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Background: The development of an evolutionarily grounded analogue magnitude representation linked to the parietal lobes is frequently thought to be a major factor in the arithmetic development of humans. We investigated the relationship between counting and the development of magnitude representation in children, assessing also children’s knowledge of number symbols, their arithmetic fact retrieval, their verbal skills, and their numerical and verbal short-term memory. Methods: The magnitude representation was tested by a non-symbolic magnitude comparison task. We have perfected previous experimental designs measuring magnitude discrimination skills in 65 children kindergarten (4-7-year-olds) by controlling for several variables which were not controlled for in previous similar research. We also used a large number of trials which allowed for running a full factorial ANOVA including all relevant factors. Tests of verbal counting, of short term memory, of number knowledge, of problem solving abilities and of verbal fluency were administered and correlated with performance in the magnitude comparison task. Results and discussion: Verbal counting knowledge and performance on simple arithmetic tests did not correlate with non-symbolic magnitude comparison at any age. Older children performed successfully on the number comparison task, showing behavioural patterns consistent with an analogue magnitude representation. In contrast, 4-year-olds were unable to discriminate number independently of task-irrelevant perceptual variables. Sensitivity to irrelevant perceptual features of the magnitude discrimination task was also affected by age, and correlated with memory, suggesting that more general cognitive abilities may play a role in performance in magnitude comparison tasks. Conclusion: We conclude that young children are not able to discriminate numerical magnitudes when co-varying physical magnitudes are methodically pitted against number. We propose, along with others, that a rather domain general magnitude representation provides the later basis for a specialized representation of numerical magnitudes. For this representational specialization, the acquisition of the concept of abstract numbers, together with the development of other cognitive abilities, is indispensable. Background developmental literature: many argue that the innate It is a major question whether the representation of analog magnitude representation is a prerequisite of the approximate numerical magnitudes in children develops acquisition of arithmetics; others claim that formal edu- and sharpens independently of symbolic arithmetical cation and numerical enculturation sharpens the analog abilities, or symbolic knowledge correlates with the magnitude representation in children. On the one hand, approximate magnitude representation in some ways. several researchers assume that children have an innate, There is a sharp divide in the corresponding preverbal approximate, language-independent magnitude representation shared with other species [1-11]. Accord- ing to this account, refinement of the analogue magni- * Correspondence: fs299@cam.ac.uk tude representation correlates with math achievement Centre for Neuroscience in Education, Faculty of Education, University of [12-14] and has a predictive value for later math Cambridge, UK © 2010 Soltész et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Soltész et al. Behavioral and Brain Functions 2010, 6:13 Page 2 of 14 http://www.behavioralandbrainfunctions.com/content/6/1/13 performance [15,16]. On the other hand, others think change, independent of perceptual variables. These that the relation is reversed. Development and sharpen- results have been replicated and extended by several ing of magnitude representation is supported by lan- later experiments, leading to the conclusion that infants guage, especially by counting skills [17-19]. possess a basic understanding and representation of Number representation skills are most frequently approximate numbers [for a review, see [9]], providing tested by quantity discrimination tasks (or by number the basis for the acquisition of later arithmetics. line estimation [12]; however magnitude discrimination Contrary to the above statement, other researchers and estimation are strongly interrelated [20,21]). In who investigated the co-development of the number these tasks, infants are expected (and older children are representation and verbal counting skills in young chil- explicitly asked) to discriminate between perceptual dis- dren, arrived to the conclusion that verbal counting plays showing a certain number of items (e.g. dots). The knowledge is inevitable for the abstraction of numerical most general finding is that quantity discrimination magnitudes [5,17-19,39-42]. For example, Mix and col- depends on the ratio of to-be-compared quantities. It is leagues [39,41,42] found that 3-4-year-old children harder to compare quantities when their ratio is closer could not match cross-modal stimuli based on numeros- to 1 than when their ratio is further away from 1. The itybefore theywereabletomasterthe verbal counting ratio effect has been consistently shown in infants (sym- system. Brannon and Van de Walle [17] and Rousselle bolic stimuli: [22,23]; non-symbolic stimuli: et al [18] also found that only children who already [4,5,10,11,16,18,24-30]), and also in animals and human mastered and understood the verbal counting system adults. Hence, it is thought that numbers are coded in and were able to use the role of cardinality, were able to analogous, approximate fashion by an evolutionarily discriminate numerical magnitudes independent of their grounded pre-verbal magnitude representation [31-34]. perceptual properties, like overall size. However, the The most important methodological challenge in mag- relationship between number discrimination and verbal nitude discrimination experiments is that perceptual counting knowledge disappears after the very first stages variables are inevitably correlated with number. These of the acquisition of the latter. This suggests that the variables correlate both with each other and with acquisition of verbal counting abilities enables children numerosity and it is impossible to control for all of to understand that numerical quantities are independent them at the same time. For instance, if intensive proper- of objects’ physical properties, like size and luminance. ties (individual item properties, like