As mentioned above, visual perception may be related to not only language processing through its impact on orthographic processing, but also mathematical performance through its impact on the processing of mathematical symbols. However, only a few studies used visual perception tasks to show the relationship between visual perception and mathematical performance (e.g., Rosner, 1973; Solan, 1987; Kulp, 1999; Kurdek and Sinclair, 2001; Sigmundsson et al., 2010).

For example, Rosner (1973) measured visual perception with a visual perception test (VAT) (Rosner, 1969) in which the children were asked to copy designs drawn on dot matrices to match a target stimulus. The results showed that visual perception explained the variance in computation performance after controlling for auditory perception. A longitudinal study by Kurdek and Sinclair (2001) showed the importance of visual perception in children's math development. They used a visual discrimination task (perception and discrimination of similarities and differences between various shapes and geometric forms) as well as other tasks in a kindergarten readiness test (KRT) and found that visual discrimination ability in kindergarten children could uniquely predict mathematical achievement in fourth grade.

Parallel with the studies on the role of visual coherent motion detection in language performance, Sigmundsson et al.'s study 2011 on children with low mathematical performance found that the children were less sensitive to visual coherent motion than age-matched controls. Boets et al. (2011) further showed that coherent motion sensitivity predicted individual differences in simple subtractions. They reasoned that subtraction highly relies on quantity processing, which is subserved by regions along the intraparietal sulcus, a region in the visual dorsal pathway which underlies coherent motion detection (Boets et al., 2011).

The approximate number system (ANS) is shared by human and non-human animals. It functions by estimating the number of items (e.g., the number of dots in a dot array) without relying on counting one by one. The ANS acuity or accuracy has been shown to be closely associated with mathematical performance (e.g., Halberda et al., 2008; Libertus et al., 2011; Lyons and Beilock, 2011). For example, children with developmental dyscalculia also suffer from ANS impairment (e.g., Landerl et al., 2004; Geary et al., 2009; Piazza et al., 2010). Sensitivity of the ANS correlated with symbolic mathematical performance for normally developing children (e.g., Halberda et al., 2008, 2012; Mundy and Gilmore, 2009; Inglis et al., 2011; Libertus et al., 2011, 2013; Mazzocco et al., 2011; Bonny and Lourenco, 2013). Halberda et al. (2008) found that 14-year-old children's ANS ability was closely associated with their earlier symbolic mathematical performance in third grade, after controlling for a series of cognitive processes including general intelligence, language processing, and working memory.

Furthermore, ANS training (such as approximate arithmetic task) has been found to promote the development of symbolic computation abilities (e.g., Park and Brannon, 2013, 2014; Hyde et al., 2014), while a comment from Lindskog and Winman (2016) doubted the effect of Park and Brannon (2013, 2014) due to their methodology.

However, some studies failed to show the association (e.g., Sasanguie et al., 2012, 2013; Vanbinst et al., 2012; Zhou et al., 2015; see a review by de Smedt et al., 2013). Recently, Zhang et al. (2016) found that ANS acuity only correlated with mathematical fluency but not with mathematical reasoning.

The close relationship between the ANS and symbolic math performance was traditionally attributed to the domain-specific processing in the ANS (e.g., Barth et al., 2005, 2006, 2008; Nieder and Dehaene, 2009; Gilmore et al., 2010; Park and Brannon, 2013; Hyde et al., 2014). For example, Park and Brannon (2013) argued that humans' ANS permitted estimation and rough calculation of numerical quantities without symbols. Gilmore et al. (2010) treated ANS as a type of core numerical ability. The ANS is important for the acquisition of symbolic numerical skills, such as counting and arithmetic, because both ANS and symbolic numerical skills involve quantity processing (e.g., Gilmore et al., 2007; Inglis et al., 2011). Indeed, previous correlational studies have shown a significant relation between symbolic mathematics and quantity processing (e.g., de Smedt et al., 2009; Lefevre et al., 2010; Sasanguie et al., 2013; Zhang et al., 2016).

An alternative hypothesis was proposed by Zhou and colleagues (Zhou and Cheng, 2015; Zhou et al., 2015), who argued that visual form perception accounts for the close relation between the ANS and exact computation. Zhou et al. (2015) found that, after controlling for the score on geometric figure discrimination as well as the scores on other general cognitive processing measures (i.e., Raven's Progressive Matrices, mental rotation, choice reaction, visual tracing, and digit span), the close relation between the ANS and exact computation was not existent (e.g., Zhou et al., 2015). Zhou and Cheng (2015) also showed that the ANS difference between children with dyscalculia and typically-developed children was associated with difference in geometric figure discrimination. These results suggested a possibility that the ability of figure discrimination might account for the relation between the ANS and exact computation because the two previous studies have excluded some types of general cognitive factors (Zhou and Cheng, 2015; Zhou et al., 2015). That is, visual form perception is one of the possible reasons to interpret the positive relation between ANS and symbolic mathematical performance.

