Cognition 118 (2011) 32–44

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Individual differences in children’s mathematical competence are related to the intentional but not automatic processing of Arabic numerals Stephanie Bugden, Daniel Ansari ⇑ Numerical Cognition Laboratory, Department of Psychology, The University of Western Ontario, Canada

a r t i c l e

i n f o

Article history: Received 19 May 2010 Revised 24 September 2010 Accepted 27 September 2010

Keywords: Numerical cognition Ratio effect Distance effect Size congruity effect Mathematical achievement

a b s t r a c t In recent years, there has been an increasing focus on the role played by basic numerical magnitude processing in the typical and atypical development of mathematical skills. In this context, tasks measuring both the intentional and automatic processing of numerical magnitude have been employed to characterize how children’s representation and processing of numerical magnitude changes over developmental time. To date, however, there has been little effort to differentiate between different measures of ‘number sense’. The aim of the present study was to examine the relationship between automatic and intentional measures of magnitude processing as well as their relationships to individual differences in children’s mathematical achievement. A group of 119 children in 1st and 2nd grade were tested on the physical size congruity paradigm (automatic processing) as well as the number comparison paradigm to measure the ratio effect (intentional processing). The results reveal that measures of intentional and automatic processing are uncorrelated with one another, suggesting that these tasks tap into different levels of numerical magnitude processing in children. Furthermore, while children’s performance on the number comparison paradigm was found to correlate with their mathematical achievement scores, no such correlations could be obtained for any of the measures typically derived from the physical size congruity task. These findings therefore suggest that different tasks measuring ‘number sense’ tap into different levels of numerical magnitude representation that may be unrelated to one another and have differential predictive power for individual differences in mathematical achievement. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction In recent years there has been a growing amount of research into the typical and atypical developmental trajectories of basic numerical magnitude processing, such as estimation and number comparison (Berch & Mazzocco, 2007; Landerl & Kolle, 2009; Noel, Rousselle, & Mussolin, 2005; Rubinsten & Henik, 2005). Efforts to better understand how children develop an understanding of numerical magnitude is partially driven by a growing body of data ⇑ Corresponding author. Address: Numerical Cognition Laboratory, Department of Psychology, University of Western Ontario, London, Ontario, Canada. Tel.: +1 519 661 2111x80548; fax: +1 519 850 2554. E-mail address: [email protected] (D. Ansari). 0010-0277/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cognition.2010.09.005

suggesting basic numerical magnitude tasks are processed atypically in children with developmental disorders of numerical cognition, such as Developmental Dyscalculia (Ashkenazi, Mark-Zigdon, & Henik, 2009; Landerl, Bevan, & Butterworth, 2004; Landerl & Kolle, 2009; Mussolin, Mejias, & Noel, 2010). Furthermore, there are several studies that have revealed that individual differences in children’s performance on numerical magnitude processing tasks, such as number comparison, are correlated with and predictive of individual differences in the mathematical ability scores of typically developing children (De Smedt, Verschaffel, & Ghesquiere, 2009; Halberda, Mazzocco, & Feigenson, 2008; Holloway & Ansari, 2009; Mundy & Gilmore, 2009).

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Notwithstanding, there currently exists a lack of systematic investigation into the similarities and differences between tasks that are presently all considered to measure ‘‘basic” numerical magnitude processing or ‘‘number sense”. It is important from both diagnostic and theoretical perspectives to understand the commonalities and differences in the effects used to study numerical magnitude processing, their developmental trajectory and relationship to mathematical competence. In light of this, the aim of the present study is to investigate the developmental changes and relationship to individual differences in higher level skills of two task effects that have commonly been used to investigate the development of numerical magnitude processing: the numerical distance/ratio effect and the physical size congruity effect. Prior to presenting the current study’s hypotheses, these two effects will be described below.

1.1. Distance and ratio effects When asked to compare the relative magnitudes of two digits, participants are faster and more accurate at identifying the numerically larger number as the distance between the two digits increases; this is known as the numerical distance effect (Dehaene, Dupoux, & Mehler, 1990; Moyer & Landauer, 1967; Sekuler & Mierkiewicz, 1977). The numerical distance effect is thought to arise from noisy representations of numerical magnitude that are placed on an internal mental number line. It is difficult to choose the larger of two numbers when they are close together because of the distributional overlap of their representational features. Furthermore, it has been shown that this effect changes over developmental time. Specifically, Sekuler and Mierkiewicz (1977) revealed that the effect of numerical distance on response times decreases over developmental time. In a cross-sectional study, these authors demonstrated that older children and adults have significantly shallower slopes relating distance and response times than do their younger peers. These results have since been replicated multiple times (Holloway & Ansari, 2008, 2009; Rubinsten, Henik, Berger, & ShaharShalev, 2002). A complementary effect is the so-called ratio or magnitude effect. The ratio effect is manifested in a systematic relationship between the ratio of two numbers and reaction times (Moyer & Landauer, 1967). It has been known since the original report by Moyer and Landauer, that, in accordance with Weber’s Law (as the ratio between two numbers increases, the time to discriminate between them also increases), the ratio between the two numbers being compared is more closely related to reaction times than the absolute difference between them. So although the number pairs 1 and 2 and 8 and 9 both have a numerical distance of 1, their ratio is significantly different and it takes longer to discriminate the relative magnitude of 8 and 9 than it does for 1 and 2. It should be noted that there is a high correlation between the numerical distance effect and the ratio effect, but the ratio effect explains more variance in number comparison reaction times and accuracy data (Moyer & Landauer, 1967).