item size) are kept Children who have not yet experienced this conceptual equal in a particular trial, extensive properties (proper- shift do not understand the abstract nature of numbers ties of the set, like summed surface of all items in a and rely on analogue perceptual features in number group) will inevitably co-vary with number, and vice comparison tasks [17,18]. versa. With a simple example, a collection of 6 apples is The inability of 3-4 year-old children to avoid the not only more, but physically also larger than a collec- effect of perceptual variables apparently contradicts find- tion of 3 apples. In nature, ‘more’ usually correlates with ings according to which even infants are able to discri- ‘bigger’ (number of individuals in a group, number of minate dot patters based purely on their numerosity pieces of food, etc.). Infants can rely on these simple when perceptual variables are controlled for [e.g., perceptual features of sets, instead of the more abstract [4,38,43]]. However, there is a perceptual confound still property of numerosity. Several of the early studies did unaccounted for in Xu and Spelke’s[4] paradigm.Con- not control for these perceptual correlates of the stimuli trolling, i.e. keeping constant overall surface, will cause [26,27] making infants’ putative numerical performance item size to covary with number. In fact, the distribution indistinguishable from their perceptual performance. In of item sizes across trials is very different from the dis- fact, when overall surface [35] or circumference [36,37] tribution of sum surface across trials and it is correlated is controlled during experiments, infants are more sensi- with the numerosity of the dots. More precisely, and tive to the continuous perceptual variable than to num- according to the authors’ stimuli description, the dia- ber. It was also shown that infants habituated to total meter of an item varies between 1.06-2.37 cm in the 8- surface area but not to number when these two dimen- item displays and between 0.75-1.67 cm in the 16-item sions were pitted against each other, i.e. when the displays. As item diameter on test displays is 1.5 cm, numerically ‘more’ set was smaller in physical size [38]. item size of dots in an 8-item test display is larger than Xu and Spelke [4] devised a habituation paradigm in the average item size in a 16-item habituation display which they attempted to control for the non-numerical (1.3). It is possible that infants reacted to the change in perceptual variables. They varied sum surface and den- dot size, instead of the number of the dots. sity of the trials in a way that nothing but the number Here, we set out to explore the developmental rela- changed from habituation to test trials. They showed tions between magnitude representation, number knowl- that 6-month-old infants were sensitive to number edge and counting skills in preschool children, using an Soltész et al. Behavioral and Brain Functions 2010, 6:13 Page 3 of 14 http://www.behavioralandbrainfunctions.com/content/6/1/13 improved number comparison paradigm and also mea- based on numerosity, independently of the competing suring reaction time in addition to accuracy. We utilized perceptual information, would or would not correlate a number discrimination paradigm similar to the one with children’s verbal counting knowledge. Based on the used by Rousselle and Noël [19]. They not only equated literature, we expect that number discrimination perfor- some perceptual variables across trials, but in some mance and verbal counting knowledge are independent instances, pitted perceptual properties against number. of each other after three years of age [e.g., [17,18]]. In one third of the trials, number and physical proper- Further, we expect to find a developmental change ties were congruent: the numerically larger set was phy- across age groups in the ability to resist task-irrelevant sically larger as well. Another third of the trials were and conflicting perceptual information and in the ability incongruent: the numerically larger set was smaller in to discriminate close magnitudes. These developmental physical size. The last third of trials were neutral: sum factors probably reflect the maturation of more general surface was equated among the dot sets. We consider abilities (i.e. executive functioning, attention and mem- the manipulation of congruency as the most optimal ory, for an overview see [55]). way to control for perceptual properties. As the equa- We predict that the congruency effect will weaken by tion of any of the perceptual properties (e.g. sum sur- age because inhibitory control substantially develops in face) yields that another property (e.g. item size) will children during the age range examined. A ratio effect is correlate with number, probably the best solution is to also expected, reflecting the approximate nature of mag- explicitly oppose the physical and numerical dimensions. nitude representation. The most interesting question is In the incongruent situations, most of the perceptual whether counting/number knowledge and markers of variables will be the opposite of the numerical property: the magnitude representation, i.e. the ratio effect, corre- density, item size, sum surface, sum circumference etc. late with each other. One possibility is that there is such will all be smaller in the numerically larger set. Mean- correlation.Thiswould supportthat1)eitherthe accu- while, in the congruent situations item size, sum surface, racy of the non-symbolic magnitude representation pre- density etc. will all be larger in the numerically larger dicts arithmetic performance [e.g., [16]], or 2) that set. If children were relying on any of the perceptual verbal counting knowledge supports non-symbolic num- properties, these properties would lead to the incorrect ber representation [17,18]. Another possibility there is answer in a significant portion of the trials. no such correlation. This would suggest that counting The co-development of the number representation and abilities and number knowledge follows a developmental verbal counting skills is mostly measured by correlating track independent of that of non-symbolic magnitude performance on a non-symbolic magnitude comparison comparison and the two form two independent develop- task and the performance on some verbal tasks. We used mental factors. the most commonly used verbal counting measures, the ‘how many’ task, the ‘give a number’ task and the ‘how Methods high’ task, measuring the understanding of one-to-one Subjects correspondence, counting and cardinality [8,18,44]. 