Visuospatial pathways to mathematical achievement

Learning and Instruction 62 (2019) 11–19

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Learning and Instruction journal homepage: www.elsevier.com/locate/learninstruc

Visuospatial pathways to mathematical achievement Winnie Wai Lan Chan, Terry Tin-Yau Wong

T

∗

Department of Psychology, The University of Hong Kong, Hong Kong

1. Introduction How well children perform in mathematics is in part related with their visuospatial working memory – i.e., the capacity to store and manipulate visuospatial information. In particular, better visuospatial working memory is associated with higher mathematical performance (Henry & MacLean, 2003; Kyttälä, Aunio, Lehto, VanLuit, & Hautamaki, 2003). Yet, little is known about the underlying mechanism. In what ways does visuospatial working memory support mathematical performance? Understanding the specific pathways between the two can help educators to identify ways to improve mathematical performance among children with poor visuospatial working memory. This study proposed the first pathway model to clarify the relation between visuospatial working memory and mathematical achievement. 1.1. Visuospatial working memory and mathematical achievement Visuospatial working memory plays an important role in mathematical achievement (Berg, 2008; Bull, Espy, & Wiebe, 2008; Simmons, Singleton, & Horne, 2008). It is significantly correlated with mathematical performance throughout the school years (Gathercole & Pickering, 2000 for 7-year-olds; Maybery & Do, 2003 for 10-year-olds; Jarvis & Gathercole, 2003 for 11- and 14-year-olds; Reuhkala, 2001 for 15- to 16-year-olds). In particular, children who perform poorly in mathematics appear to have poorer visuospatial working memory than those who do well in mathematics (Henry & MacLean, 2003; Kyttälä et al., 2003; Maybery & Do, 2003). While mathematically gifted adolescents tend to have better visuospatial working memory (Dark & Benbow, 1990; Leikin, Paz-Baruch, & Leikin, 2013), children with mathematical difficulties tend to have a deficit in it (Kyttälä, 2008; McLean & Hitch, 1999; Van DerSluis, Van DerLeij, & DeJong, 2005). Indeed, one subtype of dyscalculia is caused by a specific deficit in visuospatial working memory (Geary, 1994). Yet, the specific role of visuospatial working memory in mathematical cognition remains unclear. In adults, visuospatial working memory is related with multi-digit calculation (Heathcote, 1994) and complex algebraic and geometric problem solving (Reuhkala, 2001). In children, strong visuospatial working memory is associated with superior performance in counting tasks (Kyttälä et al., 2003; Lefevre et al., 2013),

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number line estimation (Geary, Hoard, Byrd-craven, & Nugent, 2007; Thompson, Nuerk, Moeller, & Cohen Kadosh, 2013), and problem solving (nonverbal: Lefevre et al., 2010; Rasmussen & Bisanz, 2005; verbal: Meyer, Salimpoor, Wu, Geary, & Menon, 2010). Children with deficits in visuospatial working memory have difficulties in written calculation (Szucs, Devine, Soltesz, Nobes, & Gabriel, 2013). Notwithstanding these findings, there is a lack of systematic investigation into the specific linkage between visuospatial working memory and mathematical performance. One exception is the study by Krajewski and Schneider (2009a, 2009b), in which visuospatial working memory at 5 years of age predicted mathematical achievement at 8 years of age through the mediation of quantity-number competence at 6 years of age. Yet, what quantity-number competence referred to was not clearly explained. Such competence was measured by a variety of tasks such as number conservation, magnitude comparison, matching between magnitude and number word, and word problem solving. To better understand what specifically bridge(s) the gap between visuospatial working memory and mathematics achievement, we need to identify the underlying skill(s) shared among these tasks. Based on the natures of the tasks, it seems that they might tap into at least two different underlying skills: magnitude representation and problem representation. In the following, we reviewed the findings on the potential mediating roles of magnitude representation and problem representation in relating visuospatial working memory and mathematical achievement, which led to the proposed pathway model in the present study. 1.2. Magnitude representation as a potential mediator Better visuospatial working memory predicts more accurate computation (Cirino, Tolar, Fuchs, & Huston-Warren, 2016; Dehaene, 1992; Seron, Pesenti, Noël, Deloche, & Cornet, 1992; Van DerVen, Van DerMaas, Straatemeier, & Jansen, 2013). One suggestion is that visuospatial working memory serves as a mental workspace for mental calculation (Heathcote, 1994). Another possibility is that visuospatial working memory supports representation of number magnitudes, which is visuospatial in nature and required in computation. A neuroimaging study has shown that parietal areas associated with number and magnitude processing are situated near brain regions that support

Corresponding author. Room 6.25, The Jockey Club Tower, Centennial Campus, The University of Hong Kong, Pokfulam Road, Hong Kong. E-mail address: [email protected] (T.T.-Y. Wong).

https://doi.org/10.1016/j.learninstruc.2019.03.001 Received 28 November 2018; Received in revised form 13 February 2019; Accepted 1 March 2019 0959-4752/ © 2019 Elsevier Ltd. All rights reserved.

