Florence Labrell
email: florence.labrell@univ-reims.fr

Yann Mikaeloff

Hervé Perdry

Georges Dellatolas

Time knowledge acquisition in children aged 6 to 4 11 years and its relationship with numerical skills 5 6 7

Keywords:

a b s t r a c t Acquisition of time knowledge (TK; the correct representation and use of time units) is linked to the development of numerical abilities, but this relationship has not been investigated in children. The current study examined the acquisition of TK and its association with numerical skills. A total of 105 children aged 6 to 11 years were interviewed with our Time Knowledge Questionnaire (TKQ), developed for purposes of this study, and the Zareki-R, a battery for the evaluation of number processing and mental calculation.

The TKQ assessed conventional time knowledge (temporal orientation, temporal sequences, relationships between time units, and telling the time on a clock), estimation of longer durations related to birthday and life span, and estimation of the duration of the interview. Time knowledge increased with age, especially from 6 to 8 years, and was strongly linked to numerical skills.

Regression analyses showed that four numerical components were implicated in TK: academic knowledge of numbers and number facts (e.g., reading Arabic numerals, mental calculation), number line estimation (e.g., correspondence between a number and a distance), contextual estimation (e.g., many/few leaves on a tree, children in a family), and numerical tasks involving verbal working memory (e.g., comparison of numbers presented orally).

Introduction

Many philosophers have explored the relationship between time and number-from Aristotle (384-322 B.C.), who wrote that ''time is the number of movement in respect to the before and after" (Physics IV), to more recent thinkers such as Bergson (1889[START_REF] Bergson | Essais sur les Données Immédiates de la Conscience[END_REF], who claimed that number is related to space but not to our experience of time. The relationship between time and number has been less widely investigated by developmental psychologists.

Dealing with time is an absolute necessity to organize life in society (for work, meals, leisure, etc.) [START_REF] Fraisse | Psychologie du temps[END_REF], and this has led to the creation of conventional unit systems. Children need to develop some knowledge about these systems even if they are starting to have a primitive ''sense of time" linked to actions-time to go to bed, duration of a pleasant/unpleasant event, and so forth. Both aspects, the conventional system of time units and the sense of time, have been assimilated to time knowledge (Friedman, 1990a). Time knowledge (TK), precisely defined here as the correct representation and use of the various time units (e.g., seconds, minutes, hours, days, weeks, months, seasons, years), has not been widely studied, nor has its development. Whereas using time units implies a proper representation and use of numbers (e.g., ''It is 10 o'clock on the 25th day of the 9th month in the year 2015"), children's development of time knowledge is logically linked to the development of their numerical competencies.

However, up to now, most studies have been devoted to time perception over short durations without using verbal answers involving time units (for a review, see [START_REF] Droit-Volet | Time perception in children: A neurodevelopmental approach[END_REF]. These studies reveal that the ability to judge time precisely in a wide variety of temporal tasks improves during childhood via the development of selective attention and working memory [START_REF] Droit-Volet | Development of time sensitivity and information processing speed[END_REF]. As far as we know, the relationship between TK and numerical competencies has not been systematically examined in children, probably because of the lack of a tool for TK evaluation. Therefore, our aim was twofold: first, to study TK development during childhood and, second, to investigate the links between TK and children's numerical abilities. For the purposes of this study, we developed a Time Knowledge Questionnaire to evaluate the correct representation and use of the various time units.

Evaluation of TK: previous and new components

Different aspects of TK, such as present-time awareness, using a calendar [START_REF] Fraisse | Psychologie du temps[END_REF]Friedman, 1990a;[START_REF] Godart | Le développement de la localisation dans le temps chez des enfants de 5 à 9 ans de milieux socio-économiques différents[END_REF], and telling the time on a clock [START_REF] Burny | Towards an agenda for studying learning and instruction focusing on time-related competences in children[END_REF][START_REF] Cohen | Developmental progression of clock face drawing in children[END_REF][START_REF] Friedman | Children's analog and digital clock knowledge[END_REF][START_REF] Vakali | Clock time in seven-to ten-year-old children[END_REF], have already been investigated in children. Present-time awareness, also called time orientation, is the awareness of the present hour, day, month, year, or season. Awareness of the present day is thought to be acquired at the end of kindergarten at around 6 years, whereas that of the present month, season, or year occurs later in primary school at 7 or 8 years (Friedman, 1990a).