65 children attending a public kindergarten participated Further, we added control measures of verbal fluency, in the experiment which was carried out in Hungary short term memory for numbers, short-term memory (Nyíregyháza). Children came from a working- or mid- for words, arithmetic problem solving (thought to be dle-class background. Children assigned to different age based on memory retrieval), line halving and number groups entered kindergarten in consecutive years. There knowledge, in addition to measuring counting abilities. were 14 4-year olds (7 boys, mean = 4 years, SD = 0.14 We were motivated by a growing literature emphasizing years), 17 5-year-olds (7 boys, mean = 5.6 years, SD = the role of memory in the aetiology of numerical disabil- 0.28 years), 17 6-year-olds (8 boys, mean = 5.96, SD = ities [e.g., [45-47]]. For example, children with mathe- 0.24) and 17 7-year-olds (9 boys, mean = 6.88 SD = matical disabilities have difficulties in rehearsing verbal 0.28). Written informed consent was obtained from par- information and in control processes attributed to the ents and the study was approved by the institutional central executive [48]. They also have verbal fluency dif- ethics committee of the Research Institute for Psychol- ficulties [45,49,50]. Memory and control processes are ogy at the Hungarian Academy of Sciences. also important in normal numerical development [[51,52]; however, see [53,54] for an opposite opinion]. Procedure We also aimed to identify the relevant and possibly Data were collected in two sessions. During the first ses- interrelated developmental factors behind number dis- sion behavioural tests were administered. During the crimination performance and counting knowledge. We second session the computerized magnitude comparison were interested whether the ability to judge pairs of sets test was administered. Soltész et al. Behavioral and Brain Functions 2010, 6:13 Page 4 of 14 http://www.behavioralandbrainfunctions.com/content/6/1/13 Tests presented verbally and had to be answered verbally. There were twelve behavioural tasks. The tasks were Children were scored according to the number of addi- grouped into thematic sets for presentation purposes. tions that they could solve. The difficulty of additions The order of sets was counterbalanced during adminis- was also increased by increasing the problem size (e.g. tration in order to avoid systematic effects arising from the simplest problem was 1+1, then 2+2, then 3+3 and general fatigue or boredom. The first thematic set con- so on). The seventh measure characterized children’s sisted of two tests measuring children’s number knowl- understanding of halving and their estimation abilities. edge: “Say as many numbers as you can” (number According to the literature, estimation also reflects the recitation) and a number recognition task (Arabic magnitude representation and correlates with perfor- numerals from 1 to 10 were shown in random order). mance on magnitude discrimination tasks [20,21]. In Children were scored for each number they said and for this task children were asked to share a salty stick each Arabic numeral they named correctly. The second (approximately 13 cm in real life) with their peer set of two tests measured verbal fluency, including two equally. A drawing of ten sticks (10 to 20 cm) was common tasks in categories which are familiar to young shown and they were asked to mark the point where children [for example, [56]]: “Sayasmanyanimals as they would break the real stick to share it equally. Per- you can” and “Say as many colours as you can”.Chil- formance was measured in terms of deviation in milli- dren were given a score for every word that they gener- metres from the middle point. ated in one minute. The third set of two tasks assessed children’s verbal short term memory: auditory short Magnitude comparison task term memory for numbers and auditory short term The thirteenth task was the magnitude comparison task. memory for words. Children’s score was the largest Black dots on a light yellow background were used as number of items they recited correctly. The fourth set stimuli. Two sets of dots were presented simultaneously contained three tasks measuring counting abilities: on a computer screen (see Figure 1). “Count as far as you can” (knowledge of the verbal We used only congruent and incongruent trials, so counting system), “How many toys are there” (one-to- that perceptual features were pitted against number in one correspondence and cardinality) and “Give me a exactly 50% of the trials. We omitted the neutral situa- number” (counting) tasks. These measures for assessing tion because neutral trials always contain at least one children’s counting abilities followed the design of physical feature which is correlated with number (for Wynn [8,44] and Rousselle et al. [18]. Children were example, density provided a reliable cue for numerosity scored on the “Count as far as you can” task according both in the congruent and in the neutral trials in the to the maximum number that they could count to with- Rouselle and Noël [19] study). Congruent and incongru- out committing an interchange or an omission. The ent trials were intermixed with each other. In conse- “How many toys are there” task score reflected the max- quence, attending to a certain perceptual cue would lead imum number of elements (toy cars) children could systematically to the correct answer in the congruent count correctly. The “Give me a number” task score was condition and to the incorrect answer in the incongru- determined by the maximum number of toys children ent condition. By having incongruent condition, we do could select correctly according to the instruction. The not need to worry about all the perceptual variables number of elements in “How many toys are there” and which are impossible to control for at the same time: “Give meanumber” was extended (to maximum 100) when for example the sum surface is incongruent with until two consecutive errors were made by the child. number, item size, or density are also incongruent with After presenting/asking for 2-5 items, item number was number. We also controlled for two perceptual features randomly extended (e.g. 8, 13, 15, 20). The fifth set con- at the same time, so that both perceptual controls were sisted of two tasks measuring verbal problem solving included in the same experiment, intermingled in a abilities: easy problems and