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W.W.L. Chan and T.T.-Y. Wong

visuospatial processing, and that damage to these parietal areas disrupts the ability to form visuospatial representations and construct a mental number line (Zorzi, Priftis, & Umiltà, 2002). Hence, it seems that magnitude representation and visuospatial processing are somewhat related. Indeed, the triple-code theory of mathematical cognition (Dehaene, Piazza, Pinel, & Cohen, 2003; Dehaene, Spelke, Pinel, Stanescu, & Tsivkin, 1999) suggests that magnitude representation is visuospatial in nature. In particular, number magnitudes are represented in the form of a mental number line running from left to right (Dehaene, Bossini, & Giraux, 1993). This mental number line hypothesis has been supported by two phenomena. First, when given two numbers and asked to choose the larger one, both humans (Moyer & Landauer, 1967) and primates (Brannon & Terrace, 1998) would be quicker and more accurate if the numbers are numerically farther apart (e.g., 2 vs. 6) rather than closer (e.g., 2 vs. 3). Second, smaller numbers are responded to faster and more accurately with the left hand and larger ones with the right hand (spatial numerical association of response codes, or SNARC, effect, Bächtold, Baumüller, & Brugger, 1998; Dehaene et al., 1993). The tendency to relate small numerosity to the left and large numerosity to the right is found in infants as well (DeHevia, Girelli, Addabbo, & Cassia, 2014). Hence, it seems that number magnitudes are represented in a visuospatial manner, and the visuospatial representation of numerical magnitude appears to be innate. Given the nature of magnitude representation, visuospatial skills are expected to play a crucial role in processing number magnitudes. Indeed, visuospatial working memory is a unique predictor of number comparison (single and multi-digit numbers) and number writing among elementary school children (Simmons, Willis, & Adams, 2012). In number comparison (single-digit), children with visuospatial disability are slower and less accurate than their peers in the control group (Bachot, Gevers, Fias, & Roeyers, 2005). Moreover, the former group does not show SNARC effect whereas the latter group does (Bachot et al., 2005). Furthermore, while children with specific language impairment (showing a deficit in visuospatial skills) perform more poorly on number comparison tasks than typically developing children (Donlan, Cowan, Newton, & Lloyd, 2007), those with developmental dyslexia (having a specific phonological deficit) perform similarly to their peers (Simmons & Singleton, 2009). Together, these findings seem to suggest that visuospatial skills are required for magnitude representation. Magnitude representation in turn appears to predict computational ability (Lyons, Price, Vaessen, Blomert, & Ansari, 2014; Zhang, Chen, Liu, Cui, & Zhou, 2016), and such relation is supported by a recent meta-analysis (Schneider et al., 2017). There are different proposals concerning the mechanism behind such relation. Booth and Siegler (1996) have suggested that a good grasp of numerical magnitudes can narrow the range of candidate answers when solving arithmetic problems, thus resulting in increasingly accurate performance and errors that are close misses. It also supports advanced counting procedures among children, such as the counting-from-larger strategy in which children add up two numbers by counting up from the larger addend (Siegler, 1996). To do so, the children need to compare the two addends and decide on the larger one, which draws on the understanding of numerical magnitudes. The importance of numerical magnitude representation in computation has been well supported by neuroimaging studies showing that the intraparietal sulcus for the processing of magnitudes in children (Ansari, Garcia, Lucas, Hamon, & Dhital, 2005; Kaufmann et al., 2008; Temple & Posner, 1998) appears to be consistently active during arithmetic tasks (Dehaene et al., 2003; Rivera, Reiss, Eckert, & Menon, 2005; Ruxandra Stanescu-Cosson et al., 2000). Given that magnitude representation appears to be related with visuospatial working memory on one hand and computation on the other hand, it seems possible that the association between visuospatial working memory and mathematical achievement may reflect a