The geometric figure matching task used in the above two studies was adopted from the Manual for the Kit of Factor-Referenced Cognitive Tests (Ekstrom et al., 1976) and the visual perception test (VMI-VP) in the Beery-Buktenica Developmental Test of Visual-Motor Integration (VMI 4th edition, Beery et al., 1997). This task was used to measure the ability of geometric figure discrimination, which involves the abilities to attend to and to identify a figure's distinguishing features and details. The figure could be treated as a form or shape, consisting of abstract lines. The geometric figure matching task was used in the two studies because it could be associated with visual form perception, alphanumeric symbols, and numerosity.

Some previous studies found close relationships between the ANS and symbolic mathematical performance even after controlling for visual perception (Halberda et al., 2008). Halberda et al. (2008) controlled 4 types of visual processing tasks, including visual working memory, visual segmentation, object perception, and visual motor integration, and they still found significant relationships between the ANS and symbolic mathematical performance. Piazza et al. (2010) found that math education level did not relate to the precision of visual size discrimination in a size comparison task. Tibber et al. (2012) showed that the sensitivity to the orientations of two clusters of Gabor blobs could predict math abilities, but the sensitivity to size and density could not. The visual perception tasks used in these studies, however, were mostly focused on size, color, or brightness, which are different from visual form processing (e.g., Milner et al., 1991; Cavina-Pratesi et al., 2015). Thus, these tasks might not fully measure visual form processing. Given that visual form processing appears to be a critical component of visual perception with regard to relationships with the ANS and symbolic mathematics performance, tasks that are focused on visual form processing, such as the geometric figure discrimination task (Zhou and Cheng, 2015; Zhou et al., 2015) may be more proper to be utilized in studies of the relationships between visual form perception, the ANS, and symbolic mathematics performance.

Although previous studies have shown the relation between visual perception and symbolic mathematical performance, they seldom differentiated between different categories of mathematical performance. Given that cognitive correlates for different types of mathematical performance differ greatly (e.g., Fuchs et al., 2008; Petrill et al., 2011), the role of visual perception might also differ among different types of mathematical performance. To address these two issues, the current study focused on the role of visual form perception, measured with the figure matching task which involves geometric figure discrimination (Zhou and Cheng, 2015; Zhou et al., 2015), in five typical categories of mathematical performance: numerosity comparison, digit comparison, exact computation, approximate computation, and curriculum-based math achievement.

According to previous studies on the relations between visual perception, ANS, and mathematical performance, it is hypothesized that visual perception is more closely associated with the performance in lower level math categories which rely more on visual features. The lower level math categories could include the ANS, digit comparison, and simple arithmetic. Previous studies have shown that visual form perception or the ANS was not significantly associated with higher level math categories, such as curriculum-based math achievement, which relies on math concept and math problem solving instead of lower level abilities (e.g., Sasanguie et al., 2012, 2013; Vanbinst et al., 2012; Zhou et al., 2015; Zhang et al., 2016). We thus predict that visual perception has minimal contribution to higher level math categories including approximate computation and curriculum-based math achievement.

Following the general hypothesis, we propose five specific hypotheses for the five math measures:

Hypothesis 1: visual form perception has a unique contribution to numerosity comparison. Numerosity comparison tasks usually use dot arrays to express quantity information. Patient studies showed that a patient suffering from visual form agnosia had deficit in numerosity processing (Milner et al., 1991). To extract numerical quantity information expressed by a dot array, the participants may transform the dot array into some type of visual form or pattern, which is related to visual form perception.

In addition, Hypothesis 5 is supported by a recent finding that, after controlling for some general cognitive processes (i.e., non-verbal IQ, mental rotation, visual tracing, verbal working memory, and choice reaction time), visual form perception has an independent contribution to exact computation, but not curriculum-based math achievement for third to fifth graders in primary schools (Zhou et al., 2015).

Several general cognitive processes, such as processing speed, attention, spatial working memory, and intelligence have been shown to be associated with mathematical performance (Berg, 2008; Krajewski and Schneider, 2009). To control for these general cognitive processes, these processes were measured and used as covariates in the current study.

