^{*}

^{a}

^{b}

^{c}

^{a}

Recent studies indicate that Arabic digit knowledge rather than non-symbolic number knowledge is a key foundation for arithmetic proficiency at the start of a child’s mathematical career. We document the developmental trajectory of 4- to 7-year-olds’ proficiency in accessing magnitude information from Arabic digits in five tasks differing in magnitude manipulation requirements. Results showed that children from 5 years onwards accessed magnitude information implicitly and explicitly, but that 5-year-olds failed to access magnitude information explicitly when numerical magnitude was contrasted with physical magnitude. Performance across tasks revealed a clear developmental trajectory: children traverse from first knowing the cardinal values of number words to recognizing Arabic digits to knowing their cardinal values and, concurrently, their ordinal position. Correlational analyses showed a strong within-child consistency, demonstrating that this pattern is not only reflected in group differences but also in individual performance.

The remarkably creative and flexible use of a vast array of symbols is a unique and distinctively human talent (

Many studies have looked at whether children map newly acquired symbolic representations of number onto their pre-existing non-symbolic representations of number (the approximate number system;

These findings are consistent with the view that the link between non-symbolic and symbolic representations of number becomes weaker at some point in development. In a study by

Together, these recent studies indicate that the acquisition of Arabic digits, especially the ability to access and translate exact magnitude information (cardinality) between number words and Arabic digits is a key foundation for arithmetic proficiency at the start of a child’s mathematical career. However, to our knowledge, no study has yet investigated the developmental trajectory of children’s acquisition of Arabic digit knowledge.

To date, basic developmental achievements with regard to Arabic digit knowledge have been investigated in different studies using varying number ranges, age groups and tasks, such as recognizing Arabic digits (e.g.

Using a cross-sectional design, we examined how 4- to 7-year-old children handle Arabic digits in a series of tasks differing in magnitude manipulation requirements that range from digit recognition to implicit as well as explicit access to magnitude information. There were two main objectives. First, to have a closer look at Arabic digit knowledge during the preschool period (4- and 5-year-olds) and school entry (6- and 7-year-olds) by using a variety of different tasks. Second, to reveal developmental trajectories towards Arabic digit knowledge. We choose this particular age range because children start to learn about Arabic digits in this time period; hence, developmental shifts with regard to Arabic digit knowledge likely occur during this time window. Since most children in Germany enter school at either 6 or 7 years of age, we included both age groups, anticipating possible performance differences between these two age groups.

Five tasks were used to assess children's Arabic digit knowledge. These tasks are presented here in the order in which they were presented to the children, together with the rationale behind their choice and ordering. First, in the

The study by

Third, in the

Fifth, in the

To summarize, the LBT measures implicit access to magnitude information of Arabic digits, whereas the ADGive-N and the SCT measure explicit access to magnitude information of Arabic digits. The SCT may well be more difficult compared to the ADGive-N task, since children have to ignore the perceptually salient physical size of the digits in order to give the correct answer. Based on previous results (

As shown in

Gender | Mean age (months) | Handedness | |||
---|---|---|---|---|---|

23 | 13 female | 51.4 | 3 | 21 right, 1 ambidextrous | |

23 | 13 female | 65.4 | 3.2 | 15 right | |

22 | 11 female | 77.4 | 3 | 17 right, 1 ambidextrous | |

20 | 10 female | 87.9 | 3.8 | 17 right |

Of the 6-year-old children, there were 15 children who already had entered primary school (_{age} = 77.9 months,

Nine additional children were tested but excluded from analyses, because of technical failure (two 4-year-olds, one 5-year-old), shyness (two 4-year-olds, one 5-year-old), unwillingness (one 4-year-old, one 7-year-old) or insufficient knowledge of German language (one 5-year-old). Two children had to be excluded from individual tasks because of unwillingness (LBT: one 4-year-old) or because of experimenter error (ADGive-N, for the verbal measure only: one 5-year-old).