deployment of visuospatial processes to manipulate magnitude representation, which in turn facilitates computation. 1.3. Problem representation as a potential mediator Another possible mediation pathway between visuospatial working memory and mathematical achievement may be related to problem representation in solving word problems. Previous studies have shown that visuospatial skills are probably required in word problem solving. Visuospatial skills at 4 years of age contribute to children's performance in word problems at 5 years of age (Zhang & Lin, 2015). Among the children with mathematical difficulties, those with better visuospatial skills are better problem solvers (Booth & Thomas, 1999). In particular, visuospatial working memory is the key predictor of children's word problem solving (Henry & MacLean, 2003; Zheng, Swanson, & Marcoulides, 2011). Yet, why are visuospatial skills involved in word problem solving? A crucial step in word problem solving is problem representation, which entails comprehending the problem and mathematically interpreting the relations among the problem parts to form a structural representation (Mayer, 1985). The ability to form appropriate problem representations can be reflected by the way in which children visualize the problems. Some children tend to construct visual images that are pictorial in nature (i.e., focusing on the visual appearance of the problem parts); while others tend to construct images that are more schematic in nature (i.e., focusing on the relations between problem parts; Hegarty & Kozhevnikov, 1999; Van Garderen, Scheuermann, & Jackson, 2013). Although both kinds of visual images are visuospatial in nature, only the use of schematic representations was positively correlated to spatial visualization skills, suggesting that children with higher spatial visualization skills tend to produce more schematic, instead of pictorial, representations (VanGarderen, 2006). Because schematic representations capture the most essential elements of the problem (i.e., the relations between problem parts), children who tend to construct schematic representations during word problem solving are better able to understand and visualize the relations between different problem parts, and thus are better problem solvers (Boonen, van derSchoot, vanWesel, deVries, & Jolles, 2013; Hegarty & Kozhevnikov, 1999; Krawec, 2014). Hence, another possible pathway between visuospatial working memory and mathematical achievement is through the deployment of visuospatial processes to build schematic mental representations to solve mathematical problems. Although such relation has been suggested previously (Holmes & Adams, 2006), it has rarely been empirically tested. The present study would fill this gap in the literature. 1.4. Present study Although the linkage between visuospatial working memory and mathematical achievement is somewhat well established, the underlying mechanism remains unclear. In this study, we proposed the first pathway model to clarify the relation between the two. In particular, we asked how children's visuospatial working memory in Grade 1 would predict their mathematical achievement in Grade 2. In the model, visuospatial working memory and mathematical achievement are linked in two pathways: magnitude representation pathway and problem representation pathway. In the magnitude representation pathway, the association between visuospatial working memory and mathematical achievement is mediated by the deployment of visuospatial processes for magnitude representation, which in turn facilitates computation. In the problem representation pathway, the association between the two is mediated by the deployment of visuospatial processes for problem representation in solving mathematical problems. We tested the proposed model using path analysis. 12

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Table 1 Means, standard deviations, range, skewness and kurtosis of the variables. Grade

Variable

Mean

S.D.

Min.

Max.

Skewness

Kurtosis

1

Raven's matrices Word reading Digit naming Back. digit span Math achievement Visuospatial working memory Num. comp. Computation Word problems Math achievement

117.13 51.67 22.81 4.29 560.10 5.14 869.10 46.03 12.74 726.91

12.33 17.54 5.90 1.63 265.96 2.18 205.27 3.76 3.46 251.51

85 4 11.80 1 60 1 457.40 32 2 133

135 103 42.39 9 1000 11 1908.40 50 19 1050

-.20 .08 .80 .71 .39 -.20 1.26 −1.54 -.66 -.36

-.76 -.11 .54 .23 −1.25 -.72 2.70 2.37 -.09 −1.16

2

2. Method

based on reaction time and accuracy were 0.95 and 0.77 respectively. Inverse efficiency score was used as the index for children's performance, which was the mean reaction time divided by the mean accuracy rate (the higher the score, the less efficient the performance).

2.1. Participants Five hundred and sixty-eight first graders (mean age = 7.0 years; 323 boys and 245 girls) were recruited from six elementary schools in Hong Kong in the first phase as part of another large-scaled longitudinal study. Due to school absence or withdrawal, twenty-seven of them did not complete the second phase in Grade 2. The remaining 541 children (mean age = 7.0 years; 306 boys and 235 girls) completed both phases. All children were Chinese-speaking and had normal intelligence (with IQ 85 or above, see Table 1). Children with an IQ of 84 or lower were excluded because of their potential difficulties in understanding the instructions of the tasks. All of them participated with written parental consent.