Other aspects of TK, more related to the experience of time, have been less widely envisaged. For example, the representation of time units might depend on implicit learning, such as experiencebased knowledge (e.g., ''A familiar activity takes x minutes/seconds/hours"), which could be clinically relevant. Indeed, children with autistic spectrum disorders have been shown to perform less well than typically developing children when durations were to be compared (e.g., eating a banana, going to a swimming pool) [START_REF] Janeslätt | Patterns of time processing ability in children with and without developmental disabilities[END_REF]).

Children's estimation of time during a long ongoing activity has rarely been studied (e.g., in the case of an interview duration, ''For how long have we been here together?"), unlike children's percep- [START_REF] Droit-Volet | Time perception in children: A neurodevelopmental approach[END_REF] without a verbal answer in the form of a number of time units.

In addition, children's representations of the life span have rarely been investigated. Life-span representations have been explored in studies of children's conceptual development in biology. For instance, the understanding of irreversibility of growth for humans, animals, and plants has been explored in typically developing children [START_REF] Inagaki | Young children's recognition of commonalities between animals and plants[END_REF][START_REF] Labrell | The development of diachronic thinking between 6 and 11 years: The case of growth and death[END_REF][START_REF] Maurice-Naville | The development of diachronic thinking: 8-12-Year-old children's understanding of the evolution of forest disease[END_REF][START_REF] Rosengren | As time goes by: Children's early understanding of growth in animals[END_REF] and in pediatric populations such as survivors from cerebellar tumors [START_REF] Labrell | Conceptions of time in children treated for malignant cerebellar tumors[END_REF]. However, investigations examining how children estimate the duration of growth and ageing (e.g., between infancy and childhood, between childhood and adulthood, between adulthood and old age) are lacking.

Lastly, few studies have investigated time intervals based on birthdays, for instance, how children progressively orient themselves with respect to their own birthdays. According to [START_REF] Godart | Le développement de la localisation dans le temps chez des enfants de 5 à 9 ans de milieux socio-économiques différents[END_REF], 33% of a sample of 9-year-olds made a correct estimation of the present-next birthday time interval versus 16% for the present-last birthday time interval.

As far as we know, none of the above dimensions (i.e., classic present-time awareness, telling the time on a clock, and explicit knowledge of time units), nor any of the less investigated dimensions (e.g., the estimation of (a) durations longer than a few seconds, (b) human life span intervals, and

(c) birthday-based intervals), have ever been combined in the same tool. Consequently, correlations between all of these different aspects of TK during development are not well known, and it remains unclear how many different dimensions of TK (e.g., one or more than one) should be considered.

Time knowledge: relationships with numerical skills and underlying mechanisms TK acquisition enables everyday time judgments. For instance, TK enables children to locate themselves in time, from the present moment to the more or less close past or future. The verbal sequences of days and months are learned during early primary school [START_REF] Fraisse | Psychologie du temps[END_REF]Friedman, 1990a;[START_REF] Godart | Le développement de la localisation dans le temps chez des enfants de 5 à 9 ans de milieux socio-économiques différents[END_REF]. According to [START_REF] Friedman | Image and verbal processes in reasoning about the months of the year[END_REF][START_REF] Friedman | Analog and semantic models of judgments about the months of the year[END_REF][START_REF] Friedman | The development of children's knowledge of temporal structure[END_REF][START_REF] Friedman | Children's representations of the pattern of daily activities[END_REF], children first develop a list-based representation of the days and the months, relying on lexical working memory, before being able to use an analogous spatial representation of the time intervals between days and months. Moreover, these time sequences rely on logical reasoning, particularly seriation performance [START_REF] Friedman | The development of children's understanding of cyclic aspects of time[END_REF],

and involve an understanding of the linear and cyclic nature of time given that days and months not only occur but also recur in a specific order.