difficult problems. Easy pro- pseudo-random way within each block and within each blems were additions with the same numbers, e.g. 1+1, subject. We chose to control for overall surface area and 2+2; these additions often come up in kindergarten circumference, because these were found to be more activities (6 problems; following the curriculum in Hun- salient features than number for children of this age garian kindergartens). Here the correct answer could be (surface: [18,38,57]; circumference: [36]). We were inter- simply recited from memory (as children have already ested if either of these perceptual controls were more overlearned them in short rhymes). More difficult pro- salient to children. The two different types of perceptual blems could not be easily recited from memory. These control are also shown in Figure 1. problems consisted of additions using the same numbers The sets were separated by 7.5 cm, and were visually as the easy problems but were not overlearned by chil- easily distinguishable from each other. The overall dren (15 problems, e.g. 2+5). The problems were envelope (corresponding to contour length in Rousselle Soltész et al. Behavioral and Brain Functions 2010, 6:13 Page 5 of 14 http://www.behavioralandbrainfunctions.com/content/6/1/13 Figure 1 Example stimuli. et al. [18]) of a set was kept constant at 9 × 9 cm, as the sum of items in the distance 6 condition in the 2:3 overall envelope has been found to help children to ratio condition. make appropriate judgements, even in conditions where Two different physical variables of the dot groupings overall surface is incongruent with number [10]. The were manipulated as perceptual controls: overall surface children’s task was to find out which set contained (hence, luminance) and overall circumference (sum of more dots, and to respond by pressing the button at the individual items’ circumferences). These perceptual thesideofthe ‘winner’ set. Response side was controls were intermixed during stimulus presentation. counterbalanced. The ratios of the overall physical sizes (surface in half of The size of dots was constant within set and varied the trials, and circumference in the other half of the between sets. The individual size of dots and the pattern trials) of the dot sets were congruent or incongruent of dots were varied randomly through pairs of sets. Sets with the numerical ratio of the dot sets. In the congru- with the same number of items never had the same dot ent condition, the more numerous set was larger in size. Only numerosities above the subitization range were overall physical size than the less numerous set. In the used in order to exclude object-based attention respond- incongruent condition, the more numerous set was ing (based on the object file system, [see [58-61]]. smaller in overall physical size then the less numerous The following factors were varied systematically for set. Congruent and incongruent trials were pseudo-ran- the purposes of analysis: (1) The ratio of the number of domly intermixed (no more than three of each could dots in the two sets. (2) The type of the physical control occur consecutively in a sequence). variable. (3) The congruity of physical control variables In each trial the ratio of perceptual features of the two and numerosity. We also attempted to control for both dot patterns was kept the same as their numerical ratio. problem size (the total number of items) and for numer- This was done to ensure that the influence of perceptual ical distance (the numerical difference between the two variables did not differ for different numerical ratios. For sets) in order to avoid that either of these parameters example, if the ratio between the numbers was 1:2 and would distort the ratio effect, as ratio effect is the com- posite of numerical distance and size. The ratios and Table 1 The dot number pairs per each ratio. numerical distances for all combinations of numerosities 1: 2 3 : 5 2: 3 are summarized in Table 1. There was no extreme large numerical distance for the 4 8 * 6 10 * 8 12 * 2:3 ratio because the sum of items would have been 6 12 ** 9 15 ** 12 18 ** ° much larger than in other conditions (at least 16:24, sum 10 20° 12 20° is 40; or 20:30, sum is 50). Rather, we decided to keep the The numerical distance between dot numbers are the same in dot number pairs marked with * and ** (the numerical distances are 4 and 6, respectively). sum of items for the largest numerical distances in the Ratios are in columns. The overall sum is almost equal for dot number pairs 1:2 and 3:5 ratio conditions to be approximately equal to marked with ° (30, 32 and 30). Soltész et al. Behavioral and Brain Functions 2010, 6:13 Page 6 of 14 http://www.behavioralandbrainfunctions.com/content/6/1/13 the ratio between perceptual variables was 2:3, the per- Age was significant in 10 of the 12 tests. Post-hoc Scheffé ceptual difference would be less salient than the numeri- tests showed that in 6 out of 10 tests (number recognition, cal difference. This would result in better numerical counting abilities and in verbal arithmetic abilities) there discrimination performance solely because the percep- was an apparent developmental change between the ages tual variables would be less distracting. Similarly, if the of 5 and 6 years: children aged 4 and 5 did not differ from perceptual ratio were 1:2 and the numerical ratio were each other significantly; and neither did children between 2:3, the perceptual difference would be more salient. 6 and 7. However, the difference between these two age Each child was presented with 64 test stimuli pairs, ranges was significant: both 6- and 7-year-olds performed preceded by 8 practice pairs. Trials were separated by better than any age group in the younger range. funny smiley figures and friendly pictures of animals and soft toys, to retain children’s attention and motiva- Magnitude comparison task tion. Encouraging verbal feedback was given after each As accuracy data is binary in nature, summarized as trial, independently of performance. Trials began when proportions, the arcsine transformation was applied to children attended to the screen and indicated that they the data. Statistical analyses were carried out on both were ready for the next task. All children enjoyed the raw proportions and on arcsine-transformed accuracy tasks and actively sought participation. data. As the results were mostly identical, we report The exceptionally large number of trials allowed for a p-values from the transformed data in