Supplementary Material
Method
Participants
Twenty-six students from Ben-Gurion University of the Negev participated in the experiment. Thirteen of them were diagnosed as having DD and the other thirteen were age and sex matched controls. The controls did not have any learning or other disabilities. All students were paid 60 NIS ($15) for participation in the experiment.
DD group. Before the beginning of the experiment every candidate was tested individually for DD, dyslexia (i.e., reading disabilities), IQ and ADHD. All the participants in the clinical group were diagnosed as having DD according to Rubinsten and Henik's (2005, 2006) battery, which is based on Shaelv’s work (Shalev, Manor, Amir, & Gross-Tsur, 1993) (see Table 1 and 2 for more details).
For reading assessment we used a reading test that was composed and published by Shalev and colleagues (Shalev et al., 1993; Shalev, Manor, Auerbach, & Gross-Tsur, 1998) and standardized in a separate study (Shalev et al., 2001). Our sample of DD students did not have any reading problems and there were no differences in their scores and those of the control group in any of the reading tests. The number of errors for the controls was 11.8 (SD 6.3) and the number of errors for the DD group was 9.34 (SD 6.9).
We converted participants’ scores on the Raven’s Progressive Matrices to IQ scores. All the participants achieved average or above average IQ scores. There were no differences between the groups in the Raven's raw scores, which were 51.7 (SD 2.35), and 51.5 (SD 2.3), for the control and DD groups, respectively.
We used a Hebrew version of the Conners Self-Report Ratings to help assess ADHD and evaluate problem behavior in children and adolescents. None of the participants were diagnosed as having ADHD according to the results of the Conners Ratings.
Control group. Participants in this group were never diagnosed as having DD or any other learning disability. All of them took the arithmetic, reading, Raven’s Progressive Matrices, and Conners tests and did not show any learning disability.
Stimuli and Design
Each trial was composed of two Arabic digits at the center of a computer screen and two figures (7/8ths of a circle) at the periphery of the screen (5 degrees from the center of the screen, one on each side). Each participant performed two kinds of tasks in two separate blocks; in one task the relevant dimension was physical size, and in the second, numerical value. Participants performed both tasks under two different attentional conditions: 1) no-load condition, and 2) attentional load condition (load condition).
In each trial the two digits could differ in their numerical value and physical size. The digits were presented in Arial font, size 18 or 24. Each block contained equal numbers of congruent, incongruent and neutral stimuli. A congruent stimulus was defined as a pair of digits in which one digit was larger than the other on both the relevant and irrelevant dimensions (e.g., 6 3). A neutral stimulus was defined as a pair of digits that differed only on the relevant dimension (e.g., 6 6 in the size comparison, 6 3 in the numerical comparison). An incongruent stimulus was defined as a pair of digits in which a given digit was simultaneously larger on one dimension and smaller on the other dimension (e.g., 3 6). The digits 1 through 9 were used with the digit 5 excluded. The two digits in each pair could be of the same numerical value (in which case the pair served as a neutral for the size comparison) or could differ in numerical distance. There were three numerical distances: 1 (the digits 1-2, 3-4, 6-7, and 8-9), 3 (the digits 1-4, 3-6, 4-7, and 6-9) or 5 (the digits 1-6, 2-7, 3-8, and 4-9).
In every trial two open circles appeared on the screen; the open circles could be identical or different. They could differ in color or in orientation of the opening (e.g., the color could be different but the shape identical- , or the color could be identical but the shape different- , or the color and shape could be identical- ).
The following variables were manipulated: group (DD vs. control), task (physical, numerical), numerical distance (1, 3 or 5), congruity (incongruent, neutral or congruent), and attention (load vs. no-load). Thus, we had a 2 X 2 X 3X 3 X 2 factorial design. Group was the only between-participants variable, task and attention conditions were manipulated within subjects but between blocks, and the other two variables (distance, congruity) were manipulated within subjects and within block.
Every task (physical or numerical) contained 288 trials: 3 distances x 3 congruity conditions x 2 (same or different) responses x 16 repetitions for every condition. Every participant performed 1,152 trials—288 trials x 2 tasks (physical, numerical) x 2 load conditions (no-load, load).
Procedure
Every participant was tested twice, once in the no-load condition and once in the load condition, with a minimum of three days and maximum of two weeks break between the two sessions. The order of the sessions was counterbalanced between participants for each group (DD and control).