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Recent evidence suggests that the numerical distance effect not only changes over developmental time, but that individual differences in the size of this effect are related to variability in children’s performance on standardized tests of mathematical competence. Specifically, Holloway and Ansari (2009) investigated the relation between the numerical distance effect as measured by the number comparison task and 1st and 2nd grade children’s standardized math scores. The findings indicated that between-subjects variability in the size of the numerical distance effect is negatively correlated with individual’s ability to perform simple arithmetic. Thus, children with relatively larger distance effects also have comparatively lower math scores. In a longitudinal study, De Smedt et al. (2009) studied whether the distance effect at the beginning of grade one would predict individual differences in math achievement 1 year later (beginning of grade 2). The findings from this longitudinal study demonstrated that individual differences in the size of the distance effect on the number comparison tasks at the beginning of first grade were significantly correlated with the participants’ standardized math scores at the beginning of second grade. In other words, children with a smaller distance effect in grade one had higher math scores in grade two. In the present study we seek to extend this work by investigating both how individual differences in the ratio effect change over developmental time, as well as how this effect relates to individual differences in mathematics achievement and how it relates to another measure of basic numerical magnitude processing: The size congruity effect. 1.2. The size congruity effect In order to be fluent in calculation and numerical magnitude processing, it is important to be able to access the semantic meaning of numerical symbols effortlessly and quickly. Automaticity is defined as processes that occur without conscious monitoring (Tzelgov, 1997); therefore, automatic processing can occur even when the particular stimulus dimension being processed is not relevant to the task at hand. In contrast, intentional processing refers to processes that occur in response to a particular task requirement, are task-relevant and are engaged in pursuit of a particular goal. In the literature on number processing, the so-called ‘Number Stroop’ paradigm or ‘Size Congruity Task’ is thought to be a marker of the automatic processing of numerical magnitude. In this paradigm, participants are asked to either choose the numerically or physically larger number and to ignore the task-irrelevant dimension. In studies of numerical magnitude processing, the physical version of the task is considered to be a measure of the automatic activation of numerical magnitude, since participants only have to explicitly process the relative physical sizes of Arabic numerals and ignore their numerical magnitude. In this task, the participants are presented with congruent (numerically larger is physically larger, e.g. 8 2), incongruent (numerically larger is physically smaller e.g. 8 2) or neutral (during the physical task e.g. 2 2) trials. The size congruity effect, thought to be an measure of

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automatic processing of the task-irrelevant dimension (numerical magnitude), is characterized by shorter reaction times for congruent trials compared to neutral and incongruent trials (Girelli, Lucangeli, & Butterworth, 2000; Rubinsten et al., 2002). The task-irrelevant numerical dimension facilitates the participant’s decision to choose the physically larger number when both dimensions match in size and, conversely, the numerical magnitude interferes with the relative comparison of physical size when the physical size and numerical magnitude associated with the correct response conflict with one another. The majority of studies of automaticity in numerical magnitude processing have been carried out with adult subjects (Besner & Coltheart, 1979; Henik & Tzelgov, 1982; Tzelgov, Meyer, & Henik, 1992). However, there also exist a series of studies revealing that children can be found to exhibit the same effects as those observed in adults, but that there are significant developmental changes in the automaticity of numerical magnitude processing. Using the size congruity paradigm, Girelli et al. (2000) investigated the development of the size congruity effect in first, third and fifth graders as well as university students. The results indicated that in the physical task, where numerical magnitude is the irrelevant dimension, the size congruity effect was absent in first grade children, evident in grade three, and was significantly robust in grade five. In view of these findings, the authors concluded that the automatic activation of numerical magnitude in the physical size congruity task emerges gradually over the course of developmental time. A study by Rubinsten et al. (2002) conducted in parallel to the Girelli et al. (2000) study, revealed a similar pattern of results. However, these researchers investigated both the interference (incongruent–neutral) and facilitatory (congruent–neutral) components of the size congruity effect as well as the numerical distance effect in students at the beginning and end of grade one, as well as third grade, fifth grade and university students. It has been suggested that facilitation is an indicator of automaticity, because congruent items are processed faster and more efficiently than neutral trials. Thus, the task-irrelevant numerical magnitude facilitates the relative judgment of physical size when both dimensions for the stimuli associated with the correct response are congruent with each other. Interference, on the other hand, might reflect attentional processing (Posner, 1978) since the irrelevant numerical dimension does not match the physical size of the number. Therefore, incongruent trials are processed more slowly and less efficiently than neutral pairs. To examine the facilitatory effect of the congruency between numerical magnitude and physical size, Rubinsten et al. (2002), compared the congruent stimuli to the neutral stimuli (such as 2 2). They also manipulated the numerical distance between the pairs in addition to their congruency. Rubinsten et al. (2002) found that in the physical version of the task, a size congruity effect does not appear at the beginning of first grade. However, about eight months later, numerically irrelevant information significantly interferes with physical judgments. Thus, the size congruity effect starts to appear only at the end of grade one and is interference based. Grade one students did not show a

facilitation effect. However, grades three and five students and adults show both an interference effect and facilitation effect. Interestingly, the numerical distance effect was only significant in all groups when the numerical dimension was relevant. The results from both the Girelli et al. (2000) and the Rubinsten et al. (2002) studies demonstrate that the ability to process numerical magnitude when it is the irrelevant dimension develops with age. Girelli et al. (2000) did not show a significant size congruity effect until children were in grade five. In contrast, Rubinsten et al. (2002) concluded that children at the end of grade one could automatically process numerical magnitude. These studies, therefore, show inconsistent findings when it comes to developmental changes of the size congruity effect, which may be the result of the fact that slightly different age groups were tested in both studies and thus, the exact developmental time points at which significant interference and facilitation effects can be detected remains to be established. While developmental changes in the size congruity effect have been found, it remains unclear how these developmental changes in automaticity relate to explicit processing of numerical magnitude (as measured, for example, by the number comparison paradigm). It is also unclear whether automaticity in processing of numerical magnitude in the physical size congruity paradigm is related to individual differences in children’s math achievement. If indeed the interference or facilitation effects in the physical size congruity effect reflect the development of fluent (automatic) processing of the numerical magnitudes represented by Arabic numerals, then those children who exhibit greater automaticity in processing numerical magnitude should have relatively higher levels of performance on standardized tests of mathematical achievement, since the quick activation of the semantic referents (numerical magnitude) of Arabic numerals is essential for calculation. 1.3. The current study In view of the above, the aim of the present study is to provide an investigation into the relationship between the above discussed intentional (number comparison) and automatic (physical size congruity) processing of numerical magnitude. Currently, these measures are frequently cited under the umbrella term of ‘basic numerical magnitude processing tasks’ or tasks thought to tap into the broad construct ‘number sense’, but the relationship between these measures has not been thoroughly investigated. In other words, it is unclear whether children with smaller distance effects are also those who are more likely to automatically process numerical magnitude when it is task-irrelevant. If both tasks tap into the same numerical magnitude processing competencies, then a correlation between them should be obtained. If, however, these tasks tap into different levels of numerical magnitude processing that are characterized by different developmental trajectories, no correlations should be revealed. In the present study we assess developmental changes in these effects between 1st and 2nd grade children and investigate the relationship between these effects.