square brackets full factorial ANOVA. We also measured high-precision only when the results were slightly different. Figures reaction time. To our knowledge, RT has never been show row accuracy proportions for the sake of analyzed (or reported) in previous studies of magnitude intelligibility. discrimination with children of this age. Information First, in order to see whether accuracy was signifi- from RT analyses may offer significant advantages in cantly different from chance at the group level, one- understanding cognitive processing in children, for sample t-tests were run against 50%. Second, accuracy example serving as a complementary source of informa- and median reaction time (RT) were calculated and tion (in fact, RT was found to be more informative than entered into ANOVAs, taking Side (of the response) × accuracy in school children’s magnitude comparison Congruency × Type (of perceptual control) × Ratio as performance [15,62]). within-subject factors. Age and Gender were the between-subject factors. The main effect of Age was Results highly significant with respect to accuracy (Figure 3, Behavioural tests F(1,58) = 11.9, p < 0.0001). A multiple analysis of variance (MANOVA) was per- Post-hoc Scheffé tests indicated that the youngest formed on the 12 different behavioural tests (number group made significantly more errors than the other knowledge (recitation and recognition), verbal fluency three age groups in the Incongruent condition (animals and colours), verbal short term memory (num- (p < 0.001). There were no significant differences among bers and words), counting abilities ("Count as far as you the three older groups (accuracy and RT data is shown can”, “How many toys are there” and “Give meanum- in Table 3). The main effect of Gender was marginally ber”), problem solving abilities (easy and difficult), and significant for accuracy (F(1, 58) = 3.8, p = 0.057 line halving). Raw scores on each task were used as [p = 0.062]). Boys committed fewer errors than did girls dependent variables with Age and Gender as indepen- (89.4% vs. 83.5%). Neither of the above main effects was dent variables. Results from the twelve tests were significant in RT. entered into univariate F-tests. A Bonferroni-type The main effects of Congruency (Accuracy: F(1,58) = adjustment was performed in order to lower the possibi- 31.8, p < 0.0001; RT: F(1,58) = 11, p < 0.002) and Ratio lity of an inflated type I error due to multiple compari- (Accuracy: F(1,58) = 23.4, p < 0.0001. RT: F(1,58) = sons. The critical alpha-level set by the Bonferroni-type 22.5, p < 0.0001) were highly significant. Children adjustment was 0.0042 (0.05/12). For further analysis of responded faster in congruent trials and with fewer significant MANOVA and F-test results, post-hoc errors than in incongruent trials (2880 vs. 3452 ms, 94% Scheffé tests were implemented. vs. 78.9%). More-different ratio pairs were responded to Normalized test results are plotted in Figure 2. The data more accurately and faster than less-different ratio pairs were normalized so that all tests could be visualised on a (2422, 3062, 4014 ms and 90.8%, 85.5%, 83.2% correct common scale, independently of the metrical differences for Ratios 1:2, 3:5 and 2:3, respectively). The Type of between the different tests. The MANOVA demonstrated perceptual control stimulus was not significant for accu- that age was a significant factor (Wilks’Ë = 0.11, racy nor RT (p > 0.4). F(48,175.38) = 2.83, p < 0.0001). The results of follow-up The Age × Congruency interaction was significant univariate ANOVAs are given in Table 2. The effect of both in accuracy (Figure 3/A, F(3,58) = 7.5, p < 0.001) Soltész et al. Behavioral and Brain Functions 2010, 6:13 Page 7 of 14 http://www.behavioralandbrainfunctions.com/content/6/1/13 Figure 2 Standardized scores (units in SD) for the 12 tests separately for each age group. The critical p value for significance was 0.0042 (see text and Table 2). See text and Table 2 for abbreviations of tests. Significant age group differences are denoted by red lines. See text for more details. and in RT (Figure 3/B, F(3,58) = 3.96, p < 0.02). Pair- Ratios 1:2, 3:5 and 2:3: 59, 50.5 and 45.5% correct. wise post-hoc comparisons revealed that the effect of T-test results in the same order: t(13) = 0.99, p > 0.3, t Congruency on accuracy decreased with increasing age, (13) = 0, p = 1 and (13) = -0.42, p > 0.6) Post-hoc tests reaching statistical significance only in the youngest yielded non-significant results for RT data (p > 0.1). group (p < 0.001). One-sample t-tests confirmed that all The Gender × Congruency interaction was also sig- age groups performed significantly above chance in both nificant (Accuracy: F(3,58) = 5.36; p < 0.03. RT: F(3,58) Congruent and Incongruent conditions except for the 4- = 5.9; p < 0.03). Girls’ accuracy was significantly affected year-old group, whose performance did not differ from by Congruency (94.1% vs. 72.9% in congruent and chance in the Incongruent condition (subsequently for incongruent trials respectively, p < 0.0002). In contrast, Table 2 Univariate F-tests for the 12 tests. Task Age Gender F(4, 56) p F(1, 56) p Number knowledge - NRT: Say as many numbers as you can 3.18 0.0199 0.61 0.4 -*NRE: Written (Arabic) number recognition 14.58 <0.0001 0.91 0.3 Verbal knowledge -*VA: Say as many animals as you can 8.66 <0.0001 4.7 0.033 -*VC: Say as many colours as you can 9.53 <0.0001 0.8 0.4 Working memory (verbal) -*MN: Short term memory for numbers 10.7 <0.0001 3.12 0.08 -*MW: Short term memory for words 8.19 <0.0001 4.78 0.033 Counting abilities -*CCT: Count as far as you can 12.54 <0.0001 1.1 0.3 -*CHM: How many objects are there 13.82 <0.0001 0.8 0.8 -*CGV: Give me a number 15.37 <0.0001 0.9 0.3 Verbal counting abilities -*PR1: Problems - simple 28.85 <0.0001 2.1 0.2 -*PR2: Problems - difficult 14.88 <0.0001 2.9 0.09 Fractions - H2: Halving 1.23 0.3 0.04 0.8 The critical p value for significance was 0.0042 (set by the Bonferroni adjustment). Significant p levels are denoted by bold italic typesetting. Soltész et al. Behavioral and Brain Functions 2010, 6:13 Page 8 of 14 http://www.behavioralandbrainfunctions.com/content/6/1/13 Figure 3 Congruency effect. A. Congruency effect in accuracy across age groups. One-sample t-tests showed that performance in the Incongruent condition did not differ from chance in this age group (the non-significant one-sample t-test is denoted by #). B. Significant Age × Congruency interaction in RT (p < 0.02). the Congruency effect was not