2.2.2.2. Computation. The computation task consisted of two untimed subtests (addition and subtraction), each containing 25 problems in a columnar format. For each subtest, 10 problems involved two-digit numbers (half requiring carrying-over or borrowing) and 15 involved three-digit numbers (all requiring carrying-over or borrowing). The total number of correct answers represented the final score. Cronbach's α was 0.86. 2.2.2.3. Word problems. Seven arithmetic word problems were developed based on Wong and Ho (2017), which involved addition, subtraction, multiplication, or division. For each problem, the children were asked to write down the number sentence and then solve it. Take the following problem as example: A teacher distributes some coloured pencils evenly among 7 students. Each student receives 2 coloured pencils. How many coloured pencils does the teacher have in total? The children would need to write down the number sentence “7 × 2” and the solution “14”. The number sentence and the solution were scored separately. Each correctly written number sentence scored 1 mark if it required one operation (e.g., 7 × 2), or 2 marks if it required two operations (e.g., 8 + 2–3). For the four questions requiring two operations, 1 mark would be given if the number sentence was partially correct. Each correct solution scored 1 mark. The sum of the scores of the number sentences and solutions indicated the ability to represent word problems. Cronbach's α was 0.67.

2.2. Materials and procedure All children completed assessments once in Grade 1 and again in Grade 2. In Grade 1, they completed a nonverbal intelligence test, a Chinese word reading test, a digit naming test, the Corsi block task, and a backward digit span test. In Grade 2, they completed a number comparison task, a computation test, and a word problem test. In both grades, they completed a grade-appropriate mathematical achievement test. As the data were drawn from a large-scaled longitudinal study, the children also completed other tasks in both grades, which were irrelevant to the study here and thus not reported. All the tasks were administered in a fixed order in each grade. 2.2.1. Independent variable: visuospatial working memory Children's visuospatial working memory was assessed by the Corsi block task (Corsi, 1972). Nine blocks were mounted on a wooden board. In each trial, the experimenter tapped some blocks one by one and the children were asked to repeat the sequence in the backward order. The task started with a span of two blocks, then three blocks, and so on. The longest span contained all nine blocks. There were two trials for each span. The task was terminated when the children failed both trials of the same span. Each correct sequence scored one point. Cronbach's α was 0.75.

2.2.3. Dependent variable: mathematical achievement test (Grade 2) Children's mathematical achievement was measured by the Learning Achievement Measurement Kit 3.0 Mathematics (LAMK 3.0; Hong Kong Education Bureau, 2015), which was the standardized, grade-appropriate mathematical achievement test in Hong Kong. There were 36 items covering the topics in the local mathematics curriculum for Grade 2 children (i.e., number and algebra, measures, shape and space, and data handling, see Appendix 1 for examples). The children completed the test within 45 min. Standard scores were used as an index of the children's mathematics achievement. Cronbach's α was 0.91.

2.2.2. Mediators 2.2.2.1. Number comparison. Children's magnitude representation was assessed by the number comparison task, which was adapted from Rousselle and Noël (2007). In each trial, the children saw two digits (from 1 to 9) on the screen and were asked to choose the larger one by pressing the key on that side. There were 48 trials, half were close pairs (the two digits differing by 1) and half were far pairs (the two digits differing by 3 or 4). The digit pairs were presented in a pseudo-random order. In half of the trials, the larger digits were on the right, whereas in half they were on the left. The children went through four practice trials with feedbacks. Afterwards, no feedback was provided. Cronbach's αs

2.2.4. Control variables Various abilities including nonverbal intelligence (Krajewski & Schneider, 2009a), reading ability (Berg, 2008; Koponen, Aunola, Ahonen, & Nurmi, 2007; Lee, Ng, Ng, & Lim, 2004), processing speed (Krajewski & Schneider, 2009a), and verbal working memory (Berg, 2008; Lee et al., 2004) are related with mathematical performance. To take into account the individual differences in these abilities, we included them as control variables in the pathway model. We also included the mathematical achievement in Grade 1 as a control of the 13

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autoregressive effect of mathematical achievement.