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A related question pertains to the functional significance of these effects. In previous work it has been shown that the distance effect is correlated with individual differences in children’s mathematical competence (De Smedt et al., 2009; Holloway & Ansari, 2009; Mundy & Gilmore, 2009), and it has been shown that participants with dyscalculia differ from their typically developing peers on measures obtained from the size congruity paradigm. To the best of our knowledge, there is only one study which briefly addresses the relationship between the numerical distance effect and the size congruity effect conducted by Landerl and Kolle (2009); however, the sample used to estimate such correlations included both children with and without Developmental Dyscalculia and relationships were estimated by collapsing across these group and therefore the results are not necessarily generalizable to unselected sample. In view of the above, the present study investigates the extent to which both ratio and size congruity effects are correlated with variability in children’s mathematical achievement scores. Investigating the relationship between, on the one hand, the size congruity and ratio effects and, on the other, individual differences in children’s math achievement will help to better understand the relationship between these intentional and automatic measures of numerical magnitude processing in children. In other words, correlations between these effects and standardized math scores will elucidate their predictive power and will also serve as a probe for the similarities and differences in the processes measured by these two effects. In this way, such correlations are both informative from diagnostic and theoretical standpoints. Against the background of the extant literature (De Smedt et al., 2009; Holloway & Ansari, 2009), we hypothesize that 6–7 year old children who have a smaller ratio effect (higher reaction times due to small ratios between digits) will have lower standard math scores. Secondly, we predict that the individual differences in the influence of the irrelevant numerical magnitude on physical size judgments will be positively related to children’s mathematical competence. Therefore children who seem to automatically activate the irrelevant dimension during incongruent trials are slower and less accurate at choosing the physically larger number and will have higher standardized math scores.

2. Method 2.1. Participants One hundred and seventy-four children in grades one and two were recruited from three elementary schools in London, Ontario, Canada. Forty-four children were excluded because they received a different task. Three boys were excluded from the analyses because they were unable to complete the numerical tasks properly. Five ESL students, who were learning English as a second language, were also excluded from the study so as to avoid a language of testing proficiency confound. One child with Autism, one child with down syndrome and one child with developmental dyspraxia was excluded from the study,

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and as a result, the sample of 119 participants represents a group of typically developing students in 1st and 2nd grade (M = 88.40 months/7 years 4 months, SD = 6.87 months). There were 74 girls and 45 boys with various ethnic backgrounds who participated in this study. Sixty-one participants were in grade one and 58 participants came from grade two classrooms. Parent consent forms, approved by the University of Western Ontario’s Research Ethics Board, were completed by the participants’ parents or legal guardians before the study was conducted. Permission was granted by the Thames Valley District School Board, as well as from the principals and teachers of each public school. 2.2. Materials 2.2.1. Numerical comparison task To measure children’s explicit number processing abilities, a numerical comparison task was used, in which participants were presented with two single digit numbers (ranging from 1 to 9) on a computer screen, and were asked to choose the numerically larger number as fast as they could without making any errors. Both numbers had a font size of 58 and appeared on a 15 inch computer screen on either side of a centrally-located fixation dot until the participants made a response. There were 108 trials in which the ratio between the two numbers was manipulated and fell between 0.11 and 0.89, for example the ratio between two and three is 0.67 (Appendix A for a list of pairs and ratios). There were 27 levels of ratio for the numerical comparison task. Each ratio was repeated four times in random order, and each number was counterbalanced for the side of presentation. Each participant received a break halfway through the task. 2.2.2. Physical size congruity task To measure children’s implicit processing of numerical magnitude, the ‘Number Stroop’ paradigm (also commonly referred to as the physical size congruity effect) (Girelli et al., 2000; Rubinsten et al., 2002) was administered. In this task, participants were presented with two single digit numbers with one number physically larger than the other on the computer screen. The participants were asked to choose the physically larger number as fast as they could without making any errors. The stimuli remained on the computer screen until a response was made by pressing the left or right computer key. Each participant’s reaction times and accuracy scores were recorded once they made their response. A total of 72 trials were administered to the participants: 24 incongruent, congruent and neutral trials. Congruent trials are those in which the physically larger number was also numerically larger (e.g. 4 8). In an incongruent trial, the physically larger number was numerically smaller (e.g. 8 4). In the congruent and incongruent trials, six trials had relatively small ratios, 0.11, 0.13, 0.14, 0.17, 0.2 and 0.22 and six trials had relatively large ratios, 0.78, 0.8, 0.83, 0.85, 0.88 and 0.89 (Appendix B for a list of pairs and ratios for incongruent and congruent trials). The six smallest and six largest ratios were presented twice for the congruent and incongruent trials with the order randomized and the larger number appearing

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equally on both sides of the computer screen. For the neutral trials, each digit was paired with itself and the physical size of one digit was larger (e.g.: 8 8). Pairs for the neutral trials were created from 1 to 9 and were presented twice with digits 1, 2, 4, 5, 7 and 8 was presented for a third time to equal 24 trials. The font sizes of the large and small digits were 58 points and 30 points respectively. Thus the font size of the larger numeral in each pair was approximately double in size, which is similar to the stimulus parameters used by Girelli et al. (2000), whose font sizes were 48 and 24. Both numbers appeared equidistant from the middle of the screen. Participants were given a break halfway through the task. 2.2.3. Mathematical skills Two mathematics subtests from the Woodcock Johnson III subtests of achievement (Woodcock, McGrew, & Mather, 2001) were administered to each participant. First, the calculation subtest was completed to measure the ability to perform mathematical computations. The children were asked to complete as many simple arithmetic facts as they could without any time restraints. The test got progressively more difficult; therefore, the children were asked to tell the experimenter when he or she was finished. Second, in order to measure the ability to solve simple addition, subtraction, and multiplication facts quickly, the math fluency subtest was administered. In this task, the participants were required to answer as many simple arithmetic problems as they could without make any errors in three minutes. 2.3. Procedure Once permission was granted by the school board and the principal of each school, information letters and parent/participant consent forms were sent home with all grades one and two students in the schools. Testing began once a large number of consent forms were returned to each school. Testing was administered in a quiet room within the school such as, the special education room, learning support teacher room, empty classroom or the library. First, the participants completed the number tasks on the computer; the numerical comparison task and the size congruity task were counterbalanced across participants. Second, the participants completed the math calculation subtest, followed by the math fluency subtest. Each testing session took approximately 30 min to complete per participant. After the session was finished the child was given a $15.00 gift card for a local book store in appreciation for their participation in the study. Following the session any questions by the students were answered and the child returned to class.