In the load condition participants were instructed to perform two tasks: the primary task was to compare two Arabic numbers. The secondary task was to compare two figures, each presented on a different side of the periphery of the screen, and to report whether the two figures were identical or different. Participants were asked to first indicate, by a key press, their decision regarding the digits and next to vocally report whether the two open circles were identical or not. In the no-load condition, stimuli were identical to those in the load condition (the figures appeared on the screen) but the participants were instructed to perform only the primary task (i.e., to compare the digits) (see Figure 1).
Each trial began with a fixation asterisk for 300 ms. The fixation disappeared, and two open circles appeared in the periphery. One hundred ms after the onset of the circles, two digits appeared at the center of the screen. Participant responded with a key-press to the numbers, followed by a vocal response to the circle, after which the number and circles disappeared. The next trial began 1,000 ms after the participant’s response. Reaction time (RT) was measured in milliseconds from target onset until the participant’s key-press.
Results
Error rates in the primary task for the physical and numerical comparisons were generally low and therefore were not analyzed (in the no-load condition—physical DD: 1%, physical control: 1%, numerical DD: 2%, numerical control: 2%; and in the load condition—physical DD: 1%, physical control: 1%, numerical DD: 3%, numerical control: 2%). Two participants in the DD group and 2 participants in the control group were excluded from the analysis due to error rates higher than 30% in the primary task. The error rates in the secondary task were 13% for the control group and 14% for the DD group.
Only the 'different' responses in the secondary task were analyzed in order to prevent inclusion of trials with a pop-up effect. The secondary task in the attentional load condition requires comparison of two 7/8 circles (see Figure 1 for more details) and to give a vocal same/different response. In the 'same' response the figures were identical and less attention was needed in order to give this response in comparison to the condition where the figures were different. In the different condition the figures varied in two features—orientation and color. According to Treisman and Schmit (1982), attention is needed in order to process a multi-featured object as opposed to processing of a single-featured object that can be processed without attention. More attention is needed in the 'different' condition compared to the 'same' condition. Thus, to maximize the attentional effect of the secondary task, we analyzed only the 'different' response.
For every participant in each condition the mean RT was calculated (only for the correct trials) and subjected to a five-way analysis of variance (ANOVA): group X load X task X congruity X numerical distance.
There were main effects for load [F(1, 24) = 67.73, = 0.73, MSE = 627,480, p < 0.01], as expected, since participants were slower to respond in the load condition compared to the no load condition, and for task [F(1, 24) = 52.8, = 0.68, MSE = 108,126, p < 0.01], since participants were slower to respond in the numerical task compared to the physical task. There was also a main effect for congruity [F(2, 48) = 104.25, = 0.81, MSE = 5,299, p < 0.01], as participants responded faster to the congruent condition (663 ms) than to the neutral condition (677 ms) and were faster in the neutral condition compared to the incongruent condition (742 ms). In addition, there was a main effect for numerical distance [F(2, 48) = 31.26, = 0.62, MSE = 3,121, p < 0.01].
Four two-way interactions were found to be significant. First, there was the interaction between task and numerical distance [F(2, 48) = 22.96, =0.49, MSE = 3,514, p < 0.01]. In the numerical task, there was a distance effect [F(1, 24) = 68.73, MSE = 5,032, p < 0.01], whereas RTs in the physical task were not influenced by the numerical distance [F < 1]. Second, task and group interacted [F(1, 24) = 4.6, =0.16, MSE = 108,126, p < 0.05], with the difference in RT between the groups being larger in the numerical task (814 ms for the DD group and 730 ms for the controls) than in the physical task (611 ms for the DD group and 620 ms for the controls). Third, an interaction between load and congruity was found [F(2, 48) = 8.66, = 0.27, MSE = 4,523, p < 0.01]. The interference components got larger as a result of the load (43 ms with no-load, 87 ms with load) [F(1, 24) =15.68, MSE = 4,821, p < 0.01]. Fourth, the interaction between numerical distance and group was significant [F(2, 48) = 5.64, = 0.19, MSE = 3,121, p < 0.01]. The difference between the RTs of numerical distances 1 and 3 was larger in the control group compared to the DD group. The opposite pattern was observed in relation to the difference between distances 3 and 5 (i.e., larger difference for the DD group compared to the controls). This interaction was modulated by load: namely, the interaction between numerical distance, group, and load was significant (see Figure 2) [F(2, 48)=5.9, = 0.20, MSE=2,882, p 0.01]. Both in the load and the no-load conditions, the slope of the numerical distance effect was similar in the two groups [F 1]. However, in the load condition the difference between the RTs of the numerical distance of 1 and 3 units was larger in the control group compared to the DD group [F(1, 24) =16.12, MSE =4,216 , p < 0.01]. The opposite pattern was observed in relation to the difference between distances 3 and 5 units [F(1, 24) =5.7, MSE =3,577, p < 0.01] (i.e., larger difference for the DD group compared to the controls). No such pattern was found in the no-load condition.
