, temporal processing, and number processing goes above and beyond domain-general abilities, we performed four multiple regression analyses. The first multiple regression analysis used a composite measure of mathematical ability as a dependent variable in which we z-transformed all of the scores from each mathematical task. Three additional multiple regression models were created, with one model corresponding to each mathematical task. The order in which the blocks were added into the models corresponded to the amount of prior support in the literature identifying them as important predictors of mathematical achievement. For each model, we inserted the same five blocks. Age was included as a separate block and predictor to account for mathematics experience, after which domain-general skills were added in the second block given their well-established role in supporting mathematical achievement. The third block consisted of number processing skills, and the fourth and fifth blocks consisted of spatial processing and temporal processing. Temporal processing was added last due to the absence of prior support. Hierarchical regression model of overall mathematical ability Table 2 summarizes the results of the multiple regression analysis of the composite score of mathematical ability. Age accounted for 30% of the variation in mathematical ability, R2 = .30, F(1, 131) = 54.92, p < .001. Adding the domain-general block added an additional 26%, DR2 = .26, Fchange(5, 126) = 14.62, p < .001. The number processing block accounted for only 3%, DR2 = .03, Fchange(2, 124) = 4.77, p = .010, and the spatial processing block accounted for 2%, DR2 = .02, Fchange(2, 122) = 3.18, p = .045. The temporal processing block did not add significantly to the model. The complete model explained a total of 61% of the variance, R2 = .61, F(11, 121) = 17.01, p < .001. Analysis of individual predictors of mathematical ability is summarized in Table 2. Age accounted for 6% of unique variance. Among the domain-general skills, general intelligence contributed 2% of unique variance and shifting ability con-

Task

M

SD

2

3

4

5

6

1. Age (in years) 2. Overall math score (z-score) 3. Multidigit calculation 4. Arithmetic equations 5. Arithmetic fact retrieval 6. General intelligence 7. Processing speed (RAN) 8. Trail making 9. Visuospatial WM (dot matrix) 10. Verbal WM (listening span) 11. Symbolic number comparison 12. Non-symbolic number discrimination (w) 13. Mental rotation 14. Paper folding 15. Time discrimination (w)

9.69 0.07 5.15 5.87 9.10 22.82 52.86 91.72 17.98 20.50 1432 0.51 12.55 10.29 0.39

0.97 0.90 2.70 4.01 7.33 5.73 12.64 50.38 5.74 4.76 424 0.56 5.99 3.37 0.29

.54 –

.52 .85 –

.50 .93 .75 –

.49 .94 .70 .81 –

.10 .43 .42 .41 .37 –

Note. Significant correlations are in bold (p < .05). RAN, rapid automatized naming.

7

8 .23 .41 .33 .41 .37 .22

9 .23 .46 .39 .47 .41 .25 .25

–

10 .31 .36 .32 .36 .31 .15 .22 .19

–

11

.21 .22 .22 .20 .19 .16 .13 .08 .12

–

12

.45 .48 .39 .40 .49 .05 .46 .28 .18 .08

–

13

.01 .18 .15 .19 .15 .17 .24 .02 .24 .08 .05

–

14

.34 .42 .43 .59 .52 .39 .34 .42 .37 .11 .36 .14

–

15

.20 .42 .39 .43 .36 .55 .23 .15 .25 .14 .15 .24 .46

–

.16 .35 .40 .32 .28 .37 .25 .26 .14 .01 .18 .25 .29 .32

– –

K. Skagerlund, U. Träff / Journal of Experimental Child Psychology 143 (2016) 85–101

Table 1 Descriptive statistics and correlations among the tasks used in the study.