The participants included 223 seventh grade students (90 boys and 133 girls, mean age 12.9 years-old (SD = 0.66), ranging from 11.2 to 13.7) in a middle school in Shijiazhuang municipality of Hebei Province in China. All participants were native Chinese speakers.

The school attended the study in order to assess the basic learning ability of its seventh graders and to perform remedial instruction partially based on the assessment. The program including the cognitive testing was fully explained to students' guardians (typically parents) in the school's parent meeting in the semester. During the meeting, the guardians gave written consent forms for the remedial instruction program. Among a total of 248 students, 223 students finally attended the cognitive testing. Testing results were provided only to the school's psychological counselors who possess psychological counseling certificates of the third or second level (highest level in China). No other teachers or administrators had access to the results. All tests were fully explained to the counselors, who were then able to use the results as evidence, along with the students' achievement scores and the teachers' subjective assessments to provide instructional suggestions for the students.

The school is public, with a slightly above-average academic level in the city. There were ~30–40 students per class. The study was approved by the institutional review board (IRB) at the State Key Laboratory of Cognitive Neuroscience and Learning at Beijing Normal University.

A total of 11 tests were used. All except for the curriculum-based mathematical achievement test were computerized using web-based applications in the “Online Psychological Experimental System (OPES)” (www.dweipsy.com/lattice). Besides a figure matching test, which measured visual perceptual ability, other tests were also included to measure general cognitive processes, including choice reaction time, mental rotation, spatial working memory (adapted from Corsi Blocks Task, Corsi, 1972), visual tracing, and non-verbal matrices reasoning. The tests covered a wide range of cognitive processes that involve visual perception, attention, working memory and general intelligence. Previous studies have shown that these cognitive processes are closely associated with mathematical performance (e.g., Spinath et al., 2006; Deary et al., 2007; Berg, 2008; Krajewski and Schneider, 2009; Wei et al., 2012b).

For most tests, the participants responded to two-choice options by pressing the Q and P keys on a computer keyboard to choose the correct answers. The other response modes were explained below in more details. In all tests except for the spatial working memory test, the participants were encouraged to respond as quickly and accurately as possible. The schematic representations of all tests, except for the curriculum-based mathematical achievement test, are displayed in Figure 1. The tests are introduced as follows.

The battery of tests was administered in two 40-min sessions. The practice session included four to six trials, which were similar to those used in the formal testing. The feedback for correct answers for all cognitive tasks was “Correct! Can you go faster?,” and that for incorrect answers was “It is wrong. Try again.” During the practice session, the children could ask the experimenters any questions related to the testing. The formal testing began after the children had finished the practice session.

Computerized tests were administered to the students (one class at a time) in a computer classroom. Each class was monitored by two experimenters as well as the teacher of that class. The experimenters explained the instruction with slides for each task. The teacher was present only for the purpose of discipline (e.g., maintaining silence during the formal testing). After all the students completed one test, the experimenter started to administer the next test. For each test, the students were first given instructions and then completed a practice session, followed by the formal testing. Only when all students understood the procedure in practice, could they begin the formal testing. After the main experimenter said “Start!,” all children in the computer classroom pressed a key to begin the formal testing. All students performed the tasks in the same order.

The data collection took place between March and June, 2014.

First, descriptive statistics was performed for all tests. Mean, standard deviation and split-half reliability were calculated for each test. Spearman-Brown formula was used to adjust for the test length when calculating split-half reliabilities. Pearson's correlation coefficients were calculated between the scores of all cognitive and mathematical tests.

Second, a series of hierarchical regression analyses were conducted to test the role of visual form perception in the five categories of mathematical performances, while controlling for age, gender, and all other 5 types of general cognitive processes.

Third, a path model was used to test the structural relationships among all measures in the current study. The path analysis was performed in the IBM SPSS Amos 20 software, using maximum likelihood estimation. The chi-square, root mean square error of approximation(RMSEA), comparative fit index (CFI) and standard root mean square residual (SRMR) were used to assess model fit. According to Kline (2005), good model fit was indicated by a non-significant chi-square, RMSEA values less than 0.05, CFI values greater than 0.95, and SRMR values less than 0.10. The hypothesized path model assumes that the five general cognitive processes (i.e., choice reaction time, visual tracing, mental rotation, spatial working memory, and non-verbal matrix reasoning) contribute to the five categories of mathematical performance, and that figure matching only contributes to numerosity comparison, digit comparison, and exact computation. The above model was compared with a full-connection model, which added two additional links (figure matching and approximate computation, and figure matching and curriculum-based math achievement).