Location in time implies a correct use of time units (e.g., seconds, minutes, hours, days, weeks, months, seasons, years), relying both on the mastery of the verbal sequences of the time units and on the numerical abilities of counting (using forward and backward sequences) and mental calculation. Indeed, the sequences of time units are closely related to numbers, and especially to counting (e.g., days of the week, months, years, hours), but also to subtracting (e.g., ''How long is it to my birthday from now?"). The Chinese calendar in particular is based on an arithmetical labeling of each day or month (e.g., March is ''month three"), but many other languages also use arithmetical labels for the days of the week (e.g., in Portuguese: second, third, fourth, fifth, and sixth for Monday-Friday; in modern Greek: second, third, fourth, and fifth for Monday-Thursday).

Numbers are also involved in telling the time on a clock, which is a complex cognitive activity relying on mathematics, [START_REF] Andersson | Mathematical competencies in children with different types of learning difficulties[END_REF][START_REF] Burny | Clock reading: An underestimated topic in children with mathematics difficulties[END_REF][START_REF] Cohen | Developmental progression of clock face drawing in children[END_REF][START_REF] Friedman | Children's analog and digital clock knowledge[END_REF][START_REF] Vakali | Clock time in seven-to ten-year-old children[END_REF] as well as on several domains of competence such as language [START_REF] Bock | Minding the clock[END_REF], visuospatial skills [START_REF] Eden | Clock drawing in developmental dyslexia[END_REF], and conceptual representations [START_REF] Burny | Towards an agenda for studying learning and instruction focusing on time-related competences in children[END_REF]. In particular, explicit knowledge of the relationships between time units (e.g., how many hours in a day, how many minutes in an hour) has been considered as a prerequisite for correct use of analogue and digital clocks [START_REF] Burny | Towards an agenda for studying learning and instruction focusing on time-related competences in children[END_REF][START_REF] Burny | Clock reading: An underestimated topic in children with mathematics difficulties[END_REF][START_REF] Cohen | Developmental progression of clock face drawing in children[END_REF][START_REF] Friedman | Children's analog and digital clock knowledge[END_REF]. It has been shown that most children are able to read both analogue and digital clocks correctly at the age of 8 to 10 years [START_REF] Burny | Towards an agenda for studying learning and instruction focusing on time-related competences in children[END_REF].

Correct representation and use of time units is probably related not only to knowledge of numbers and computation but also to the so-called ''cognitive estimation" of quantities [START_REF] Harel | It takes nine days to iron a shirt: The development of cognitive estimation skills in school age children[END_REF]. This cognitive estimation has at least two components: the first relating to the First, it has been proposed that the estimation of the magnitude of a number and the estimation of a time interval rely on the same process (for a review, see [START_REF] Allman | Developmental neuroscience of time and number: Implications for autism and other neurodevelopmental disabilities[END_REF]-a mode control model generating an estimate of the magnitude of a duration or nonsymbolic numerosity from an accumulation of ''pulses" compared with stored criteria of a given duration or number [START_REF] Meck | A mode control model of counting and timing processes[END_REF][START_REF] Meck | Temporal integration in duration and number discrimination[END_REF]. [START_REF] Walsh | A theory of magnitude: Common cortical metrics of time, space, and quantity[END_REF] proposed a theory of magnitude postulating that the same cognitive system, located in the inferior parietal cortex, is devoted to common processing mechanisms for the evaluation of time, number, and space. Several studies on the nonsymbolic ''sense" of number and time showed a primitive time and number sense during infancy that is based on ratio (rather than absolute) differences in magnitude [START_REF] Provasi | Temporal bisection in 4-month-old infants[END_REF].

One and the same developmental progression for temporal discrimination and number discrimination during infancy has been suggested [START_REF] Brannon | Temporal discrimination increases in precision over development and parallels the development of numerosity discrimination[END_REF]. The ''triple code" model for numbers (i.e., verbal, Arabic, and analogical) also supposes an analogical representation associating a quantity with a number [START_REF] Cohen | Cerebral networks for number processing: Evidence from a case of posterior callosal lesion[END_REF][START_REF] Dehaene | Varieties of numerical abilities[END_REF].