Two additional triple interactions were found to be significant. First, an interaction between congruity, task, and group was found [F(2, 48) = 4.59, = 0.16, MSE = 5,835, p < 0.05]; in the control group the size congruity effect was similar in the physical (68 ms) and numerical (71 ms) tasks, whereas in the DD group, the size congruity effect in the numerical task (125 ms) was larger than in the physical task (49 ms) [F(1, 24) = 7.03, MSE = 7,556, p < 0.05]. Second, an interaction between congruity, task, and load was found [F(2, 48) = 5.6, = 0.19, MSE = 4,989, p < 0.01]. The interference component was more influenced by the load in the physical task compared to the numerical task. Actually, the interference component in the physical task was larger in the load condition compared to the no-load condition [F(1, 24) = 18.2, MSE = 6,353, p < 0.01], whereas the interference component in the numerical task was similar in the no-load and load conditions [F < 1].
References
Rubinsten, O., & Henik, A. (2005). Automatic activation of internal magnitudes: A study of developmental dyscalculia. Neuropsychology, 19, 641-648.
Rubinsten, O., & Henik, A. (2006). Double dissociation of functions in developmental dyslexia and dyscalculia. Journal of Educational Psychology, 98, 854-867.
Shalev, R. S., Manor, O., Amir, N., & Gross-Tsur, V. (1993). The acquisition of arithmetic in normal children: Assessment by a cognitive model of dyscalculia. Developmental Medicine and Child Neurology, 35, 593–601.
Shalev, R. S., Manor, O., Auerbach, J., & Gross-Tsur, V. (1998). Developmental dyscalculia: What counts? Results from a 3-year prospective follow-up study. Journal of Pediatrics, 133, 358-362.
Shalev, R. S., Manor, O., Kerem, B., Ayali, M., Badichi, N., Friedlander, Y., et al. (2001). Developmental dyscalculia is a familial learning disability. Journal of Learning Disabilities, 34, 59–65.
Treisman, A., & Schmidt, H. (1982). Illusory conjunctions in the perception of objects. Cognitive Psychology, 14, 107-141.
Table 1
Arithmetic Battery Part 1: Number Comprehension and Production. Mean number of correct answers
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Test / DD / Control / TotalMatching written Arabic numbers to quantities / 5(0) / 5 (0) / 5
Comprehension of quantities / 5(0) / 5 (0) / 5
Serial order / 1.96(0.27) / 2 (0) / 2
Counting / 4.77(0.44) / 5 (0) / 5
Production (writing) of numbers / 5(0) / 5 (0) / 5
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Note. Standard deviations are in parentheses. Total = total number of exercises. There are no significant differences between the groups.
Table 2
Arithmetic Battery Parts 2-4: Calculation
DD / ControlMean RT / Mean ACC / Mean RT / Mean ACC
Simple operations
(one digit)
total of 20 equations
/ 5.12 (2.22) / 18.83 (1.27) / 3.4 (1.43) / 19.4 (1.43)
Complex operations (2 or 3 digits)
total of 32 equations / 27.7 (10.5) * / 25.6 (3.17) * / 7.37 (2.12) / 31.1 (0.3)
Decimals
(e.g., 1.43 - 0.59)
total of 8 equations / 14.19 (4.4)* / 6.9 (1.16)* / 8.21 (1.8) / 7.6 (0.3)
Fractions
(e.g., ½ - ½)
total of 20 equations / 12.08 (4.2) / 17.16 (3.57)* / 10.3 (0.9) / 19.5 (0.2)
Note. Standard deviations are in parentheses. Mean RT in seconds. ACC = mean number of correct problems. * = Significant difference between performance of the DD group and the control group.