Fourth, mediation analyses were conducted with the bootstrapping method (Preacher and Hayes, 2008) to investigate the contribution of visual form perception in mediating the relation between numerosity comparison and exact computation, after controlling for the general cognitive processing variables (choice reaction time, visual tracing, mental rotation, spatial working memory, and non-verbal matrix reasoning), as well as age and gender.

Last, a series of Bayesian regression analyses using Matlab were conducted to test the two null hypotheses (4 and 5), to test Bayesian estimation relying on the probability beyond linear regressions. According to Kleiman et al. (2014), Beta weights and 95% confidence intervals (also called credibility or probability intervals) were produced, and the 95% confidence intervals without including zero were considered as significance at p < 0.05.

Means and standard deviations of scores for all the general cognitive processing and the 5 categories of mathematical performance are displayed in Table 1. The split-half reliability for each cognitive test is also shown in Table 1. All tests have acceptable reliabilities, ranging from 0.84 to 0.96.

Performance in the numerosity comparison task was first examined with a partial correlation analysis to ensure that the participants performed the task in a ratio-dependent manner. After controlling for the five visual properties (i.e., total surface area, envelope area or convex hull, item size, density, and circumference, see Gebuis and Reynvoet, 2011), accuracy across all trials was still ratio-dependent, r₍₁₁₃₎ = 0.41, p < 0.001.

Pearson's correlation coefficients among all measures with Bonferroni correction are displayed in Table 2. A Bonferroni correction was used to maintain the p-value < 0.05 across the 78 correlations in Table 2. Thus, a conservative p-value of < 0.00064 (=0.05/78) was considered statistically significant.

Linear hierarchical regression analyses for numerosity comparison, digit comparison, exact computation, approximate computation, and curriculum-based mathematical achievement were conducted separately to test the 5 hypotheses. Tables 3–7 show the results. According to Tables 3–7, visual form perception had unique contributions to numerosity comparison, digit comparison, and exact computation, but had no significant contribution to approximate computation and had only a minor contribution to curriculum-based math achievement. These results are presented in the following five sections. Thus, subsections in Section Results include five hierarchical regression analyses, as shown in Tables 3–7.

We also performed Bonferroni correction on the 5 regression analyses. Thus, a conservative p-value of <0.01 (=0.05/5) was considered statistically significant.

The above linear hierarchical regression analyses have shown that visual perception might have significant linear correlation to numerosity comparison, digit comparison, and exact computation, but not to approximate computation and curriculum-based mathematics achievement, after controlling for other types of general cognitive processes. In order to test the two null hypotheses (4 and 5), we used Bayesian regression analyses to check whether visual perception would have some non-linear relationship with approximate computation and curriculum-based mathematics achievement. The results were shown in Tables 8, 9, using the same steps as those in Tables 3–7.

Table 8 showed the results of the Bayesian regression analysis for Hypothesis 4. Only general cognitive processing abilities excluding visual form perception were significant predictors for approximate computation. Visual form perception also had no additional contribution to approximate computation (ΔR² = 0.005), similar to the result from the above linear regression analysis.

Table 9 showed the results of the Bayesian regression analysis for Hypothesis 5. Only general cognitive processing abilities excluding visual form perception were significant predictors for curriculum-based math achievement. Visual form perception also had no additional contribution to curriculum-based math achievement (ΔR² = 0.013), similar to the result from the above linear regression analysis.

Figure 2 shows the path model for the structural relationships among all six types of cognitive abilities and five types of mathematical performances, and for the contribution of lower level of mathematical performance to higher level of mathematical performance. The hypothesized model was a good fit for the data, χ(3)2 = 3.73, p = 0.292, RMSEA = 0.03, CFI = 1.00, and SRMR = 0.02. The model showed that only visual form perception in all six types of cognitive abilities had significant direct associations with numerosity comparison, digit comparison, and exact computation. Numerosity comparison did not display any significant relation with the other four categories of mathematical performance in the path model. All the significance of loading in the path model was under Bonferroni correction. A Bonferroni correction was used to maintain the p-value < 0.05 across the 35 links in Figure 2. Thus, a conservative p-value of < 0.0014 (=0.05/35) was considered statistically significant.

After removing all the non-significant relations in Figure 2, the new model was also a good fit for the data, χ(5)2 = 2.27, p = 0.881, RMSEA = 0.00, CFI = 1.00, and SRMR = 0.02.