Second, the correct representation of time units might also be related to another ''cognitive estimation" of quantities linked to everyday and contextualized activities. This concerns, for instance, the estimation of the time needed to carry out an activity, as in the Biber Cognitive Estimation Test [START_REF] Bullard | The Biber cognitive estimation test[END_REF], also studied during childhood [START_REF] Harel | It takes nine days to iron a shirt: The development of cognitive estimation skills in school age children[END_REF]. These estimations are thought to rely on mental control and planning, which are generally considered to be frontal lobe functions [START_REF] Shallice | The involvement of the frontal lobes in cognitive estimation[END_REF], and not exclusively on ''magnitude" given that the same quantity (e.g., 10) may be ''a lot" (e.g., 10 children in the same family) or ''a few" (e.g., 10 leaves on a big tree).

Aims and hypotheses

We investigated the development of most aspects of TK, from 6 to 11 years, and its link with children's numerical competencies. We hypothesized that TK development is not linear because the different numerical components are thought not to be acquired at the same age. Indeed, TK is likely to be linked to at least three aspects of numerical abilities: (a) the knowledge of numbers and number facts involved in devices that measure time such as clocks and calendars, (b) analogical estimation, and (c) contextual estimation. For example, we hypothesized that knowledge of numbers might be acquired earlier than contextual estimation, which is thought to rely on children's experience of everyday and contextualized activities. It can be recalled that the (b) and (c) aspects above are thought to be processed in two different brain areas.

Consequently, the links between TK and numerical abilities might be different according to children's age. Educated adults are expected to perform perfectly on TK when it concerns the usual time units (i.e., from seconds to years); however, most adults do not know that 10 -24 s is a yoctosecond and that 10 24 s is a yottasecond. As a consequence, correlations between usual TK and numeracy cannot be estimated in adults. In children, some subcomponents of number processing may be perfectly acquired by older children (e.g., counting, reading and writing Arabic numerals) but not by younger children. Conversely, other subcomponents of number processing (e.g., calculation) may show floor effects in younger children, making it difficult to estimate their correlation with TK, whereas this is not so among older children.

Method

Participants

The participants were 105 school children from Grade 1 to Grade 5, 57 girls (54%) and 48 boys (46%), aged 6.2 to 11.1 years. Each school grade corresponded to a different age range (see Table 1).

Children were recruited in their schools, with informed consent to participate in the study obtained from both parents.

Time Knowledge Questionnaire

The TKQ (Appendix A) comprises seven subtests. The first four subtests investigated conventional time knowledge: orientation (OR), sequences (SEQ), time units (TU), and telling the time on a clock (CL), and comprised 5, 3, 4, and 5 questions, respectively. The questions about telling the time on a clock used pictures of five analogue clocks with five different positions of the minute and hour hands.

Each answer was coded as correct (1 point) or incorrect (0 points), for a total conventional time score from 0 to 17.

The fifth subtest, life span (LS), used four pictures (a baby, a child, a young man, and an old man)

illustrating the biological process of ageing (see Fig. 1). A fair-haired man, depicted at four different stages in development from infancy to old age, was chosen. The child was asked three questions about the time required to go from one age to the next (i.e., from baby to child, from child to young man, and from young man to old man). Each answer was coded as 0, 1, or 2 points. The coding rule was based on the answers of 20 adults that were as follows: from baby to child, median = 6 years, range = 4 to 10; from child to young man, median = 22 years, range = 15 to 30; from young man to old man, med- point was given if the answer was borderline (i.e., not in the adult range but not clearly impossible), that is, within the following intervals: baby-child, 2 to 3.99 and 10.01 to 15 years; child-young man, 5 to 14.99 and 30.01 to 40 years; young man-old man, 55.01 to 90 years. No points were given for answers outside of these ranges or for ''don't know" answers (total LS score from 0 to 6).

The sixth subtest, birthday (BIR), comprised six questions. Questions 1 to 3 (''How old are you?", ''What age were you last year?", and ''How old will you be next year?") were not taken into account in the calculation of the score because the answers were invariably ''correct" (i.e., the absolute difference between the given age and the exact age was always less than 1 year). The other three questions regarding birthdays were ''When is your birthday?", ''How long ago was your birthday?", and ''How long is it to your next birthday?" For these three questions, the correct answer was available (for each child, date of birth and date of interview were recorded), which made it possible to use a coding for error. Zero points were given if the child did not know the date of his or her birthday. Otherwise, for both of the last two questions (i.e., time intervals to the previous and next birthdays), 0 points were given for ''don't know" and answers exceeding 12 months, and 1, 2, and 3 points were given for errors (i.e., absolute difference between the answer and the correct interval) greater than 3 months, between 3 months and 1 month, and less than 1 month, respectively (i.e., total BIR score from 0 to 6).