significant for boys This large discrepancy was not present in the older (93.8% and 84.6%, p > 0.16). No similar effect was found groups (2605 vs. 2325 ms, 3558 vs. 2712 ms, and 2859 in RT data (p > 0.1). According to post-hoc compari- vs. 3023 ms, respectively). None of the post-hoc com- sons, girls’ performance was significantly worse in the parisons was significant. incongruent condition than boys’ (p < 0.04 [p = 0.065]). In contrast, the performance of boys and girls did not Correlations and factor analysis differ in the congruent condition (p > 0.99). The three- In order to investigate the relationship of counting abil- way interaction of Congruency, Gender and Age was ities and magnitude comparison, correlational analyses not significant (p > 0.3) indicating that the gender dif- were carried out. Variables were the outcomes of the ference in the congruency effect is stable across age behavioural tests, and median RT and accuracy for each groups. No such result emerged from the RT data. condition of the magnitude comparison task (3 levels of The Congruency × Ratio interaction was significant Ratio and 2 levels of Congruency). Further, derived vari- in accuracy(Figure 4/A):F(2,116) =4.7;p<0.02) and ables such as the slope of the Ratio effect (RT and accu- it was marginally significant in RT (Figure 4/B, p = 0.066). racy for smaller ratio minus RT and accuracy for larger Post-hoc tests revealed that there was a steeper Ratio ratio), and Ratio × Congruency cells (accuracy and RT effect in the accuracy data in the incongruent trials (p < data for each Ratio within each Congruency condition) 0.0005 between ratio 1:2 and ratio 3:5; and p < 0.0002 were also used for analysis. Again, both raw accuracy between ratio 1:2 and ratio 2:3), than in the congruent proportions and arcsine transformed accuracy data were trials. Incongruent trials yielded more mistakes in the entered into the analyses. P-values of the arcsine trans- more difficult ratios. In RT, planned comparisons formed data are denoted by superscripts when different revealed that the ratio effect was significant in both con- from the p-values of the raw data. Age was controlled gruent and incongruent conditions (p < 0.001 between by computing partial correlations. Further, the 4 age- ratio 1:2 and 3:5; and between ratio 2:3 and 3:5). groups were also analysed separately, in order to reveal Interestingly, the Age × Gender interaction was sig- any possible group differences in correlations. nificant in RT (F(3,35) = 3.3, p = 0.029) but not in accu- The matrix of correlation coefficients (controlling for racy. Boys responded faster than girls in the 4-year-old age) for test results is shown in Table 4. The correlation group (2495 vs. 5750 ms, the difference is 3254 ms). coefficient matrix for the magnitude task results is shown in Table 5. As it can be seen in these correlation matrices, there Table 3 Performance in the magnitude comparison task. are strong correlations among some number knowledge, Group Accuracy (%) Median RT (ms) verbal fluency, counting and problem solving tasks; 4 yrs 70.5 (3.26) 4123 (586.2) among reaction time measurements; and among accu- 5 yrs 88.3 (3) 2465 (338.5) racy measurements. In order to draw a concise summary 6 yrs 93.4 (2.96) 3135 (291.7) of these interrelations, we performed a factor analysis. 7 yrs 93.6 (2.87) 2941 (276.4) The factor analysis would also show clearly that which abilities form together a common factor and which Standard errors are given in brackets. Soltész et al. Behavioral and Brain Functions 2010, 6:13 Page 9 of 14 http://www.behavioralandbrainfunctions.com/content/6/1/13 Figure 4 Congruency × Ratio interaction. A. Congruency × Ratio interaction in accuracy. Ratio effect was significant only in the Incongruent condition (*** denotes p < 0.001 significance level). B. The same interaction was marginally significant in RT data (p = 0.066). abilities are independent of each other. The factor analy- independent (predictor) variables (IVs). The regression sis (controlling for age) confirmed the above intercorre- modelwas significantand thetwo shorttermmemory lations and yielded the following factors: 1) ‘Counting, measures accounted for 21% of the variance in ‘Accu- number knowledge and verbal fluency’;2) ‘Speed’;and racy’ (adjusted R = 0.211; F(2,63) = 9.7, p < 0.001). 3) ‘Accuracy’. Table 6 summarizes the results of the fac- Neither of the memory tests accounted for a significant tor analysis. amount of variance on its own (but both showed a Consequently, global indices of the three factors were strong trend: memory for numbers: t = 1.76, p = 0.08; calculated, by averaging the performance values of the and memory for words: t = 1.95, p = 0.06). measures contributing to each factor. ‘Speed’ and ‘Accu- racy’ were then correlated with behavioural tests, and Discussion ‘Counting, number knowledge and verbal fluency’ was In this study, we set out to explore the developmental correlated with performance in the magnitude compari- relations between magnitude representation, number son task. ‘Accuracy’ significantly correlated with the two knowledge and counting skills in children aged 4 to 7 memory measures (memory for numbers: r = 0.246, p = years. In order to get around some unwanted confounds, 0.05; and memory for words: r = 0.29, p < 0.03). No we made a number of methodological innovations to the other correlations were significant. basic number comparison task. In order to investigate the unique variance explained In response to our main developmental question, we by the short term memory measures in ‘Accuracy’,a found no relationship between non-symbolic number regression analysis was carried out with ‘Accuracy’ as comparison performance and number or counting the dependent variable (DV) and short term memory knowledge. These data are consistent with recent studies for numbers and short term memory for words as reporting no relationship between non-symbolic Table 4 Partial correlations among tests. NRT NRE VA VC MN MW CCT CHM CGV PR1 PR2 H2 NRT NRE 0.27 VA 0.20 0.10 VC 0.24 0.35 0.39 MN 0.22 0.19 0.24 0.12 MW 0.16 0.28 0.21 0.16 0.46 CCT 0.37 0.25 0.33 0.08 0.12 -0.08 CHM 0.31 0.26 0.34 -0.01 0.21 0.00 0.82 CGV 0.21 0.17 0.29 0.00 0.18 0.04 0.77 0.84 PR1 0.13 0.14 0.40 0.20 0.11 -0.02 0.45 0.37 0.28 PR2 0.33 0.07 0.48 0.05 0.15 0.10 0.46 0.56 0.47 0.51 H2 -0.04 -0.06 -0.05 0.04 -0.09 0.00 -0.13 -0.16 -0.14 -0.29 -0.19 R values from partial correlations for tests. Age was controlled. Bold typesetting indicates significant (p < 