It is a major question whether the representation of approximate numerical magnitudes in children develops and sharpens independently of symbolic arithmetical abilities, or symbolic knowledge correlates with the approximate magnitude representation in some ways. There is a sharp divide in the corresponding developmental literature: many argue that the innate analog magnitude representation is a prerequisite of the acquisition of arithmetics; others claim that formal education and numerical enculturation sharpens the analog magnitude representation in children. On the one hand, several researchers assume that children have an innate, preverbal approximate, language-independent magnitude representation shared with other species [1–11]. According to this account, refinement of the analogue magnitude representation correlates with math achievement [12–14] and has a predictive value for later math performance [15, 16]. On the other hand, others think that the relation is reversed. Development and sharpening of magnitude representation is supported by language, especially by counting skills [17–19].

Number representation skills are most frequently tested by quantity discrimination tasks (or by number line estimation [12]; however magnitude discrimination and estimation are strongly interrelated [20, 21]). In these tasks, infants are expected (and older children are explicitly asked) to discriminate between perceptual displays showing a certain number of items (e.g. dots). The most general finding is that quantity discrimination depends on the ratio of to-be-compared quantities. It is harder to compare quantities when their ratio is closer to 1 than when their ratio is further away from 1. The ratio effect has been consistently shown in infants (symbolic stimuli: [22, 23]; non-symbolic stimuli: [4, 5, 10, 11, 16, 18, 24–30]), and also in animals and human adults. Hence, it is thought that numbers are coded in analogous, approximate fashion by an evolutionarily grounded pre-verbal magnitude representation [31–34].

The most important methodological challenge in magnitude discrimination experiments is that perceptual variables are inevitably correlated with number. These variables correlate both with each other and with numerosity and it is impossible to control for all of them at the same time. For instance, if intensive properties (individual item properties, like item size) are kept equal in a particular trial, extensive properties (properties of the set, like summed surface of all items in a group) will inevitably co-vary with number, and vice versa. With a simple example, a collection of 6 apples is not only more, but physically also larger than a collection of 3 apples. In nature, 'more' usually correlates with 'bigger' (number of individuals in a group, number of pieces of food, etc.). Infants can rely on these simple perceptual features of sets, instead of the more abstract property of numerosity. Several of the early studies did not control for these perceptual correlates of the stimuli [26, 27] making infants' putative numerical performance indistinguishable from their perceptual performance. In fact, when overall surface [35] or circumference [36, 37] is controlled during experiments, infants are more sensitive to the continuous perceptual variable than to number. It was also shown that infants habituated to total surface area but not to number when these two dimensions were pitted against each other, i.e. when the numerically 'more' set was smaller in physical size [38].

Xu and Spelke [4] devised a habituation paradigm in which they attempted to control for the non-numerical perceptual variables. They varied sum surface and density of the trials in a way that nothing but the number changed from habituation to test trials. They showed that 6-month-old infants were sensitive to number change, independent of perceptual variables. These results have been replicated and extended by several later experiments, leading to the conclusion that infants possess a basic understanding and representation of approximate numbers [for a review, see [9]], providing the basis for the acquisition of later arithmetics.

Contrary to the above statement, other researchers who investigated the co-development of the number representation and verbal counting skills in young children, arrived to the conclusion that verbal counting knowledge is inevitable for the abstraction of numerical magnitudes [5, 17–19, 39–42]. For example, Mix and colleagues [39, 41, 42] found that 3-4-year-old children could not match cross-modal stimuli based on numerosity before they were able to master the verbal counting system. Brannon and Van de Walle [17] and Rousselle et al [18] also found that only children who already mastered and understood the verbal counting system and were able to use the role of cardinality, were able to discriminate numerical magnitudes independent of their perceptual properties, like overall size. However, the relationship between number discrimination and verbal counting knowledge disappears after the very first stages of the acquisition of the latter. This suggests that the acquisition of verbal counting abilities enables children to understand that numerical quantities are independent of objects' physical properties, like size and luminance. Children who have not yet experienced this conceptual shift do not understand the abstract nature of numbers and rely on analogue perceptual features in number comparison tasks [17, 18].