Before conducting a path analysis, we had to handle the missing data. Because 27 children did not complete the assessment in Grade 2, we first conducted the Little's MCAR test (Little, 1988) to see if the data were missing completely at random. The result was significant, (χ2 (55) = 85.235, p = .005), indicating that the data were not missing completely at random. We therefore imputed the missing data using multiple imputations with Fully Conditional Specification implemented by the MICE algorithm with the package “mice” (vanBuuren & Groothuis-Oudshoorn, 2011) in R (R development Core Team, 2008). We obtained 10 imputed datasets and repeated the following analyses for each dataset. Because the results based on the imputed datasets were largely similar to the results obtained after listwise deletion of the missing data, we therefore reported the results using listwise deletion in the following paragraphs for easy interpretation. The data were first standardized and screened for univariate and multivariate outliers. Data points which were 3 S.D. beyond the corresponding means were considered univariate outliers. Multivariate outliers were then screened out using Mahalanobis distance in SPSS. All the outliers (n = 27; 26 univariate outliers1 and 1 multivariate outlier) were excluded from the sample.2 This resulted in a final sample of 514 children with complete data. Table 1 presents the means, standard deviations, ranges, skewness, and kurtosis of the variables, while Table 2 presents the correlations among the variables. The correlation matrix revealed that all the independent (visuospatial working memory) and control (Raven's matrices, word reading, digit naming, backward digit span, Grade 1 mathematical achievement) variables were significantly correlated with mathematical achievement in Grade 2. The correlations ranged from moderate to strong (0.26 < |r|s < 0.60). Moreover, children's visuospatial working memory was significantly correlated with all mediating (number comparison, computation, word problems) (0.20 < |r|s < 0.31) and control variables (0.17 < |r|s < 0.32). To test the two mediation pathways, the data were subjected to a path analysis using the lavaan package (Rosseel, 2012) in R (R development Core Team, 2008), with children's visuospatial working memory being the independent variable and their mathematical achievement in Grade 2 being the dependent variable. The potential mediators formed two different pathways: one through number comparison efficiency and computation (the magnitude representation pathway), while the other through word problems (the problem representation pathway, see Fig. 1). Computation and word problems were also expected to be significantly correlated with each other. Participants' performance in Raven's standard progressive matrices, Chinese word reading, digit naming, backward digit span, and the mathematical achievement in Grade 1 were included in the model as control variables. As the data were significantly skewed (skewness = 765.66, p < .001), the model was bootstrapped 5000 times to ensure the robustness of the results. The bootstrap procedure also enabled us to examine the significance of the two indirect effects. Path models were considered to fit the data if the following criteria were met: non-significant χ2 test, comparative fit index (CFI) > 0.95, root mean square error of approximation (RMSEA) < 0.06, and standardized root mean square residual (SRMR) < 0.08 (Hu & Bentler, 1999). The current model fitted the data well based on these criteria, with χ2 (1) = 449, p = .503, CFI = 1.000, RMSEA = 0.000, SRMR = 0.003. All the paths in the two proposed pathways were significant (for the magnitude representation pathway: visuospatial working memory to number comparison: β = −0.119, p = .005, 95%

2.2.4.1. Raven's standard progressive matrices. Nonverbal intelligence was measured by the abbreviated version (set A to C) of Raven's Standard Progressive Matrices (DeLemos & Raven, 1989) with local norms. There were 36 items, each of which required the children to choose a pattern to fill in the missing part of a matrix. Cronbach's α was 0.88. 2.2.4.2. Chinese word reading. Children's reading ability was assessed by the Chinese word reading subtest from the Hong Kong Chinese Literacy Assessment for Junior Primary School Students, Second Edition (CLA-P (II)) (Ho et al., 2013). Given a list of 100 Chinese words, the children were asked to read aloud as many as possible within 1 min. The score was the number of correct words divided by the time needed and multiplied by 60. Cronbach's α was 0.98. 2.2.4.3. Digit naming task. Children's processing speed was measured by the digit naming task, which assessed children's speed at retrieving phonological numerical representations from long-term memory. The task was adapted from the Second Edition of the Hong Kong Test of Specific Learning Difficulties in Reading and Writing (HKT-P(II); Ho et al., 2007). The children were shown eight rows of five randomlyordered digits (2, 4, 6, 7, and 9) each on A4 paper. They were asked to name the list of 40 digits as fast and as accurately as possible from left to right row by row twice. The score was the average of the time needed in the two trials. 2.2.4.4. Backward digit span. The backward digit span was used to assess children's verbal working memory capacity. They listened to a sequence of digits in each trial and repeated it in the backward order. The task started with a span of two digits, then three digits, and so on. The longest span contained eight digits. There were two trials for each span. The task was terminated when the children failed both trials of the same span. Each correct sequence scored one point. Cronbach's α was 0.64. 2.2.4.5. Mathematical achievement (Grade 1). The Grade 1 version of the LAMK was conducted, which served as a control of the autoregressive effect of mathematical achievement. There were 41 items covering the topics in the mathematics curriculum for Grade 1 children in Hong Kong (i.e., number and algebra, measures, shape and space, and data handling, see Appendix 1 for examples). The children completed the test within 45 min. Standard scores were used as an index of the children's mathematics achievement. Cronbach's α was 0.90. 3. Results We hypothesized that the children's visuospatial working memory in Grade 1 predicted their mathematical achievement in Grade 2 via two pathways: the magnitude representation pathway in which the children's performance in magnitude representation and computation in Grade 2 were the mediators, and the problem representation pathway in which the children's performance in the representation of word problems in Grade 2 was the mediator. Although computation was also involved in word problem solving process, previous studies suggested that errors in word problem solving were more likely to come from errors in problem representation instead of errors in magnitude representation (Lewis & Mayer, 1987; Muth, 1991; Wong & Ho, 2017). Furthermore, in the current path model, the magnitude pathway was also included, and thus any variance of word problem solving that could be explained by numerical magnitude representation should have already been taken into account. Therefore, the paths in the problem representation pathway are believed to be largely capturing problem representation.