Mussolin & Noel, 2007). For example, Holloway and Ansari (2009) defined ‘small’ numerical distance trials as those pairs of numbers separated by either a distance of 1 or 2 and ‘large’ distance trials as those pairs where the numerical distance was either 5 or 6. In the present study, we adopt a different approach. Specifically, in order to enumerate the relationship between ratio and reaction time (RT), the slope and intercept of the regression line that relates ratio and RT was calculated using reaction times for each correct trial for each participant. The slope and intercept represent a more fine-grained assessment of the ratio effect that moves the analysis beyond an arbitrary separation of groups of trials into ‘small’ versus ‘large’ numerical ratios (or numerical distances). Furthermore, while the slope indicates the strength of the relationship between ratio and reaction time, the intercept can be used as a measure of participants’ reaction times.

3.2. Reaction time data In order to establish whether our measures of slope and intercept changed between first and second grade and if they did so differentially, a Mixed Factorial Analysis of Variance using measure (intercept and slope) as a within subjects variable and grade (1st and 2nd grade) as a between subjects variable was conducted on reaction times. We found a significant main effect of measure [F(1, 117) = 274.12, p < .0001, g2 = .70], and a significant interaction between measure and grade [F(1, 117) = 3.98, p = .048, g2 = .03]. In order to investigate the locus of this interaction, independent samples t-tests were calculated to investigate differences between grades for intercept and slope separately. The results from these analyses illustrate that the intercept is significantly higher in grade one than in grade two [t(104) = 5.39, p < .0001], showing that children in grade two are significantly faster at choosing the larger number in a numerical comparison task than grade one participants (see Table 1). Although there is no significant difference in slope between both grades, this difference is approaching significance, [t(117) = 1.88, p = .062] with students in grade one showing a non-significantly steeper slope than grade two students. Correlational analyses between age and linear slope revealed a significant negative correlation [r(117) = .24, p = .009], indicating that the older children exhibit smaller effects of ratio on reaction times, resulting in shallower slopes. Further analyses show that a negative correlation

Table 1 Mean intercept and slope data for grades 1 and 2 participants. Grade 1 RT

3. Results 3.1. Ratio effect In most existing research, the numerical distance effect is quantified by collapsing reaction time into small and large distance categories (Holloway & Ansari, 2009;

Linear slope Mean SD

Grade 2 ACC

RT

ACC

513.03 363.86

0.23 0.12

400.12 282.87

0.21 0.13

Linear intercept Mean 1139.71 SD 297.89

1.03 0.05

892.01 195.14

1.02 0.09

Note: RT, response time; ACC, accuracy.

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is also significant between age and intercept [r(117) = .48, p < .0001]; demonstrating that older children are faster at the number comparison task. To determine whether the ratio effect is related to individual differences in math achievement scores, partial correlations were conducted using math fluency subtest and math calculation subtest raw scores and intercept and slope of the ratio effect, while controlling for chronological age. A significant negative correlation between math fluency raw scores and intercept was found [r(116) = .42, p < .0001] as well as with slope [r(116) = .37, p < .0001] (see Fig. 1). Additionally, negative correlations were also found between math calculation raw scores and intercept [r(116) = .22, p = .018] and math calculation raw scores and slope [r(116) = .21, p < .025] (see Fig. 2). As can be seen in Fig. 2, one participant had both an extremely low math score as well as a large slope (slope = 2112.66 ms, math calc raw score = 7), and as can be seen in Fig. 3, another participant had an extremely high math score and an average slope (slope = 677.51 ms, math calc raw score = 26). Once both potential outliers were removed, significance levels of the correlation between slope and math fluency did not change [r(114) = .39, p < .0001] and neither did the significance levels between math calculation and slope [r(114) = .27, p < .01]. Using a similar formula presented in the Holloway and Ansari (2009) study to calculate the ratio effect, we subtracted the correct RT for small ratio (0.11–0.33) from the correct RT for large ratio (0.67–0.89) and divided it by the correct RT for small ratio. This measure of the ratio effect was also found to significantly correlate with the math fluency raw scores [r(116) = .25, p < .01], while controlling for age. However, this measurement of the simple ratio effect did not significantly correlate with math calculation raw scores while controlling for age [r(116) = .15, ns]. To investigate the extent to which the ratio effect explains unique variance in the math fluency and calculation subtest raw scores, hierarchical regression analyses were conducted (see Table 2). Three steps were included in the

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Fig. 2. Scatter plot showing the significant negative correlation between Math Calculation raw scores from the Woodcock Johnson III battery and linear slope of reaction times for all participants.

Fig. 3. Mean reaction times (ms) for incongruent and congruent trials when the ratio is small and large for 1st and 2nd grade students.

analysis in order to establish whether the slope explained unique variance in math fluency and calculation over and above age differences (step 1) and individual differences in the basic reaction times during number processing. For this we used entered neutral trials in the size congruity task (step 2). The results of this model show that after controlling for these factors, the ratio effect (slope) explained a significant amount of unique variance in math fluency, DR2 = .081, F(1, 115) = 14.79, p < .0001, as well as a significant amount of unique variance in calculation, DR2 = .027, F(1, 115) = 4.66, p = < .05. 3.3. Accuracy data

Fig. 1. Scatter plot showing the significant negative correlation between math fluency raw scores from the Woodcock Johnson III battery and linear slope of reaction time for all participants.

An independent samples t-test was conducted to investigate accuracy rates, and we found no significant difference between mean accuracy for grade one and two students [t(117) = .15, p = .881]. Intercept and slope were calculated for the accuracy data and we found no significant differences between intercept [t(117) = .87, p = .388] and slope [t(117) = 1.19, p = .235] for grade one and two students (see Table 1). For both t-tests, equal variances were assumed. Due to mean accuracy reaching ceiling or close to ceiling levels (M = .90, SD = .06), further analyses of the accuracy data were not conducted.