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Table 2 Hierarchical multiple regression analysis of overall mathematical ability. B

ß

t

p

pr2

Overall mathematical ability Step 1 Age

.50

.54

7.41

<.001

.29

Age General intelligence (RPM) Trail making Verbal WM Visuospatial WM Processing speed (RAN)

.35 .04 .00 .01 .02 .01

.38 .27 .24 .04 .11 .17

5.80 4.31 3.77 0.70 1.78 2.70

<.001 <.001 <.001 .486 .078 .008

.12 .07 .05 .00 .01 .03

Age General intelligence (RPM) Trail making Verbal WM Visuospatial WM Processing speed (RAN) Symbolic number comparison Non-symbolic number discrimination (w)

.29 .04 .00 .01 .02 .00 .00 .16

.31 .28 .23 .05 .10 .08 .21 .10

4.57 4.47 3.57 0.79 1.50 1.08 2.87 1.57

<.001 <.001 .001 .434 .137 .284 .005 .118

.07 .07 .04 .00 .01 .00 .03 .01

Age General intelligence (RPM) Trail making Verbal WM Visuospatial WM Processing speed (RAN) Symbolic number comparison Non-symbolic number discrimination (w) Mental rotation Paper folding

.27 .03 .00 .01 .01 .00 .00 .13 .02 .02

.29 .20 .19 .05 .06 .06 .18 .08 .15 .08

4.26 2.82 2.99 0.85 0.90 0.91 2.46 1.33 1.97 1.04

<.001 .006 .003 .397 .372 .367 .016 .187 .052 .302

.06 .03 .03 .00 .00 .00 .02 .01 .01 .00

Age General intelligence (RPM) Trail making Verbal WM Visuospatial WM Processing speed (RAN) Symbolic number comparison Non-symbolic number discrimination (w) Mental rotation Paper folding Time discrimination (w)

.27 .03 .00 .01 .01 .00 .00 .12 .02 .02 .13

.27 .19 .19 .06 .06 .06 .18 .08 .15 .07 .04

4.20 2.61 2.85 0.92 0.91 0.86 2.40 1.17 1.95 0.97 0.64

<.001 .010 .005 .359 .363 .389 .018 .244 .053 .332 .527

.06 .02 .03 .00 .00 .00 .02 .00 .01 .00 .00

Step 2

Step 3

Step 4

Step 5

Note. R2 = .30 for Step 1 (p < .001); DR2 = .26 for Step 2 (p < .001); DR2 = .03 for Step 3 (p = .010); DR2 = .02 for Step 4 (p = .045); DR2 = .01 for Step 5 (p = .527). pr2, squared part correlations (represents the unique contribution for each predictor).

tributed an additional 3% of unique variance. Number processing skills contributed as a block, but only symbolic number comparison contributed uniquely with 2% of the variance. Spatial abilities also explained unique variance, but only mental rotation (1%) and not paper-folding.

Hierarchical regression model of arithmetic fact retrieval Table 3 summarizes the results of the multiple regression analysis of each mathematical aspect. Age accounted for 24% of the variation in performance on arithmetic fact retrieval, R2 = .24, F (1, 131) = 41.78, p < .001. Adding domain-general abilities added an additional 20%, DR2 = .20, Fchange(5, 126) = 8.75, p < .001. The number processing block accounted for 5%, DR2 = .05,

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K. Skagerlund, U. Träff / Journal of Experimental Child Psychology 143 (2016) 85–101 Table 3 Hierarchical multiple regression analysis of specific aspects of mathematics. B

ß

t

pr2

p

Arithmetic fact retrieval Age General intelligence (RPM) Trail making Verbal WM Visuospatial WM Processing speed (RAN) Symbolic number comparison Non-symbolic number discrimination (w) Mental rotation Paper folding Time discrimination (w)

1.78 0.23 0.02 0.07 0.05 0.02 0.00 1.06 0.19 0.09 0.02

.24 .18 .15 .05 .04 .03 .25 .08 .15 .04 00

3.09 2.20 2.06 0.67 0.52 0.42 3.06 1.15 1.79 0.50 0.01

.003 .030 .042 .503 .605 .677 .003 .254 .075 .622 .993

.04 .02 .02 .00 .00 .00 .04 .01 .01 .00 .00

Multidigit calculation Age General intelligence (RPM) Trail making Verbal WM Visuospatial WM Processing speed (RAN) Symbolic number comparison Non-symbolic number discrimination (w) Mental rotation Paper folding Time discrimination (w)

0.89 0.10 0.01 0.05 0.04 0.01 0.00 0.15 0.00 0.05 1.49

.32 .22 .15 .08 .08 .03 .12 .03 .00 .07 .16

4.21 2.63 2.05 1.19 1.07 0.41 1.44 0.44 0.02 0.81 2.20

< .001 .010 .043 .236 .287 .679 .152 .660 .987 .332 .030

.07 .03 .02 .01 .01 .00 .01 .00 .00 .00 .02

Arithmetic equations Age General intelligence (RPM) Trail making Verbal WM Visuospatial WM Processing speed (RAN) Symbolic number comparison Non-symbolic number discrimination (w) Mental rotation Paper folding Time discrimination (w)