1 The univariate outliers had extreme scores on the following variables: 1 for Chinese word reading, 6 for rapid digit naming, 1 for backward digit span, 1 for visuospatial working memory, 7 for number comparison, 10 for computation, and 3 for word problems. Three participants were outliers in two tasks. 2 The results of the analyses excluding the outliers did not differ significantly from the results of the analyses when the outliers were included.

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Table 2 Correlations among the variables. Variable

1. Raven's matrices 2. Word reading 3. Digit naming 4. Back. digit span 5. Math achievement (G1) 6. Visuospatial working memory 7. Num. comp. 8. Computation 9. Word problems 10. Math achievement (G2)

Correlations 1

2

3

4

5

6

7

8

9

10

– .29*** -.24*** .24*** .52*** .31*** -.17*** .32*** .50*** .48***

– -.59*** .23*** .49*** .17*** -.26*** .29*** .48*** .45***

– -.18*** -.39*** -.17*** .34*** -.29*** -.33*** -.34***

– .31*** .30*** -.14** .18*** .26*** .26***

– .31*** -.31*** .44*** .56*** .56***

– -.22*** .20*** .31*** .33***

– -.33*** -.24*** -.29***

– .41*** .50***

– .60***

–

*p < .05, **p < .01, ***p < .001.

Fig. 1. The proposed mediation model between visuospatial working memory and mathematics achievement. The first mediation pathway involves number comparison and computation, while the second pathway involves word problems. For simplicity, control variables are not shown in the figure.

CI = [-0.124, −0.022]; number comparison to computation: β = −0.183, p < .001, 95% CI = [-0.342, −0.120]; computation to mathematical achievement: β = 0.216, p < .001, 95% CI = [0.169, 0.375]; for the problem representation pathway: visuospatial working memory to word problems: β = 0.098; p = .008, 95% CI = [0.024, 0.160]; word problems to mathematical achievement: β = 0.262, p < .001, 95% CI = [0.184, 0.349]). The significance of the two indirect effects were further confirmed through the examination of the confidence intervals of the indirect pathways. The indirect effect through the magnitude representation pathway was .005 (95% CI = [0.001, 0.010]) and the indirect effect through the problem representation pathway was .026 (95% CI = [0.006, 0.045]). Both indirect effects were significant as the 95% confidence interval did not include 0, and both were considered to be in small to medium effect sizes.3 On top of these two mediation pathways, computation was also significantly correlated with word problems at r = .167 (p = .001, 95% CI = [0.032, 0.119]). Besides these major pathways, number comparison was significantly predicted by digit naming speed and mathematical achievement in Grade 1 but not by other variables (see Table 3 for the relevant statistics for the paths); computation was significantly predicted by Raven's matrices and mathematical achievement in Grade 1 but not by other variables; word problems were significantly predicted by Raven's matrices, word reading, and mathematical achievement in Grade 1, but not by backward digit span and digit naming speed; and mathematical