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Table 2 Hierarchal regression analyses predicting math fluency and calculation scores. Step

Math fluency Predictor

1 2 3

Age Neutral trials (mean RT) Linear slope RT

Calculation b 0.434** .048 0.265**

R2

DR 2

Predictor

b

R2

DR2*

0.274 0.292 0.373

0.274** 0.018 0.081**

Age Neutral trials (mean RT)r Linear slope RT

0.500** 0.00 .179*

0.295 0.298 0.325

0.295** 0.00 0.027*

Note. RT: reaction time. * p < .05. ** p < .01.

3.4. Size congruity effect 3.4.1. Reaction time data The size congruity effect was examined by comparing the correct reaction times for the incongruent and congruent trials (Besner & Coltheart, 1979; Henik & Tzelgov, 1982; Tzelgov, Meyer, et al., 1992). The mean RTs for correct responses was calculated for the incongruent and congruent trials and entered into a Mixed Factorial Analysis of Variance using congruency (incongruent and congruent) and ratio (small and large) as within subjects variables and grade (1st and 2nd grade) as a between subjects variable. The small ratio is defined by the six smallest ratios (.11–.22) and large ratio is defined by the six largest ratios (.78–.89). Results indicated that a significant main effect of congruency on RT was found [F(1, 117) = 6.06, p < .05, g2 = .049] revealing that participants were slower at determining the physically larger number during incongruent trials than congruent trials. A significant interaction of congruency and ratio was also observed [F(1, 117) = 7.62, p < .01, g2 = .061] (see Fig. 3). To investigate the source of the interaction further, a paired samples t-test was carried out and revealed that the reaction times for incongruent small ratio trials were significantly larger (M = 773.05, SD = 215.39) than congruent small ratio trials (M = 730.97, SD = 181.53) [t(118) = 3.51, p = .001] (see Fig. 3). However, there was no significant difference in RT between the incongruent and congruent trials when the ratio was large [t(118) = .74, p = .462]. No effect of grade was found demonstrating that grade one and two students do not significantly differ in the size congruity effect. The interference and facilitation effects were also examined by investigating the reaction times from correct trials for incongruent and congruent conditions in comparison to correct trials from the neutral condition (Girelli et al., 2000; Henik & Tzelgov, 1982; Rubinsten et al., 2002). The interference effect was calculated by subtracting the mean neutral RTs from the mean incongruent RTs. The facilitation effect was calculated by subtracting the mean neutral RTs from the mean congruent RTs. To explore these effects as a function of ratio, a Mixed Factorial Analysis of Variance was carried out with the task effect (facilitation and interference) and ratio (small and large) as within subjects variables and grade (1st and 2nd) as a between subjects variable. A significant main effect of task effect (facilitation versus interference) was found [F(1, 117) = 6.06, p < .05, g2 = .049] indicating that there is a significant difference between facilitation and interference effect, with the facilitation effect being, on average, negative (M = 12.91,

SD = 101.29) and mean interference being positive (M = 4.12, SD = 86.50). Moreover, a significant interaction between effect and ratio was found [F(1, 117) = 7.62, p < .01, g2 = .061]. In order to investigate the locus of the interaction, a paired samples t-test was conducted and a significant difference was found between the interference and facilitation effect for small ratio [t(118) = 3.51, p = .001], which reveals that the interference effect was larger when the ratio between the numbers was small (see Fig. 4). However, there is no significant difference between the facilitation and interference effect when the ratio is large [t(118) = .74, ns], demonstrating that the difference between interference and facilitation effects is modulated by ratio. There was no significant main effect of grade [F(1, 117) = .56, p = .568] and no significant interaction between grade and ratio [F(1, 117) .032, p = .858]. There was no significant interaction between effect and grade either [F(1, 117) = 0, p = .996]. Correlational analyses also showed that the interference effect with small [r(117) = .024, ns] and large ratio [r(117) = .028, ns] does not correlate with children’s chronological age; and the facilitation effect with small [r(117) = .008, ns] and large ratios [r(117) = .068, ns] was also not found to correlate with age. Partial correlational analyses were explored to determine whether the facilitation and interference effects were related to individual differences in math achievement scores, while controlling for age. These analyses revealed that no significant correlations were found between the

Fig. 4. The mean interference and facilitation effects when the ratio is either large or small for both 1st and 2nd grade students.

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significantly correlated, is important in order to control for reaction times derived from processing Arabic numerals in a different task (i.e. the size congruity task in the present study). Therefore, hierarchical regression analyses were conducted to investigate the extent to which the ratio effect explains unique variance in the math fluency and calculation subtest raw scores once the mean size congruity effect (calculated mean RT for incongruent, congruent and neutral trials) was entered into the analysis (see Table 4). Four steps were included in the analysis in order to establish whether the slope of the ratio effect explained unique variance in math fluency and calculation over and above age differences (step 1), the size congruity effect (calculated using the same formula as above) (step 2) and the mean reaction time for neutral trials in the size congruity task (step 3). The results of this model show that after controlling for these factors, the ratio effect (slope) explained a significant amount of unique variance in math fluency, DR2 = .081, F(1, 114) = 14.64, p < .0001, as well as a significant amount of unique variance in calculation, DR2 = .027, F(1, 114) = 4.59, p < .05. Thus, even after accounting for the variance in math scores explained by individual differences in the size congruity task reaction times, there still exists variance that can be uniquely explained by between-subjects variability in the ratio effect. Conversely, after controlling for age, the size congruity effect did not explain a significant amount of unique variance in the math fluency subtest, DR2 = .017, F(1, 116) = 2.74, ns, or the math calculation subtest, DR2 = .003, F(1, 116) = .47, ns. (see Table 4).

interference or facilitation effects and math fluency or math calculation raw scores. (see Table 3 for correlations). 3.4.2. Accuracy data Due to the mean accuracy data reaching ceiling for incongruent small (M = .964, SD = .058) and large ratios (M = .972, SD = .056) as well as congruent small (M = .976, SD = .063) and large ratios (M = .977, SD = .048), further analyses were not conducted. 3.4.3. Relationships between the ratio and size congruity effects In order to explore the relationship between the ratio effect and the size congruity effect, correlational analyses were carried out between intercept and slope for reaction time and the interference and facilitation effects. We found a marginal significant correlation between the interference effect for small ratio and the intercept RT [r(117) = .179, p = .051] and no significant correlation between interference effect and the slope of the ratio effect [r(117) = .047, ns]. The interference effect with large ratio did not significantly correlate with the intercept of the ratio effect [r(117) = .049, ns] or slope RT [r(117) = .053, ns]. Correlational analyses also showed that the facilitation effect for the small ratio trials did not significantly correlate with the intercept of the ratio effect [r(117) = 173, p = .06] or slope RT [r(117) = 034, ns]. Lastly, there is no significant correlation between facilitation effect with large ratio and intercept of the ratio effect [r(117) = .078, ns] and slope RT [r(117) = .087, ns] (see Table 3). We correlated the ratio effect (large ratio-small ratio/ small ratio) with the size congruity effect (incongruent– congruent/congruent) and found no significant correlation [r(117) = .002, ns]. The simple ratio effect does not significantly correlate with the interference effect [r(117) = .015, ns] or the facilitation effect [r(117) = .013, ns]. Determining whether the slope of the ratio effect accounts for unique variance in the math scores independently of the size congruity effect, despite not being