A single testing session comprised the five tasks in the following order:

As shown in

Left panel: Child and experimenter during the

Children were shown six cards (laminated sheets of A5-sized paper with numbers printed in portrait format in the center using Arial font 280) depicting the Arabic digits 1 through 6 (one digit per card) twice in random order. They were asked for each digit ‘What is this?’ The mean number of digits children were able to identify was calculated.

In each of two trials children were handed over a set of cards (digits 1 through 6, cards were the same as in the DNT) and asked to lay the cards out on the table in correct order. In each set of cards (one set per trial), the digits were shuffled (A: 2-6-1-3-4-5 and B: 6-1-5-4-3-2). The order of sets (shuffling A first, shuffling B first) was randomized across children in each age group. The mean length of correctly ordered sequences was compared across age groups, as well as the number of children per age group who were able to lay out all six digits in a correct order. Additionally, for those children who laid out all six digits in a correct order, it was coded whether they laid out the digits left to right (L-R) or right to left (R-L).

In this version of the Give-N task (

Since children who fail to produce the correct amount of hazelnuts when seeing a written numeral may well know its cardinal value when hearing the number word, the experimenter proceeded after two failures for a given digit by asking the child verbally (ADGive-N_v). For example, if a child succeeded two times when seeing the digit 4, but failed two times when seeing the digit 5, the child was asked ‘Can you give the bear five hazelnuts?’ Also with the verbal procedure children needed to succeed twice on a given number word to qualify as knowing the cardinal value of that number word.

The mean of correctly assigned nuts when presented (1) Arabic digits and (2) number words was compared separately as well as to each other across age groups.

As shown in

Example of an incongruent trial in the

For all analyses, a significance level of α = 0.05 was used and all

Participants with a mean bias of two standard deviations below or above the group mean bias were considered outliers and excluded from analyses (one 4-year-old, two 5-year-olds, two 6-year-olds and two 7-year-olds).

The mean deviation from zero in cm for trials with the digit 5 on the left of the line was compared to the mean deviation from zero in cm for trials with the digit 5 on the right of the line. As depicted in

Line Bisection Task. Mean bias (mean deviation from zero in cm) for bisected lines with larger digit to its right (5 right) and larger digit to its left (5 left) as a function of age. Error bars represent standard errors of the means.

As shown in ^{2}(3, 88) = 44.57,

Mean number of Arabic digits (DNT: number words correctly assigned to the digits; DST: amount of digits laid out in a correct order; ADGive-N: correct cardinal values produced for digits) and number words (ADGive-N_v: correct cardinal values produced for number words) across tasks as a function of age. Error bars represent standard errors of the means.

Across two trials, the mean number of Arabic digits children were able to lay out in correct order increased with age, Kruskal-Wallis, χ^{2}(3, 88) = 40.21,

Likewise, the number of children that were able to lay out all digits from 1 through 6 in a correct order in both trials increased with age, χ^{2}(3, 88) = 40.54, ^{2}(1, 46) = 7.09, ^{2}(1, 44) = 8.93,

Of all successful trials, the order (L-R, R-L) in which children laid out the digits was also coded. Of those children who laid out all digits in a correct order in both trials, the majority of children in each age group was consistent in the order in which they laid out these digits (4-year-olds: 100%, 5-year-olds: 87%, 6-year-olds: 96%, 7-year-olds: 100%). Of those consistent children, the majority in each age group laid out the digits L-R rather than R-L (4-year-olds: 83%, 5-year-olds: 85%, 6-year-olds: 86%, 7-year-olds: 95%).