achievement in Grade 2 was significantly predicted by Raven's matrices, word reading, and mathematical achievement in Grade 1, but not by backward digit span and digit naming speed. It should be noted that even after taking the mediators and control variables into account, the relation between visuospatial working memory and mathematical achievement remained significant (β = 0.088, p = .012, 95% CI = [0.017, 0.151]), suggesting that the two proposed mediation pathways did not fully explain the association between children's visuospatial working memory and their mathematical achievement. The variables altogether accounted for 50.5% of the variance in participants' mathematical achievement in Grade 2. To further examine the unique contributions of each of the pathways/variables to the children's mathematical achievement in Grade 2, the path model was repeated with the relevant mediators removed. When either the problem representation pathway or the magnitude representation pathway was removed from the model, the variance in Grade 2 mathematical achievement being accounted for was reduced to 46.7%, suggesting that the each of the pathways accounted for the same amount of 3.8% unique variance in Grade 2 mathematical achievement. Furthermore, within the magnitude representation pathway, when computation was further removed from the model, the variance in Grade 2 mathematical achievement being accounted for was further reduced to 42.6%; but when number comparison was removed from the model instead, the explained variance remained the same (i.e., 46.7%). This suggested that while computation accounted for 4.1% unique variance in Grade 2 mathematical achievement on top of number comparison and other control variables, the common variance between number comparison and mathematical achievement in Grade 2 seemed to be completely shared with computation.

3

The traditional benchmark considered 0.1, 0.3, and 0.5 to be small, medium, and large effect for a beta respectively (Cohen, 1988). For an indirect effect with n betas involved, the relevant benchmark shall be raised to the power n (i.e., squared when two betas are involved, cubed when three betas are involved; Kenny, 2018). The indirect effect through the problem representation pathway was the product of two betas, and thus, the criterion for small effect was 0.01 (0.1 × 0.1) and the criterion for medium effect was 0.09 (0.3 × 0.3). The indirect effect through the magnitude representation pathway was the product of three betas, and thus the criterion for small effect was 0.001 (0.1 × 0.1 x .1) and the criterion for medium effect was 0.027 (0.3 × 0.3 x .3).

4. Discussion The current study set out to examine the potential mechanisms of the relation between children's visuospatial working memory and their mathematical achievement. Visuospatial working memory was suggested to play a role in children's mathematical performance not only 15

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Table 3 Pathways in the path model.

Key pathways are shaded. Significant pathways are bolded.

because of the spatial nature of the numerical magnitude representation (the magnitude representation pathway), but also because our visuospatial working memory provided the mental space for the representation of problem structures in mathematical word problem solving (problem representation pathway). Both pathways received support from the current data, but they did not fully account for the relation between visuospatial working memory and mathematical achievement. Theoretical and practical implications were discussed in the following.

memory provides us with a mental workspace not only for the processing of numerical magnitude, but also for the representation of the problem structure in mathematical problem solving. 4.2. Educational implications The current findings, by postulating the underlying relations between visuospatial working memory and mathematics, provide an important framework for future intervention studies to examine the causal relation between visuospatial working memory and mathematics. For instance, it would be interesting to examine whether improving visuospatial working memory would enhance children's mathematical achievement, and which pathway(s) actually drive such enhancement. Although the findings from the earlier studies suggested that the effect of working memory training did not generalize to arithmetic (MelbyLervåg & Hulme, 2013), such transfer was observed in a recent study in which kindergarteners participated in a series of memory games (Kroesbergen, van't Noordende, & Kolkman, 2014). More research is needed to identify effective and generalizable working memory training. A recent study from Nath and Szücs (2014) suggested that children's block construction ability was related to their mathematical achievement, and their visuospatial working memory capacity mediated such relation. Based on such correlational evidence, future studies may examine whether block construction activities can enhance visuospatial working memory capacity, which may further facilitate children's mathematical performance. Besides providing directions for the development of visuospatial working memory training, the current findings also offer other important educational implications. As the relation between visuospatial working memory and mathematical achievement is mediated by both magnitude and problem representations, children with poor visuospatial working memory may have difficulties in both understanding and manipulating numerical magnitude as well as representing mathematical word problems in their mind. Relevant educational activities can thus be developed to address these difficulties. For instance, educational interventions that aim at improving children's numerical magnitude representation (Kucian et al., 2011; Siegler & Ramani, 2009)