The full-connection model with two additional links (figure matching and approximate computation, and figure matching and curriculum-based math achievement) was not a good fit for the data, χ(2)2 = 12.78, p = 0.002, RMSEA = 0.16, CFI = 0.98, and SRMA = 0.28.

For the mediation analyses, the dependent variable was the exact computation, and the controlled covariates were the general cognitive processing variables (choice reaction time, visual tracing, mental rotation, spatial working memory, and non-verbal matrix reasoning), as well as age and gender. We tested whether visual form perception mediated the relation between numerosity comparison and exact computation (see Figure 3A). Results showed that the relation between numersoity comparison and exact computation was fully mediated by visual form perception (see Figure 3B).

Visual form perception had independent contribution to the ANS (i.e., numerosity comparison) after the other general cognitive processes (i.e., choice reaction time, visual tracing, mental rotation, non-verbal matrix reasoning, and spatial working memory) were controlled for (Table 3). The close relation between visual form perception and the ANS is consistent with results found in previous studies (e.g., Burr and Ross, 2008; Tibber et al., 2012; Anobile et al., 2013b).

Although some behavioral studies have investigated the relationships between visual perception and mathematical performance (e.g., Rosner, 1973; Rourke and Finlayson, 1978; Sigmundsson et al., 2010; Tibber et al., 2012; Anobile et al., 2013b), no study has been conducted on the relation between visual perceptual ability and symbolic numerical quantity comparison. In this study, we showed that visual perceptual ability was highly correlated with symbolic numerical quantity comparison, suggesting that visual form perception is critical for not only non-symbolic, but also symbolic numerical quantity comparison. As proposed in the Triple Code Model of Dehaene and Cohen (1995, 1997), visual code is about the representation of Arabic number forms. On the other hand, visual processing takes place in the occipito-temporal regions along the visual ventral processing path (e.g., Abboud et al., 2015). Thus, the close relation between visual form perception and symbolic numerical quantity comparison might be due to visual processing of Arabic number forms.

Moreover, the ANS is supra modal and can work with information from several modalities. Thus, there is a ratio-dependent performance in visual, auditory and haptic modalities (e.g., Barth et al., 2005, 2008; Jordan and Brannon, 2006; Jordan and Baker, 2011; Sasanguie et al., 2015; Gimbert et al., 2016). To our knowledge, no direct evidence has been found to show the link between auditory or haptic ANS acuity and mathematical performance, for which further empirical evidence is expected.

Among all six measures of general cognitive processes, visual form perception showed the highest correlation with exact computation, which is consistent with previous studies showing a close relationship between visual perception and exact computation (Halberda et al., 2008). Hierarchical regression analyses showed that the correlation between visual form perception and exact computation could not be fully accounted for by the other five general cognitive processes, suggesting that the cognitive component shared by visual form perception and exact computation is not shared by the other cognitive processes. The shared component may be number form processing, given that rapid processing of forms was involved in both visual form perception (e.g., figures) and exact computation (e.g., Arabic digits).

In contrast to its unique contribution to exact computation, visual form perception did not have a unique contribution to approximate computation after the other general cognitive processes were controlled for. Previous studies have shown a dissociation between exact and approximate computation in the brain (e.g., Dehaene et al., 1999). Exact computation is associated with brain regions involved in language processing, whereas approximate computation is associated with brain regions involved in spatial or quantity processing. It is possible that approximate computation recruits cognitive processes involved in quantity processing instead of those involved in number form processing, which might explain the absence of correlation between approximate computation and visual form perception. Future studies are needed to directly investigate why visual form perception shows a unique contribution to exact computation, but not to approximate computation.

The mediation analyses suggested that visual perception could be considered as a potential mechanism for the relationship between ANS and arithmetic. The positive effect from ANS training (e.g., Park and Brannon, 2013, 2014; Hyde et al., 2014) might originate from the enhancement of visual perception, that is, speed enhancement of visual perception might associate with arithmetic improvement. It is consistent with the viewpoint from Lindskog and Winman (2016) who suggested that the approximate arithmetic training promoted non-specific response speed. The non-specific response speed might be related with the visual perception speed.

Visual form perception did not show any significant contribution to curriculum-based mathematical achievement after controlling for gender, age and the other five general cognitive processes. However, exact computation, which was related with visual form perception, had a significant contribution to mathematical achievement. The contribution could be typically associated with some computational abilities beyond visual form perception, such as memory of computational facts and the application of computational procedures.