1.08 0.10 0.02 0.03 0.04 0.03 0.00 0.12 0.14 0.12 0.11

.26 .14 .22 .04 .06 .10 .07 .08 .21 .10 .01

3.65 1.87 3.12 0.61 0.91 1.39 0.92 1.17 2.60 1.29 0.12

< .001 .064 .002 .541 .367 .167 .362 .244 .010 .201 .908

.05 .01 .04 .00 .00 .01 .02 .00 .02 .01 .00

Note. pr2, squared part correlations (represents the unique contribution for each predictor).

Fchange(2, 124) = 6.30, p = .002. Neither spatial processing nor temporal processing added significantly to the model. The entire model accounted for 51% of the total variance, R2 = .51, F(11, 121) = 11.27, p < .001. Individual predictors of arithmetical fact retrieval, together with the other aspects of mathematics, can be found in Table 3. Age accounted for 4% of unique variance, whereas general intelligence and shifting ability both accounted for 2% of unique variance each. An examination of number processing skills, which as a block contributed significantly to the model, disclosed that ANS acuity did not predict arithmetic fact retrieval but that symbolic number comparison accounted for 4%. Hierarchical regression model of multidigit calculation Age accounted for 27% of the variation in performance on multidigit calculation, R2 = .27, F(1, 131) = 48.95, p < .001 (see Table 3). Domain-general abilities added an additional 20%, DR2 = .20, Fchange(5, 126) = 9.71, p < .001. The number processing block did not contribute significantly to the model, and neither did the spatial processing block (ps > .05). However, temporal processing explained an additional 2% of the variation in the model, DR2 = .02, Fchange(1, 121) = 4.84, p = .030. The entire model accounted for 51%, R2 = .51, F(11, 121) = 11.43, p < .001. Analysis of individual predictors

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indicated that age explained 7% of unique variance. Regarding the domain-general skills, general intelligence and shifting ability once again contributed with unique variance of 3% and 2%, respectively. Surprisingly, number processing skills did not contribute to the model, whereas temporal processing ability did. Temporal processing accounted for 2% of unique variance. Hierarchical regression model of arithmetical equations For arithmetical equations (see Table 3), age accounted for 25% of the variation, R2 = .25, F(1, 131) = 43.23, p < .001. Including the block containing domain-general cognitive abilities accounted for an additional 27%, DR2 = .27, Fchange(5, 126) = 14.04, p < .001. Number processing did not add significantly to the model (p > .05), but spatial processing explained an additional 4%, DR2 = .04, Fchange(2, 122) = 5.48, p = .005. The model in its entirety explained 57% of the variation in arithmetical equations, R2 = .57, F(11, 121) = 14.58, p < .001. Age as an individual predictor accounted for 5% of unique variance. Among the domain-general cognitive abilities, shifting ability once again contributed significantly with 4% of unique variance. Only mental rotation ability accounted for additional unique variance (3%).