4.1. The link between visuospatial working memory and mathematics The current findings suggest that visuospatial working memory is related to mathematical achievement through two different pathways: magnitude representation and problem representation. On one hand, children with higher visuospatial working memory capacity are more efficient in processing numerical magnitude. This echoes with the hypothesis that numerical magnitudes are represented spatially in the form of mental number line (Dehaene et al., 1993; deHevia & Spelke, 2010; Nuerk, Wood, & Willmes, 2005; Siegler, 2016). With higher visuospatial working memory capacity, children have more mental workspace for the representation of numerical magnitude (Simmons et al., 2012). As the representation of numerical magnitude is involved in the process of computation (Dehaene et al., 2003; Stanescu-Cosson, Pinel, van de Moortele, et al., 2000), better representation of numerical magnitude may in turn result in better computation performance (Lyons et al., 2014; Schneider et al., 2017; Y.; Zhang et al., 2016). On the other hand, children with higher visuospatial working memory capacity are also better able to represent mathematical word problems. This pathway aligns with the findings that better mathematical problem solvers usually construct schematic, instead of pictorial, representation of the problem (Boonen et al., 2013; Hegarty & Kozhevnikov, 1999). Given the visuospatial nature of these schematic representations (Boonen et al., 2013; VanGarderen, 2006), children with higher visuospatial working memory capacity are therefore better able to utilize schematic representations that capture the relations between different problem parts in solving mathematical problems, which hence results in better mathematical problem solving. Overall, our visuospatial working 16

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may help to make the processing of numerical magnitude more automatic, thus saving more visuospatial working memory capacity for the computation process. Furthermore, in solving mathematical word problems, it may be possible to reduce children's visuospatial working memory load by teaching them to construct schematic representations of the problems using diagrams and graphics (e.g., a compare problem can be represented by two lines of different lengths). Such pedagogical strategies have been shown to improve students' mathematical problem-solving performance in previous intervention studies (Fuchs et al., 2014; Jitendra et al., 2007; Swanson, Lussier, & Orosco, 2013; Willis & Fuson, 1988), and the current findings provide a theoretical framework explaining why these pedagogical strategies work.

thing, visuospatial working memory is the best predictor of mathematical performance in elementary grade among various visuospatial abilities (Mix et al., 2016). Future studies are needed to examine whether the pathway model apply to other visuospatial abilities as well. Furthermore, the current study was correlational in nature and thus causality could not be claimed. If visuospatial processing is involved in solving mathematical problems, then it seems plausible that repeated practice on mathematical problem solving may in turn improve visuospatial working memory capacities. Such bi-directional relation between visuospatial working memory and mathematical achievement remains to be tested in future studies. 4.4. Conclusions

4.3. Limitations and future directions

The current study demonstrated that children's visuospatial working memory was related to their mathematical achievement through providing a mental workspace for both the representation of numerical magnitude as well as the representation of problem structure in mathematical word problem solving. These findings not only enhance our understanding of the relation between visuospatial working memory and mathematical achievement, but also inform educators of the potential means for improving children's mathematical achievement. As the proposed pathways do not fully explain the covariance between visuospatial working memory and mathematical achievement, future studies may further investigate other mechanisms involved in this relation.

While the current findings suggest that both of the two proposed pathways explain the relation between visuospatial working memory and mathematical achievement, it should be noted that even after including the three mediators in the two mediation pathways, the direct effect between visuospatial working memory and mathematical achievement remains statistically significant, suggesting that the two proposed pathways do not fully explain the relation between visuospatial working memory and mathematics. The potential relation between visuospatial processing and some particular types of mathematical problems (e.g., geometry; Giofrè, Mammarella, Ronconi, & Cornoldi, 2013; Kyttälä & Lehto, 2008), may be responsible for the remaining covariance between visuospatial working memory and mathematics, and these potential pathways should be tested in future studies. The inclusion of only one measure for each construct due to practical constraints might also have limited the generalizability of the current findings, thus future studies should consider including multiple measures for each construct. In the current study, we particularly focused on visuospatial working memory among various visuospatial abilities. For one thing, recent analyses (Mix et al., 2016) have shown that different visuospatial abilities (including visuospatial working memory, mental rotation, visual motor integration, block design, map reading, perspective taking) converged into one single factor, indicating that these visuospatial abilities share significant overlap. For another

Declaration of interest None. Acknowledgments This study was funded by Early Career Scheme 28400214 from Research Grants Council of Hong Kong to the first author. We thank the participating children, parents, and teachers for their support of this study.

Appendix 1. Overview of topics assessed in the mathematical achievement test Topic Number and algebra

Measures

Shape and space

Data handling

Questions -

Enumerating quantities Comparing the magnitudes of two-digit numbers Questions about place-value Computation (addition and subtraction, up to three-digit numbers) Computation (multiplication and division, up to two-digit numbers) Word problems Identifying of the shortest object Labelling the time shown in a clock Questions about day of the week Identifying appropriate measurement unit of the length of an object Conversion of money Distinguishing 2-D/3-D shapes Identifying 2-D shapes in a complex figure Identifying the base-shape of 3-D shapes Questions about directions Identifying sums and differences based on a pictogram

Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.learninstruc.2019.03.001.

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