4. Discussion The aim of the current study was to investigate the developmental changes and individual differences in both automatic and intentional processing of numerical magnitude and, secondly, to examine how measures of implicit and explicit magnitude processing relate to variability in children’s math achievement. To address these aims, chil-

Table 3 Correlations between math fluency and math calculation raw scores and the ratio and size congruity effects.

1 2 3 4 5 6 7 8 9 10

Measure

1

2

3

4

Interference effect small ratioa Interference effect large ratiob Facilitation effect small ratioc Facilitation effect large ratiod Linear slope of ratio effect RT Linear intercept of ratio effect RT Ratio effecte Size congruity effectf Math fluency RS Math calculation RS

–

.138 –

.465*** .313*** –

.649*** .356*** .590*** –

5 .047 .053 .034 .087 –

.056 .033

0.07 0.042

0.01 0.059

0.007 0.05

0.366*** .207*

6

7

8

9

10

.179 .049 .173 .078 .264** –

.015 .005 .082 .054 .858*** .037 –

.133 .246** .541*** .404*** .000 .003 .002 – .068 .008

– .505***

_

.417*** .218*

Note. RT, reaction time; RS, raw score; partial correlations below the diagonal; simple correlations above diagonal. a Incongruent small ratio RT – neutral RT. b Incongruent large ratio RT – neutral RT. c Congruent small ratio RT – neutral RT. d Congruent large ratio RT – neutral RT. e Large ratio RT – small ratio RT/small ratio RT. f Incongruent RT – congruent RT/congruent RT. * p < .05. ** p < .01. *** p < .001.

252** .145

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Table 4 Hierarchal regression analyses predicting Math Fluency and Calculation scores. Step

Math fluency Predictor

1 2 3 4

Age Mean SCERT Neutral trials (mean RT) Linear slope RT

Calculation b 0.434** 0.007 0.055 0.307**

R2

DR2

Predictor

0.274 0.291 0.292 0.373

0.274** 0.017 0.002 0.081**

Age MeanSCERT Neutral trials (mean RT) Linear slope RT

b 0.500** 0.02 0.019 0.178*

R2

DR 2

0.295 0.298 0.298 0.325

0.295** 0.003 .000 0.027*

Note: SCE, size congruity effect; RT, reaction time. * p < .05. ** p < .001.

dren’s performance (for children in 1st and 2nd grade) on the physical size congruity task and the number comparison task was investigated. In addition, the participants’ performances on both tasks were correlated with their mathematical achievement measures to determine whether relationships exist between measures, on the one hand, of intentional and automatic number processing and, on the other hand, children’s performance on standardized tests of arithmetic. Based on previous developmental findings (Mussolin & Noel, 2008; Rubinsten et al., 2002; Sekuler & Mierkiewicz, 1977), it was hypothesized that the ratio and size congruity effect would be present in both 1st and 2nd grade participants. Given that both these effects are thought to be measures of basic numerical magnitude processing, it was predicted that individual differences in these two effects should be correlated. More specifically, individuals with relatively smaller distance effects could be expected to be able to process numerical magnitude automatically and thus exhibit greater interference and facilitation effects on the size congruity task. Furthermore, it was hypothesized that, consistent with previous findings (De Smedt et al., 2009; Holloway & Ansari, 2009), a significant negative correlation should be obtained between the ratio and math competence scores. In addition, given the hypothesis that automatic processing of numerical magnitude is an important prerequisite for higher-level numerical and mathematical skills (Girelli et al., 2000; Rubinsten et al., 2002), it was also predicted that individual differences in the size congruity effect would positively correlate with the math achievement scores. The presence of a large ratio effect is associated with an immature representation and processing of numerical magnitude; therefore, participants who show a large ratio effect are predicted to have low standardized math scores. In contrast, we speculate that individuals who have more precise internal mental number line, are more likely to automatically process numbers and will show a large size congruity effect (resulting from large interference of the irrelevant numerical dimension during the processing of the relevant dimension of physical size), and therefore, would have higher math standardized scores. 4.1. Ratio effect Past research findings have established developmental differences in the distance and ratio effects (Holloway & Ansari, 2009; Sekuler & Mierkiewicz, 1977). In the current

study, the ratio effect was found to be significant in both 1st and 2nd grade participants with developmental changes greatest in the intercept. These findings indicate that the grade two students are faster than their grade one peers when performing this task. The slope of the ratio effect was steeper in first grade children compared to second grade indicating that children in grade one are slower at determining the numerically larger number when the ratio is large in comparison to 2nd grade participants. Given that this finding is marginally significant, it appears to take longer for developmental changes to become significantly evident in the slope. However, significant correlations with chronological age support the decrease in the slope as children age. In addition, the present study also replicated findings of correlations between the ratio effect and the mathematics achievement scores (De Smedt et al., 2009; Holloway & Ansari, 2009). Using the slope and intercept of the regression line relating reaction time and ratio, as well as the simple ratio effect measure adapted from the Ansari and Holloway paper, as independent variables to predict variability in children’s math scores, a significant relationship between all measures of the ratio effect and children’s math fluency scores was revealed. These findings suggest, consistent with previous findings (De Smedt et al., 2009; Holloway & Ansari, 2009; Mundy & Gilmore, 2009) that explicit, intentional processing of numerical magnitude is associated with individual differences in children’s mathematical competence scores. Children who had a steeper slope relating numerical ratios and reaction times were found to have lower math scores suggesting that the ability to discriminate between numerical magnitudes is related to individual differences in performing simple arithmetic. Furthermore, the regression analyses revealed that individual differences in the size of the ratio effect (as measured by the slope) accounted for significant variability in children’s math scores independent of age and reaction times. It is unclear from the present findings and the existing literature whether the relationship between performance on simple arithmetic tasks, measured by the standardized measures of math achievement, and the ratio/distance effects are an indication of the relationship between the internal representation of numerical magnitude and math performance. It has been suggested that the distance and ratio effects are not indicators of overlapping representations of numerical magnitudes, but that instead, they are reflective of decision-related processes that occur at the level of comparison between two magnitudes (Van Opstal, Gevers, De Moor, & Verguts, 2008). Therefore, the relationship