As mentioned earlier, the achievement test was curriculum-based and covered computation (including equations), mathematical concepts, and applied problem solving. The curriculum-based computation is the computation based on rational numbers, such as, |-2|+ (cos60°- tan30°)°+ 8, x⁴÷x⁵y, x2 + x3 = 14 (x + 3), which is beyond the scope of the exact computation task used in the current study. In this study, the students just leant the mathematics knowledge in the last semester, and thus they might not have acquired enough proficiency to solve the computational problems through simple visual inspection of symbols without significant mathematical processing. Instead, the students might have to use the majority of time and mental resources to search paths from the problem preconditions to the problem solution. The search process involves processing mathematical concepts and rules, which are centered on the semantic processing of mathematical knowledge and do not rely heavily on visual form perception.

Given that mathematical achievement tests cover a wide range of topics in the math curriculum, future studies should directly test the role of visual form perception in curriculum-based computation, concepts and mathematical problem solving.

Among all the general cognitive measures available in the current study, only non-verbal matrix reasoning had unique contribution to math achievement. Non-verbal matrix reasoning might be involved in the path search during the mathematical problem solving, which is a critical step of solving the mathematical achievement test. In contrast, visual processing was most likely involved in the initial perception and decoding of mathematical symbols, which is of less importance to the mathematical achievement test than path search.

Combining the current study and our previous studies (Zhou and Cheng, 2015; Zhou et al., 2015), we can conclude that the visual form perception could contribute to lower level math categories for students in primary and middle schools. Future study is needed to investigate the role of the visual form perception for younger children (e.g., in kindergarten) and for adults. According to the relation between visual form perception and exact computation, adults might have a significant association, but younger children might not have a significant association. The speculation could have some empirical evidence from the studies on developmental dyscalculia (DD).

Noël and Rousselle (2011) summarized existing studies and found that DD typically had poorer performance only in symbolic numerical processing for under 10 year old's, and had poorer performance in both non-symbolic (ANS) and symbolic numerical processing for above 10 years old's, compared to typically developing children (Rousselle and Noël, 2007; Iuculano et al., 2008; Landerl and Kölle, 2009; Landerl et al., 2009; Mussolin et al., 2010; Piazza et al., 2010; de Smedt and Gilmore, 2011; Mazzocco et al., 2011). Based on these results, Noël and Rousselle (2011) proposed that deficits in symbolic processing is directly related to mathematical difficulties, and that the association between the ANS and mathematical abilities is due to the influence of symbolic number processing in the ANS. A large-sample study on 626 5–7-year-old unselected children from the UK, China, Russia, and Kyrgyzstan found that it was the understanding of symbolic numbers that explained variations in mathematical performance in all samples, but not the ANS (Rodic et al., 2015). This view is consistent with our results that visual form processing is important for low-level math abilities, given that symbolic processing is closely related to visual form processing.

Thus, it appears that the relation between visual form processing and math abilities varies with age. More research is needed to assess this issue.

Generally, math in China is taught in a similar way to that in Western countries. For example, Arabic digits are taught and used in math textbooks. The main difference may be that Chinese students typically spend more time on math learning (Stevenson et al., 1986). In addition, Chinese characters are used in math textbooks to explain math knowledge. The Chinese language belongs to a logographic system, different from alphabetic languages. Orthographic awareness or general visual perceptual skills are critical in the development of Chinese language abilities (e.g., Meng et al., 2002; Kuo et al., 2014). Chinese students' experience in acquisition and application of Chinese characters might transfer to the development of math knowledge, which might lead to greater reliance on visual form perception. To our knowledge, no study has directly compared degrees of reliance on visual form perception in mathematics for Chinese and Westerners.

Notably, visual form perception could be also important for participants in Western countries. Studies have shown a close relation between processing speed and mathematical performance for children in Western counties (e.g., Hitch and McAuley, 1991; Bull and Johnston, 1997; Fuchs et al., 2005). Processing speed was typically measured with tasks similar to our figure matching task. For example, Bull and Johnston (1997) used visual number matching task and geometric figure cross-out task (Kail and Hall, 1994), which were initially devised to assess the processing speed factor in the theory of fluid and crystallized intelligence (Cattell, 1963). Fuchs et al. (2005, 2008) also used similar tasks. Thus, the results in the current study may be applied to students speaking alphabetic languages, although future research is needed to directly compare the importance of visual form perception on mathematical performance between different cultural groups.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.