Children’s ability to produce correct cardinal values for the digits 1 through 6 increased with age, Kruskal-Wallis, χ^{2}(3, 88) = 40.79,

Likewise, children’s ability to produce correct cardinal values for number words (ADGive-N_v) increased with age, Kruskal-Wallis, χ^{2}(3, 87) = 25.66,

As shown in ^{2}(3, 87) = 39.54,

The order (congruent first/incongruent first) in which the trials were presented did not reliably affect performance in any of the age groups, Mann-Whitney U,

When comparing the mean number of correct congruent trials (n = 2) to the mean number of correct incongruent trials (n = 2), 4- and 5-year-olds were correct more often in congruent than incongruent trials (

As depicted in

Task | DNT |
ADGive-N |
DST |
|||
---|---|---|---|---|---|---|

Comparison level | 4 years | 5 years | 4 years | 5 years | 4 years | 5 years |

ADGive-N_v | ||||||

Group | .038 | .416 | < .001 | .041 | < .001 | .020 |

Individual | .017 (.41) [7] | .023 (.47) [17] | < .001 (.75) [6] | .004 (.59) [17] | .042 (.38) [6] | .132 (.31) [14] |

DNT | ||||||

Group | .033 | .018 | .003 | .017 | ||

Individual | .001 (.58) [7] | < .001 (.77) [16] | .028 (.39) [7] | .007 (.54) [15] | ||

ADGive-N | ||||||

Group | .096 | .205 | ||||

Individual | .014 (.46) [11] | .010 (.5) [15] |

_{b}) correlations: _{b} correlation coefficients) [number of ties]. Samples sizes varied between

A consistent developmental pattern was found on the group level as well as on the individual level. On the group level statistical comparisons revealed the following performance pattern for both age groups: children assigned more cardinal values to number words than to digits (ADGive-N_v > ADGive-N) and they assigned more number words than cardinal values to digits (DNT > ADGive-N). Finally, when children knew the cardinal value of a digit, they also knew its ordinal position (ADGive-N = DST). On the individual level, for all task pairings of interest, performance correlated positively in both age groups: the higher the score in the ADGive-N_v, the higher the score in the ADGive-N. The same positive relation holds for the DNT and the ADGive-N, the DNT and DST as well as the ADGive-N and the DST task pairings.

To control for multiple testing, a Fisher’s omnibus test was run on the ^{2}(24) = 92.62, ^{2}(24) = 113.40, ^{2}(48) = 206.02,

In this cross-sectional study we documented children’s first steps towards learning the relation between Arabic digits and the exact cardinal values they represent. To our knowledge, this is the first study to date which used a broad range of age groups and tasks, including digit recognition, cardinality and ordinality comprehension as well as tasks measuring implicit and explicit access to magnitude information of Arabic digits, to explore and document the developmental trajectory towards Arabic digit knowledge. Of particular interest was the period from preschool to school entry, in which children start to learn about the meaning of Arabic digits. For this purpose the digits in the range of 1 to 6 were chosen for all tasks. A clear pattern emerged across age groups and across tasks with a strong within-child consistency.

First, comparisons across age groups in all tasks revealed a steady growth in proficiency towards learning the cardinal values of Arabic digits. Except in the

Second, in contrast to the results of

The results of the LBT revealed one unexpected exception from the general pattern described above. In contrast to the 5-year-olds, 6-year-old children as a group did not show a bisection bias towards the larger digit in the LBT, only 6-year-old school children showed a marginally significant bias. This finding is inconsistent with the overall developmental trajectory and might therefore represent a random effect occurring in the particular sample tested in this study. However, before we can decide whether this interpretation is correct the result in question needs to be replicated, especially given the possibility that cohort-specific biases may contaminate comparisons in cross-sectional designs.

Third, whereas the LBT measured implicit access to magnitude information of Arabic digits, the ADGive-N task and the SCT measured explicit access to magnitude information of Arabic digits, albeit in different ways. In the ADGive-N task, children were asked to provide the correct amount of nuts in response to a given digit. In the SCT task children were asked to compare the digits 2 and 5 differing not only in numerical but also in physical magnitude. Of the 4-year-olds, only 45% were able to provide the correct amount of nuts in the ADGive-N task for the numbers 2 and 5; consistently, 4-year-olds did not show a bisection bias in the LBT and did not succeed in the SCT, but were at chance in choosing the digit 5 when it was physically smaller than the digit 2. In contrast, 5-year-olds provided the correct amount of nuts for both digits in the ADGive-N task and showed a bisection bias in the LBT but still failed to use their cardinality knowledge in the SCT. Also the 5-year-olds were at chance when the digit 5 was physically smaller than the digit 2, even though they were given as much time as they wanted to compare the digits.