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may be attributed to the process of relating/comparing the two numbers, which is thought to be a necessary step during the computation of simple arithmetic problems. A model consistent with the notion that the numerical magnitude comparison is important for the retrieval of single digit addition facts was proposed by Butterworth, Zorzi, Girelli, and Jonckheere (2001) called the Comp model. In this model, individuals organize arithmetic facts in long term memory in a max + min format, which implies that during the processing stage the addends must be compared. For example, 5 + 3 = 8 and 3 + 5 = 8, are stored in the same format with the max value being presented first 5 (max) + 3 (min) = 8. The Comp model predicts that solving single digit addition problems involves comparing the two digits in order to retrieve response from memory or to count up from the higher addend. The time required to compare addends during the comparison stage, influences and predicts the total time it takes to compute the addition problem. Consequently, the process of comparing and identifying the max and min during addition may be responsible for the strong relationship between distance/ ratio effect task and performance on simple arithmetic measures of math achievement. Moreover, it has repeatedly been argued that the ability to understand the relationships between numerical magnitudes is a crucial component of children’s number development (Nunes, Bryant, & Watson, 2009; Piaget, 1942). In this context, the number comparison task and the ratio effect may be an indicator of individual differences in relational processing of numerical magnitude, which may explain why the ratio and distance effect account for significant variability in children’s mathematical competence as measured by standardized tests. 4.2. The size congruity effect Given the frequently highlighted importance of automatic processing of numerical magnitude in children’s development of number skills (Girelli et al., 2000; Rubinsten et al., 2002) and the absence, thereof, in adults with Developmental Dyscalculia (DD) (Landerl & Kolle, 2009; Rubinsten & Henik, 2005), it seems logical that automaticity in processing numerical magnitude is an essential component to understanding numerical magnitude and performing simple arithmetic problems. Therefore, the investigation of the size congruity effect, which is consistently used to measure automaticity in number processing, is necessary to determine the relation between automatically comparing two numerals and the school-relevant, mental arithmetic abilities. Studies investigating the size congruity effect in children have generated inconsistent findings, and there is no strong evidence indicating at what age children begin to exhibit interference or facilitatory effects of numerical magnitude when processing physical size, which is taken as an indicator of automatically processing numbers (Girelli et al., 2000; Mussolin & Noel, 2007; Rubinsten et al., 2002). In the present study, the size congruity effect was found to be present in first and second grade participants demonstrating that both groups of children seem to automatically process the numerically irrelevant dimension.

41

Girelli and colleagues found that the size congruity effect did not emerge until grade three; however, Rubinsten et al. (2002) found that by the end of grade one, children showed a significant size congruity effect. Since the grade one participants in this study had completed half a school year at the time of testing, the current results seem to be most consistent with Rubinsten’s findings and suggest that both grade one and two students automatically process the irrelevant numerical dimension. It is important to note that there was no difference in the interference and facilitation effects between both grades. Although, the age ranges do not neatly fit the same window as both papers discussed, children were tested near the end of grades one and two and have, therefore, received some instruction and consequently have some level of familiarity with numerical symbols and mathematical computations. Thus, this time period might have provided the children with sufficient experience to achieve some automaticity in numerical information to generate both facilitation and interference effects. The irrelevant dimension of numerical magnitude was found to influence processing of the relevant physical dimension only when the ratio between the two numbers is small (distance large). The longer reaction times during these trials suggest that children have greater difficulty choosing the physically larger number when the ratio between the two numbers is small. These results are consistent with the findings reported by Mussolin and Noel (2007). These authors investigated the size congruity effect in 7 and 8 year old children and found that when the distance between the two numbers is large (and therefore their ratio relatively small), the interference effect of the task-irrelevant numerical magnitude was larger compared to when the distance was small (their ratio relatively large). Consistent with the relative-speed account of the size congruity effect (Schwarz & Ischebeck, 2003), these findings reveal that when the irrelevant dimension (numerical magnitude) has more time to interfere (corresponding to when the numerical distance is large and the ratio small) with the decision over the task relevant dimension (physical size), the resulting interference effect is larger. Thus, consistent with previous findings, the present results suggest that children’s judgment of relative physical size were affected by the task-irrelevant parameter of numerical ratio. However, contrary to the original hypotheses, the correlational results showed that individual differences in math achievement were not found to correlate with the size congruity effect (incongruent > congruent), or the facilitation (neutral > congruent) and interference effects (incongruent > neutral). These results therefore suggest that task-irrelevant or automatic processing of numerical magnitude as measured by the size congruity effect does not explain individual differences in the math achievement scores of children in 1st and 2nd grade. 4.3. Ratio effect and size congruity effect The number comparison and physical size congruity paradigms are commonly used to measure basic numerical processing (both are often referred to as measures of ‘number sense’); however, for both practical (diagnostic) and theoretical reasons, it is important to disambiguate