Discussion The objective of the current study was to examine whether processing of time, space, and number predicts mathematical skill when including previously known predictors of mathematical achievement. If it did, a second objective was to explore whether time, space, and number all predict the same mathematical abilities. The following discussion is divided into two separate parts; the first part elaborates on the findings with respect to domain-general cognitive abilities, whereas the second part focuses on how magnitude processing may relate to mathematical achievement. Contribution of domain-general cognitive abilities to mathematical abilities Five domain-general cognitive abilities comprised a domain-general block in the regression analyses. When including all predictors in our models, only two of these abilities had any predictive power of mathematical ability in any aspect of math: executive functions (i.e., shifting ability) and general intelligence. It was hypothesized that executive functions would predict arithmetic fact retrieval, written calculation, and arithmetic equations, and it did. It was also a significant predictor of overall mathematical ability. The association between executive functions and overall mathematical abilities is in line with recent research (LeFevre et al., 2013; Meyer et al., 2010), and trail making has been specifically linked to mathematical achievement (Szücs et al., 2014). The association between executive functions and arithmetic fact retrieval is consistent with Kaufmann (2002) and in line with Geary’s (2010) suggestion that deficits in successfully retrieving arithmetic facts may be a result of poor inhibition of irrelevant associations in memory (see also De Visscher & Noël, 2013). As hypothesized, the ability to carry out arithmetic calculations and equations was predicted by trail-making performance. Well-developed executive functions may facilitate the selection of appropriate strategies and allocation of attentional resources during execution of those strategies. The measures of arithmetical ability in our study involve complex multidigit calculations and equations, which plausibly require a significant load on executive functioning due to the multistep nature of these problems. General intelligence predicted all aspects of math in our study, consistent with earlier research linking intelligence to mathematical achievement (de Jong & van der Leij, 1999). Intelligence may be important primarily when a child is about to understand the systematic relationship between the numerals and to understand properties of ordinality and equidistance. Geary (2013) suggested that intelligence would be important while learning the mental number line, after which the role of intelligence is reduced. Our results indicate that the role of intelligence in mathematics continues to be a significant contributor to mathematical abilities in older children. This is in line with a recent study by Murayama, Pekrun, Lichtenfeld, and vom Hofe (2013), who used latent growth modeling in children in fifth to tenth grades and found that intelligence predicted mathematical achievement.

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Working memory abilities are consistently associated with mathematical achievement and have been found to be important during multidigit calculation (e.g., Andersson, 2008; Fuchs et al., 2005). Thus, it was surprising that WM abilities did not predict arithmetic performance on any of our arithmetical tests. In a longitudinal study, Hornung, Schiltz, Brunner, and Martin (2014) found that number sense and WM capacity predicted kindergarteners’ mathematical achievement 1 year later and suggested that WM fosters acquisition of basic number skills but that the contribution of WM decreases as children grow older and instead use fact retrieval strategies during calculations. Thus, the absence of WM contribution to math abilities in the current study may be explained by age and greater reliance on arithmetic facts. Another reason why WM did not contribute may be attributed to the fact that we included executive functions (i.e., shifting) as a predictor in the model, which turned out to be a significant predictor (ß = .19). Measures of WM are not ‘‘pure” measures given that these measures are contaminated with executive processes. Our findings do not suggest that WM is unimportant, but when including other strong predictors in our models that are partially tapping the same processes, such as shifting ability and spatial skills, WM abilities decrease in predictive power. In sum, our findings with respect to domain-general skills and their contribution to mathematical achievement are largely in line with previous research that emphasizes the role of executive abilities (e.g., Fuchs et al., 2010; LeFevre et al., 2013; Meyer et al., 2010; Szücs et al., 2014).

Processing of space, time, and number contributes to mathematical abilities Overall mathematical abilities were predicted by basic number processing, but only by symbolic number comparison. In fact, non-symbolic number discrimination (i.e., ANS acuity) did not significantly contribute to our models in any of the mathematical tasks. This is partly in line with Szücs et al. (2014), who found that number sense variables, both symbolic and non-symbolic, did not relate to mathematical performance in 9-year-old children. We partly replicated this finding insofar as ANS acuity did not add significantly to our models, but our findings are discrepant regarding symbolic number comparison. A tentative explanation is that the number comparison task differed slightly from the one used by Szücs and colleagues. They used a fixed reference number (5) that was not visible on the computer screen. Children were asked to respond whether a visual stimulus, an Arabic numeral between 1 and 9, was higher or lower than 5. Our task was, in essence, the same task as in Szücs and colleagues’ study insofar as it measured basic symbolic number affinity, but our task consisted of two variable numerals simultaneously presented on the computer screen. Thus, in Szücs and colleagues’ study the children held a fixed reference in WM during the symbolic number comparison and needed to decode only one visual numeral on the screen, whereas in our study they needed to attend to and decode two variable numerals. Hence, the added complexity of our task posed additional demands on number processing skills, which may explain why we found that symbolic number comparison predicted mathematical achievement in our sample. The lacking predictive power of ANS acuity may also be explained by the age of the children in the sample. It is likely that the explanatory role of the ANS acuity in predicting mathematical achievement is indirect, which is a notion that has received recent empirical support (Chu, vanMarle, & Geary, 2015). Thus, the ANS plays a role in acquiring basic symbolic competencies and in that way may influence mathematical abilities. Once the symbolic system is mastered, the computational demands of the ANS diminish and the functional role of the ANS recedes and become dormant but may be invoked during unfamiliar operations or during estimations. Basic number processing skills have been suggested to play an important role in aspects of mathematics that are tightly linked to the whole number system such as multidigit calculation and fact retrieval (Fuchs et al., 2010). In our study, however, symbolic number comparison predicted only arithmetic fact retrieval and not multidigit calculation, thereby replicating Träff (2013). This may be explained by the complexity of the calculation problems used in the current study compared with the ones employed by Fuchs et al. (2010). The calculation problems in our study ranged from twodigit operands (e.g., 57 + 42) to four-digit operands (e.g., 4203 + 825), whereas Fuchs and colleagues used only two-digit operands. The added complexity of our calculation problems may also explain why domain-general skills such as executive attention and general intelligence become important in predicting performance. Successful solutions will depend on selecting appropriate procedural