The results of the implicit and explicit measures discussed above fit well with the previous literature on the development of Arabic digit knowledge. Implicit measures of numerical magnitude are thought to be an indicator of automatic access to numerical magnitude. That is, by just seeing an Arabic digit its magnitude representation becomes activated (

Fourth, comparisons of performances between the ADNGive-N task, DNT and the DST of the 4- and 5-year-old children revealed the developmental trajectory preschoolers traverse towards Arabic digit knowledge. The general pattern was the same for 4- and 5-year-old children on a group level as well as on the individual level. As expected, both age groups knew the cardinal values of more number words than of Arabic digits (ADGive-N vs. ADGive-N_v). These behavioral findings are in line with the specific brain activity pattern found by

On the individual level, correlational analyses of DNT, ADGive-N and DST parings revealed significant correlations between all three task pairings, demonstrating a strong within-child consistency in the developmental trajectory towards Arabic digit knowledge: children who recognized more digits also provided more correct cardinal values and ordinal positions of digits. Furthermore, children who were more proficient in cardinality comprehension of digits (ADGive-N task) were also more proficient in ordinality comprehension of digits (DST).

Previous studies (

In recent years, the development of symbolic representations of number has been studied intensively. Especially the contribution of non-symbolic and symbolic representations to math achievement at the beginning of formal mathematical education took center stage in many studies (see introduction). From those studies evidence accumulated in favor of the view that the link between non-symbolic and symbolic representations of number becomes weaker between 4 and 6 years of age. That is, non-symbolic and symbolic number representations may overlap considerably less than previously thought. The current study adds to this literature by investigating how children acquire an understanding of the exact meaning of Arabic digits. One might expect that as soon as children develop their first exact symbolic representations of number (cardinality of number words), these representations would translate directly to symbolic representations of a different format (Arabic digits) they can identify. However, the finding that children traverse through a transitional stage in which they cannot produce the correct cardinal value for Arabic digits they do correctly identify, may provide further evidence for children developing a separate exact representational system of symbolic number representations.

Becoming symbol-minded is a crucial developmental task which involves learning about the referential nature of symbolic notations. Children need to learn that symbols, such as Arabic digits, refer to something other than themselves. For example, the Arabic digit '3' is not just two half circles on top of each other, the digit itself carries meaning and refers to the cardinal value of sets. The results of the current study revealed a clear developmental pattern through which preschoolers traverse towards Arabic digit knowledge. Preschoolers traverse from first knowing the cardinal values of number words to name Arabic digits, to knowing their cardinal values and, concurrently, their ordinal position. The main developmental shifts with regard to children’s proficiency to manipulate magnitude information of Arabic digits occur between 4 and 6 years of age. By the age of 5, children access magnitude information of digits implicitly as well as explicitly, but their explicit knowledge of magnitude of digits consolidates by 6 years of age. This improvement of explicit cardinality knowledge of digits is accompanied by an improvement in translating cardinality knowledge of number words to digits. By the age of 6, which for most children in Germany demarcates the period of school entry, children are as proficient as 7-year-old school children in manipulating magnitude information of Arabic digits.

We would like to thank the parents and children who participated in this study. We also would like to thank Marion Klein, Alexander Kirmβe and all students of the bachelor course 'Empirisches Praktikum 2012/2013' for their engagement and their contribution to the implementation of the study design, data collection and coding. We would further like to thank John Towse and two anonymous reviewers for their helpful comments on an earlier version of this manuscript.

The authors have no funding to report.

The authors have declared that no competing interests exist.