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the specific cognitive processes that are measured by the ratio and size congruity effect. Given that both tasks measure the processing of numerical magnitude it could be expected that both tasks are related to one another. However, contrary to this expectation, no relationship between the size congruity effect and the ratio effect was found in the present study, which is consistent with the findings of Landerl and Kolle (2009), who correlated the two effects in a sample consisting of both typically developing children and children with Developmental Dyscalculia. However, while Landerl and Kölle only investigated the relationship between the size congruity effect (incongruent–congruent) and the distance effect, we demonstrate here that there were also no correlations between explicit number comparison and the facilitation (congruent–neutral) and interference (incongruent–neutral) effects. In addition, the ratio effect was found to be correlated with math competence scores, whereas there was no significant relationship between individual differences in any of the measures of the size congruity effect and children’s math achievement. Landerl and Kolle (2009) found a weak correlation between the size congruity effect and performance on a multiplication subtest in a sample including both typically developing children and children diagnosed with dyscalculia. However, this finding potentially is confounded by the fact that their sample consisted of both children with and without Developmental Dyscalculia. Since the children with dyscalculia were not found to exhibit a size congruity effect, the correlation between math performance and size congruity effect may be driven by the children with dyscalculia who had both small size congruity effects and low math scores and is therefore not generalizable to typically developing children. The present study shows, that in an unselected sample no such correlation is obtained. Taken altogether, we propose that the ratio effect, which is a measure of intentional numerical magnitude processing (children have to attend to numerical magnitude), measures children’s ability to compare (or put into relationship) with one another two numerical magnitudes that are represented by Arabic numerals. We contend that it is these processes of explicitly activating the numerical magnitude represented by Arabic numerals and putting these magnitude representations into relationships with one another that explains significant variability in children’s arithmetic abilities. Despite not detecting a relationship between the ratio and size congruity effects, ratio appears to influence performance on the size congruity task. The ratio between the two digits appears to modify the size congruity effect at a level of processing that is different from the explicit numerical magnitude task. One model that is consistent with this idea is the independent encoding postulate proposed by Tzelgov, Henik, et al. (1992), Tzelgov, Meyer, et al. (1992). When numerical size is irrelevant, it is proposed that independent encoding dominates autonomous processing, which consists of a large-small classification process. During this process, a crude dichotomous representation of the irrelevant numerical size is activated; digits 2–4 are initially encoded as small, and digits 6–8 are encoded as large. During intentional processing of numerical magnitude, the relevant numerical magnitude is

mapped internally to reflect the relative location of the number. This model provides further evidence that different cognitive mechanisms are responsible for intentional and autonomous processing of numerical information. Tzelgov, Henik, et al. (1992), Tzelgov, Meyer, et al. (1992) suggest that numerical information processed automatically may be less refined than numerical information processed under intentional conditions. In the context of the present finding, this distinction is important as it suggests that the level of numerical magnitude processing measured by the size congruity effect does not explain variability in children’s mathematical achievement. This finding therefore challenges the frequently articulated hypothesis (and the one that was formulated at the outset of the study) that automaticity in number processing, as measured by the size congruity effect, is an important indicator and predictor of children’s numerical and mathematical processing skills. The current findings also have important implications in the diagnostic and remediation tools for children with Developmental Dyscalculia. Previous research has shown that adults as well as children with dyscalculia do not show a size congruity effect and it has been argued that individuals with DD have difficulties automatically processing numerical magnitude (Landerl & Kolle, 2009; Rubinsten & Henik, 2005). Landerl and Kolle found that typically developing children as young as 8 years old showed a size congruity effect, but all groups of children with dyscalculia did not show any indication of a size congruity effect. In contrast, Rubinsten and Henik (2005) found that adults with dyscalculia did show a physical size congruity effect, which was comprised of an interference component only. The present findings, however, suggest that automatic processing of numerical magnitude measured by the size congruity effect does not appear to be related to the mathematical achievement scores of typically developing children. Similarly, Heine and colleagues (2010) found that when numerical distance is large, all low, normal and high math achieving children show a physical size congruity effect; however when the distance is small, a size congruity effect was not present in either of the groups. In addition, congruity effects were found in all three math achievement groups during the numerical comparison task. The current study’s findings, in addition to Heine et al.’s findings, challenge the notion that the automatic processing of numerical magnitude as measured by the physical size congruity paradigm is related to individual differences in math achievement.

5. Conclusion The study of the typical and atypical development of basic numerical magnitude processing or ‘number sense’ has revealed that measures of magnitude processing are related to individual differences in achievement and develop atypically in children with mathematical difficulties, thereby demonstrating the crucial role played by the representation and processing of numerical magnitude in children’s development of mathematical skills. The findings from the present study demonstrate that it is crucial, for both practical and theoretical reasons, to distinguish be-

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tween different levels of numerical magnitude processing and to build a more differentiated model of the processes that constitute the development of basic numerical magnitude processing and its relationship to individual differences in mathematical achievement. Specifically, the findings reported above, demonstrate that intentional and automatic measures of numerical magnitude processing are neither correlated with one another nor do they predict individual differences in the same way. While intentional processing of numerical magnitude predicts variability in children’s performance on standardized tests of mathematical achievement, measures of automatic number processing do not. Therefore, it is important to move beyond a unified construct of basic number processing and towards a differentiated, multi-level model of the typical and atypical development of numerical magnitude processing and its relationship to the learning of arithmetic. Acknowledgements This work was supported by Grants from the Natural Sciences and Engineering Council of Canada (NSERC), The Canadian Institutes of Health Research (CIHR), The Canada Foundation for Innovation (CFI), The Ontario Ministry for Research and Innovation (MRI) and the Canada Research Program (CRC). We would like to thank Bea Goffin, Meghan Reid and Rebecca Merkley for their assistance with the data collection and coding as well as Gavin Price for helpful comments on an earlier draft.

Appendix B Table B1. Table B1 Numerical pairs and ratios for incongruent and congruent trials. Congruency

Number pair

Ratio

Times repeated

Congruent Congruent Congruent Congruent Congruent Congruent Incongruent Incongruent Incongruent Incongruent Incongruent Incongruent Congruent Congruent Congruent Congruent Congruent Congruent Incongruent Incongruent Incongruent Incongruent Incongruent Incongruent

1–9 1–8 1–7 1–6 1–5 2–9 1–9 1–8 1–7 1–6 1–5 2–9 7–9 4–5 5–6 6–7 7–8 8–9 7–9 4–5 5–6 6–7 7–8 8–9

0.11 0.13 0.14 0.17 0.20 0.22 0.11 0.13 0.14 0.17 0.20 0.22 0.78 0.80 0.83 0.86 0.88 0.89 0.78 0.80 0.83 0.86 0.88 0.89

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

References

Appendix A Table A1. Table A1 Numerical pairs and ratios for the numerical comparison trials. Number pair

Ratio

1–9 1–8 1–7 1–6 1–5 2–9 2–8 2–7 3–9 3–8 2–5 3–7 4–9 4–8 5–9 4–7 3–5 5–8 2–3 5–7 6–8 7–9 4–5 5–6 6–7 7–8 8–9

0.11 0.13 0.14 0.17 0.2 0.22 0.25 0.29 0.33 0.38 0.4 0.43 0.44 0.5 0.56 0.57 0.6 0.63 0.67 0.71 0.75 0.78 0.8 0.83 0.86 0.88 0.89

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