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strategies, switching between operations, storing intermediate results, and monitoring these processes (Geary, 1993; McCloskey et al., 1985). Symbolic number processing predicted arithmetic fact retrieval performance. The problems in this task consisted of only one-digit operands, and the task was designed to evoke automatic retrieval from semantic long-term memory. The connection between symbolic number comparison and fact retrieval fluency could reflect integrity and affinity with the whole number system that enable children to encode arithmetic facts into an arithmetic facts network. Intact executive functions may then contribute to successful retrieval of accurate facts and inhibit inaccurate but associated facts in the semantic network. We hypothesized that ANS acuity would predict proficiency at solving arithmetic equations given that Kibbe and Feigenson (2015) found that young children could rely on the ANS to compute nonsymbolic ‘‘solve for x”-type tasks prior to formal math education. However, we failed to find that the ANS could predict the ability to solve arithmetic equations. Given that our task contained formal equations, it is likely that successful problem solving relies heavily on affinity with formal numbers and operators. This setup is vastly different from the non-symbolic equations that Kibbe and Feigenson used, which instead prompted a more intuitive aspect of number sense. Thus, ANS acuity served as a scaffold during intuitive non-symbolic equations in Kibbe and Feigenson’s study, but our study suggests that this ability could not be used and translated to formal problem solving in older children. Spatial abilities have been associated with mathematical achievement (e.g., Szücs et al., 2014), and Gunderson et al. (2012) suggested that spatial abilities help children to develop a linear mental number line and enhance proficiency with the symbolic number system. We found that mental rotation ability predicted performance on overall mathematical ability (b = .15, p = .053) and specifically on arithmetic equations (b = .21, p = .010). The mechanism by which mental rotation ability supports solving arithmetic equations is not straightforward, but one tentative interpretation is that children with strong spatial skills can use visuospatial cognitive resources to mentally imagine and move and operate on operands of a mathematical expression that in turn makes the problem more intuitive to solve. One caveat is that performance of mental rotations is not devoid of other domain-general processes such as visuospatial WM and executive attention. Mental rotation ability predicted mathematical performance even when including shifting ability in the model, suggesting that the spatial component of mental rotation contributes to math abilities. Our findings indicate that spatial abilities are an important predictor of mathematical achievement in 8- to 10-year-old children, which corroborates the notion of a developmental shift in children in the capacities on which they rely. Children shift from relying on verbal capacities in second grade to increasingly relying more on visuospatial cognitive abilities by third grade (Meyer et al., 2010), which seems to be consistent with the idea that mathematics curricula in which older children and adolescents partake depend considerably on visuospatial abilities (Reuhkala, 2001). By studying a group of second and third graders, Meyer et al. (2010) noted that executive functioning and phonological loop performance predicted achievement in second grade, whereas visuospatial abilities predicted mathematical achievement in third grade. Meyer and colleagues argued that this shift can be attributed to neurocognitive maturation and practice. With practice, mathematics processing will shift to more specialized mechanisms in posterior parietal cortex, including the intraparietal sulcus (IPS) and angular gyrus (Meyer et al., 2010). These cortical loci are pivotal during number processing and arithmetic fact retrieval (Ansari, 2008), whereas prefrontal cortex is involved in executive control and attentional processing during complex calculations. Moreover, spatial abilities are also subserved by posterior parietal cortex, which would explain the increased reliance on visuospatial processing in conjunction with number processing. The shift may be part of an incremental integration of number onto a spatial representation (Holloway & Ansari, 2008). Processing of time has been implicated in children with DD (Skagerlund & Träff, 2014; Vicario et al., 2012) and predicted mathematical intelligence (Kramer et al., 2011), which is why we were interested in seeing whether this ability could predict overall mathematical achievement. Our analyses indicate that time processing did not contribute to overall mathematical abilities, but it did predict performance on multidigit calculations (b = .16, p = .030). The reason why the ability to represent time would be associated with mathematical abilities is that temporal processing may share neurocognitive resources with other analogue magnitudes, such as space and number, in the parietal cortex of the brain (cf. Walsh, 2003). Kramer et al. (2011) suggested that the link between mathematical intel-