0.05) correlations. NRT: Say as many numbers as you can; NRE: Written (Arabic) number recognition; VA: Say as many animals as you can; VC: Say as many colours as you can; MN: Short term memory for numbers; MW: Short term memory for words; CCT: Count as far as you can; CHM: How many objects are there; CGV: Give me a number; PR1: Problems - simple; PR2: Problems - difficult; H2: Halving Soltész et al. Behavioral and Brain Functions 2010, 6:13 Page 10 of 14 http://www.behavioralandbrainfunctions.com/content/6/1/13 Table 5 Partial correlations among magnitude task measurements. SPEED (RT) ACCURACY (%) CON NCON RAT1 RAT2 RAT3 ALL CON INCON RAT1 RAT2 RAT3 SPEED (RT) CON INCON 0.54 RAT 1 0.71 0.68 RAT 2 0.83 0.68 0.68 RAT 3 0.85 0.63 0.63 0.68 ALL 0.92 0.74 0.80 0.88 0.93 ACCURACY CON -0.26 0.00 -0.33 -0.32 0.00 -0.18 INCON -0.14 -0.27 -0.32 -0.26 -0.06 -0.19 0.25 RAT 1 -0.18 -0.21 -0.27 -0.35 0.02 -0.18 0.54 0.85 RAT 2 -0.17 -0.22 -0.38 -0.28 -0.04 -0.20 0.52 0.89 0.81 RAT 3 -0.24 -0.22 -0.42 -0.32 -0.10 -0.25 0.57 0.86 0.76 0.81 ALL -0.21 -0.23 -0.39 -0.34 -0.05 -0.22 0.58 0.93 0.91 0.94 0.93 Age was controlled. Bold typesetting indicates significant (p < 0.05) correlations. [Arcsine transformed data: Asterix indicates correlations where p = 0.07. Circle indicates where p > 0.1]. Table 6 Factor analysis. magnitude comparison and arithmetic performance in 12 3 6-8 year-old primary school children [62] and in second, Tests third and fourth graders with mathematical disabilities NRT 0.239429 0.539796 0.117110 [63,64]. In addition, performance measures on the non- NRE 0.190808 0.705857 0.145507 symbolic number comparison task (’Speed’ and ‘Accu- VA 0.050910 0.712302 0.258825 racy’) and performance on number/counting knowledge VC 0.120982 0.557922 0.268724 tasks (’Counting, number knowledge and verbal fluency’) MN 0.175689 0.551612 0.379478 form factors which are independent of each other. MW 0.170932 0.405931 0.409953 Again, this suggests that non-symbolic number compari- CCT 0.049560 0.909566 0.164922 son skills and symbolic counting/number knowledge CHM 0.036905 0.889526 0.213583 develop in isolation between 4-8 years of age. However, CGV 0.044615 0.867434 0.223804 we cannot exclude that there may be a relationship PR1 0.058252 0.824896 0.223152 between non-symbolic number comparison and count- PR2 0.143934 0.807914 0.217954 ing knowledge before 4 years of age [17,18]. H2 -0.022292 -0.101818 -0.107875 Regarding the non-symbolic number comparison task, Comparison - speed a significant congruency effect was found. The Con- CONGRUNENT -0.930375 -0.093792 -0.103606 gruency effect was of interest because it can attest INCONGRUENT -0.722560 -0.013829 -0.208961 whether children attended to numerosity or to irrelevant RATIO 1 -0.724875 -0.188580 -0.400200 perceptual variables. We found that 4 year-olds are not RATIO 2 -0.855544 -0.115420 -0.317704 able to perform intentional numerical judgments inde- RATIO 3 -0.924577 0.053293 0.126029 pendently of physical appearance: their performance was ALL -0.982134 -0.056095 -0.133725 at chance when the perceptual information was in con- flict with numerical information. Susceptibility to irrele- Comparison - accuracy vant perceptual features weakened with age, and the CONGRUENT 0.173563 0.117307 0.640845 congruency effect was mainly driven by 4-year-olds. INCONGRUENT 0.131973 0.220069 0.892832 This developmental trend is in agreement with previous RATIO 1 0.128044 0.162317 0.909502 studies [16,18,19]. Although Halberda and Feigenson RATIO 2 0.143302 0.232862 0.911971 [16] reported above-chance performance when the data RATIO 3 0.195263 0.224094 0.892648 for all ratios were collapsed, their Figure 2 reveals that ALL 0.167608 0.221621 0.954582 3- to 4-year-old children were at chance with more diffi- Expl. Var. 9.784019 6.247673 5.951855 cult ratios. The detailed analysis of possible ratio × con- Factor loadings. Extraction method: principal components were extracted and varimax rotation was applied. Marked loadings are > 0.7. gruency interaction is worthwhile: the main effect of Soltész et al. Behavioral and Brain Functions 2010, 6:13 Page 11 of 14 http://www.behavioralandbrainfunctions.com/content/6/1/13 congruency might be distorted by the data from condi- possibility of more complex causal relations among tions when the comparison is easier. magnitude representation, symbolic maths and general Our finding that 4-year-old girls performed more abilities (inhibition, memory, attention) should be con- poorly than boys needs to be replicated. At this point, sidered in any further research into the development of we assume that gender differences were due to differen- magnitude representation and arithmetic performance. tial familiarity with computers and computer games For example, longitudinal studies have shown that cog- between the genders. nitive-linguistic skills and also motivational factors (such Performance in the incongruent condition showed as non-verbal intelligence, preschool counting skills, task developmental progression between 4 and 5 years of orientation and social dependence) predict later maths age. This change was independent of and preceded the performance during school years [71,72]. developmental change in verbal and counting abilities It is also interesting to speculate whether the shifting and simple arithmetic, which occurred between 5 and 6 relationship between non-symbolic and symbolic magni- years of age (this latter is most probably due to start of tude comparison performance and maths performance systematic preparation for school at this ages in Hungar- reflects a shifting landscape of developmentally singular ian kindergartens). The lack of correlation between con- events. For example, correlation of non-symbolic magni- gruency effects and counting abilities and simple tude comparison performance and maths skills at age 3- arithmetic suggests that children’s sensitivity to incon- 4, but not later, may point to the importance of estab- gruent irrelevant stimulus dimensions was independent lishing numerical abstraction at this particular age from counting and arithmetic knowledge. A non-numer- [17,18,39,41,42]. Similarly, the correlation between ical explanation of these results might be that older chil- maths performance and symbolic number comparison dren had more mature cognitive and motor inhibition skills at age 6-8 [15,62] may illustrate the importance of capacities than younger children and that this contribu- establishing links between symbolic numbers and refer- ted to the more efficient