The inability of 3-4 year-old children to avoid the effect of perceptual variables apparently contradicts findings according to which even infants are able to discriminate dot patters based purely on their numerosity when perceptual variables are controlled for [e.g., [4, 38, 43]]. However, there is a perceptual confound still unaccounted for in Xu and Spelke's [4] paradigm. Controlling, i.e. keeping constant overall surface, will cause item size to covary with number. In fact, the distribution of item sizes across trials is very different from the distribution of sum surface across trials and it is correlated with the numerosity of the dots. More precisely, and according to the authors' stimuli description, the diameter of an item varies between 1.06-2.37 cm in the 8-item displays and between 0.75-1.67 cm in the 16-item displays. As item diameter on test displays is 1.5 cm, item size of dots in an 8-item test display is larger than the average item size in a 16-item habituation display (1.3). It is possible that infants reacted to the change in dot size, instead of the number of the dots.

Here, we set out to explore the developmental relations between magnitude representation, number knowledge and counting skills in preschool children, using an improved number comparison paradigm and also measuring reaction time in addition to accuracy. We utilized a number discrimination paradigm similar to the one used by Rousselle and Noël [19]. They not only equated some perceptual variables across trials, but in some instances, pitted perceptual properties against number. In one third of the trials, number and physical properties were congruent: the numerically larger set was physically larger as well. Another third of the trials were incongruent: the numerically larger set was smaller in physical size. The last third of trials were neutral: sum surface was equated among the dot sets. We consider the manipulation of congruency as the most optimal way to control for perceptual properties. As the equation of any of the perceptual properties (e.g. sum surface) yields that another property (e.g. item size) will correlate with number, probably the best solution is to explicitly oppose the physical and numerical dimensions. In the incongruent situations, most of the perceptual variables will be the opposite of the numerical property: density, item size, sum surface, sum circumference etc. will all be smaller in the numerically larger set. Meanwhile, in the congruent situations item size, sum surface, density etc. will all be larger in the numerically larger set. If children were relying on any of the perceptual properties, these properties would lead to the incorrect answer in a significant portion of the trials.

The co-development of the number representation and verbal counting skills is mostly measured by correlating performance on a non-symbolic magnitude comparison task and the performance on some verbal tasks. We used the most commonly used verbal counting measures, the 'how many' task, the 'give a number' task and the 'how high' task, measuring the understanding of one-to-one correspondence, counting and cardinality [8, 18, 44].

Further, we added control measures of verbal fluency, short term memory for numbers, short-term memory for words, arithmetic problem solving (thought to be based on memory retrieval), line halving and number knowledge, in addition to measuring counting abilities. We were motivated by a growing literature emphasizing the role of memory in the aetiology of numerical disabilities [e.g., [45–47]]. For example, children with mathematical disabilities have difficulties in rehearsing verbal information and in control processes attributed to the central executive [48]. They also have verbal fluency difficulties [45, 49, 50]. Memory and control processes are also important in normal numerical development [[51, 52]; however, see [53, 54] for an opposite opinion].

We also aimed to identify the relevant and possibly interrelated developmental factors behind number discrimination performance and counting knowledge. We were interested whether the ability to judge pairs of sets based on numerosity, independently of the competing perceptual information, would or would not correlate with children's verbal counting knowledge. Based on the literature, we expect that number discrimination performance and verbal counting knowledge are independent of each other after three years of age [e.g., [17, 18]]. Further, we expect to find a developmental change across age groups in the ability to resist task-irrelevant and conflicting perceptual information and in the ability to discriminate close magnitudes. These developmental factors probably reflect the maturation of more general abilities (i.e. executive functioning, attention and memory, for an overview see [55]).

We predict that the congruency effect will weaken by age because inhibitory control substantially develops in children during the age range examined. A ratio effect is also expected, reflecting the approximate nature of magnitude representation. The most interesting question is whether counting/number knowledge and markers of the magnitude representation, i.e. the ratio effect, correlate with each other. One possibility is that there is such correlation. This would support that 1) either the accuracy of the non-symbolic magnitude representation predicts arithmetic performance [e.g., [16]], or 2) that verbal counting knowledge supports non-symbolic number representation [17, 18]. Another possibility there is no such correlation. This would suggest that counting abilities and number knowledge follows a developmental track independent of that of non-symbolic magnitude comparison and the two form two independent developmental factors.