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ligence and temporal processing can be traced to an underlying spatial ability, much like how numerical representations are ordered spatially on a mental number line. Indeed, authors have put forward the notion of a mental time line (e.g., Bonato, Zorzi, & Umiltà, 2012), which refers to the conceptualization that time is represented along a spatial continuum and that access and accurate manipulation of temporal representations require visuospatial attention. Thus, temporal processing proficiency may be an index of how well an individual can reason and manipulate magnitudes along an ordered magnitude dimension. A key characteristic of time, as with numbers, is ordinality. The ability to represent ordered relations among units of cardinal value is a core component of the numerical system. In fact, Rubinsten and Sury (2011) found abnormal ordinal processing in adults with DD, and they went on to argue that there may be two separate cognitive systems of cardinality and ordinality. Related research found that children with DD showed more impaired WM ability than typically developing children, but only when they were required to recall the items in order (Attout & Majerus, 2015). In fact, when they simply needed to retain and recall information irrespective of order, the children with DD performed on par with their peers. These authors concluded that children with DD had a general deficit in explicit processing of ordinal information. Temporal processing may in this manner be related to mathematical abilities in that it reflects the ability to represent ordinal magnitudes along a mental spatiotemporal line, much like the mental number line. This ability may be subserved by the IPS in the parietal cortex, and neuroimaging data give empirical support to the notion that the IPS represents not only magnitude but also ordinality (Fias, Lammertyn, Caessens, & Orban, 2007). Time processing ability only predicted calculation performance in the current study. One tentative interpretation is that the written calculation task requires more prolonged and effortful processing of numerals, which requires access to the underlying spatiotemporal number line as opposed to the rather automatized and fast-paced fact retrieval task and transient, but conceptually difficult, equation task. Taken together, our results indicate that space, time, and number all contribute to mathematical abilities beyond domain-general cognitive abilities but to different aspects of mathematics. Conclusions Our multiple regression analyses indicate that domain-general skills are strong predictors of mathematical abilities. Findings include that executive functions, in this case shifting ability, and general intelligence predicted all aspects of mathematics included in the study and predicted overall mathematical ability. Working memory capacity did not contribute significantly to our models, which may be attributed to the inclusion of other strong predictors such as executive abilities and spatial abilities. Indeed, spatial ability, in terms of the capacity to mentally imagine the rotation of pictorial objects, was a strong predictor of achievement. With respect to basic number processing skills, we replicated earlier research showing that non-symbolic number processing seems to lose predictive power of mathematical abilities once the symbolic system is acquired. Novel findings also include the fact that time discrimination ability was tied to calculation ability. Therefore, we conclude that magnitude processing in general contributes to mathematical abilities. Further research is warranted to verify the relationship between these magnitude dimensions and mathematical abilities and the mechanisms by which these abilities interact. Acknowledgment This research was supported by a grant from the Swedish council for working life and social research (2010-0078) awarded to Ulf Träff and was approved by the regional ethics committee in Linköping, Sweden (2011/58-31). References Andersson, U. (2008). Working memory as a predictor of written arithmetical skills in children: The importance of central executive functions. British Journal of Educational Psychology, 78, 181–203. Ansari, D. (2008). Effects of development and enculturation on number representation in the brain. Nature Reviews Neuroscience, 9, 278–291.

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