processing of task-relevant ents at this particular age. The developmental challenge information in general. For example, there is ample evi- wouldthenbetoidentifyage-appropriatemarkers dence that older children perform better in a range of which reflect the most significant achievements of chil- Stroop-like tasks than younger children [16,65,66]. dren at particular ages. Most probably, these markers Indeed adult level performance is not reached on Stroop will change continuously during development, depend- tasks till the age of 21 [67]. Moreover, it has been ing on the most significant learning events in children’s shown that automatic processing of irrelevant physical lives. This notion is similar to Siegler’soverlapping properties is inevitable even in adults and that these waves theory [73], which suggests that the relative domi- properties exert a significant effect even on adult’sper- nance of particular strategies change continuously dur- formance in dot comparison [68]. Similarly, we ourselves ing development. Similarly, the relative importance of have shown that 5.5 to 9-year-old primary school chil- developmental markers probably also shifts in a continu- dren and adults demonstrate substantial incorrect motor ous manner. That is, while the symbolic distance effect activation in the incongruent conditions of symbolic and may be a good predictor of arithmetic development non-symbolic Stroop tasks [69,70]. Overall, the evidence between ages 6-8, this relationship may weaken later suggests that general conflict resolution skills, most (Ansari, personal communication). probably cognitive and motor inhibition abilities, sub- To illustrate how optimal developmental markers may stantially contribute to performance in judgement tasks change with age and experience, we can draw an ana- such as those used here, where stimuli have both task- logy with musical cognition. At the beginning of learn- relevant and task-irrelevant properties. ing to play the violin, for example, it is interesting to It is also worth noting that the ‘Accuracy’ factor in know whether better discrimination between violin magnitude comparison correlated with performance on strings correlates with better musical abilities. In fact, short term memory tasks in our study. This indicates playing each note is directly related to the ability to dis- that general abilities like memory may play an important criminate between the strings and string positions. role in children’s performance in non-symbolic number Hence, violin string discrimination is a necessary causal comparison. Accordingly, some studies have identified precondition of playing violin music. However, later in working memory and executive functions as significant musical development it is likely that the more music factors in arithmetic development [45-47,51,52]. We do children have played during the past 10 years, the better not yet have a strong basis for any firm conclusions on they discriminate violin strings. At this point, better the role of short term memory in number comparison. string discrimination becomes a relatively unimportant Probably, better memory abilities help children to keep consequence of playing more music. Accordingly, at this the task-relevant dimension in their minds so that they developmental time point, it is unlikely that complex can ignore the task-irrelevant features easier. The musical skills are related to violin string discrimination. Soltész et al. Behavioral and Brain Functions 2010, 6:13 Page 12 of 14 http://www.behavioralandbrainfunctions.com/content/6/1/13 In other words, after a violin string-musical note con- executive functioning, especially inhibition may play the nection has been established, practice effects on string most important role. Verbal counting knowledge and discrimination ability will no longer capture the most non-symbolic magnitude comparison abilities were inde- important developmental markers (these may now be pendent of each other in 4-7-year-old children, as the understanding of tempo, melody, interpretation, etc., shown by the factor analysis. related to much higher-level musical concepts). The argument is similar for maths and predictors of Acknowledgements school maths abilities. For example, a recent study The authors thank Prof. Usha Goswami for her most valuable advice on the reported a correlation between non-symbolic dot com- manuscript. Supported by the Hungarian Research Fund (OTKA-T04381, D. Sz.) and by the Medical Research Council, UK (G0900643). parison performance measured at age 14 and maths per- formance measured during 5-11 years of age [74]. As Author details causal directions cannot be determined by correlative Centre for Neuroscience in Education, Faculty of Education, University of Cambridge, UK. Research Institute for Psychology, Hungarian Academy of studies, the significance of these results with regard to Sciences, Budapest, Hungary. Kikelet Óvoda, Nyíregyháza, Hungary. representational development is not clear. While the authors argued that the most likely explanation for their Authors’ contributions FS took part in the planning and designing the experiment and behavioral data was that the development of non-symbolic magni- tests, performed the data analyses and drafted the manuscript. DS tude comparison enhanced arithmetic performance, it is contributed to the design and planning of experiment and behavioural equally likely that better counting/arithmetic skills, and/ tests, and helped drafting the manuscript. LS took part in the planning of behavioral tests and collected all the data. All authors read and approved or more experience with numbers, resulted in better set the final manuscript. estimation skills [75]. Competing interests The authors declare that they have no competing interests. Limitations Some limitations should be noted before we draw the Received: 5 September 2009 final conclusions. First, we did not have children Accepted: 18 February 2010 Published: 18 February 2010 younger than 4 years of age in our study. The picture of References the relationship and co-development of verbal abilities 1. Brannon EM, Terrace HS: Ordering of the numerosities 1 to 9 by and magnitude representation would be complete only monkeys. Science 1998, 282:746-749. with the examination of 2- and 3-year old children. Sec- 2. Gallistel CR, Gelman R: Preverbal and verbal counting and computation. Cognition 1992, 44:43-74. ond, more tests measuring specifically working memory 3. 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