The last subtest was interview duration (ID). At the end of the interview, the examiner asked the child, ''For how long have we been here together?" The median duration of the interview was 15 min (Q3-Q1 = 14-21 min, range = 5-42). Zero points were given for ''don't know" answers. For all other answers, 0 to 10 points were given according to the absolute difference between the log of the real duration of the interview and the log of the answer (in minutes) (see Appendix C for an illustration of this scoring method).

Two subtests from the TKQ, sequences and time units, were based on a previous study on time conceptions in children treated for malignant cerebellar tumors [START_REF] Labrell | Conceptions of time in children treated for malignant cerebellar tumors[END_REF] given that the cerebellum has been implicated in time perception. Other dimensions linked to time knowledge, such as children's estimation of the duration of several daily activities (e.g., brushing their teeth, going from home to school), were investigated in this earlier study but were not retained in the current study for psychometric reasons (insufficient internal consistency). In the previous study, we were also interested in children's understanding of time in the biological process of growth. Therefore, we used the same pictures from the life span subtest, but we did not ask for the time lapse between the different stages.

Zareki-R

The Zareki-R (Appendix B) is a battery for the evaluation of number processing and mental calculation in school children. It contains 11 subtests evaluating knowledge of the verbal sequence of numbers and acquisition of the principles of counting (counting dots and counting backward), knowledge of the oral and written forms of numbers (reading numbers and writing numbers under dictation, comparison of numbers, all written presentations of numbers in Arabic numerals, and comparison of numbers presented orally), mental calculation (orally presented additions, subtractions, and multiplications), problem solving (presented orally), number line estimation (positioning integers >0 and <100 on a 0-100 line), perceptual estimation (giving the approximate number of elements seen without counting), and contextual estimation (deciding whether a given number in a given situation is ''a few," ''a medium number," or ''a lot").

Procedure

The experimenter interviewed each child separately at his or her school in a quiet room, with the order of presentation of each tool being random. Sessions lasted approximately 50 min. For the TKQ, the experimenter asked the child to answer the interview questions from the pictures for telling the time on a clock and life span with no time limit. However, the experimenter recorded the time at the beginning of the TKQ, and also at the end, in order to compute the error for the final interview duration answer. The child's answers were fully transcribed.

For the Zareki-R, the experimenter used the test as scheduled in the handbook, that is, with visual material when required (for counting dots, reading numbers, number line estimation, perceptual estimation, and written comparison).

Results

TKQ subtests in relation to children's age: generation of a total TKQ score A total TKQ score (TKS) was computed, summing the scores on each subtest (max = 39 points; see Table 2) as well as three partial scores for conventional time (sum of the first four subtests), long durations (sum of the birthday and life span subtests), and interview duration.

Performance increased with age for all of the subtests in the TKQ (p < .001 in all cases). Table 2 shows a comparison of the means of the subtests and the total TKQ scores across the five age groups.

For the TKS, post hoc comparison of the means (Scheffé test) showed significant differences across all successive age groups except between the 8-and 9-year age groups. For the subcomponents of the TKQ, the improvement between the 6-and 7-year age groups was always significant except for life span (see Table 2 for more details of the post hoc comparison of the means).

Correlations across the seven subtests were all positive and significant at p < .001 (Cronbach's alpha = .86). A principal components analysis confirmed the unidimensional structure of the TKQ (first factor explaining 54% of the variance, loading of the seven subtests on the first factor between .63 and .83, and eigenvalue of the second factor < 1), justifying the computation of a total TKQ score. However, when age was partialed out, some partial correlations between the first four conventional time knowledge subtests and the last three subtests were close to 0 (e.g., between telling the time on a clock and birthday or life span). The age-adjusted alpha was .66 (see Table 3).

The correlation between total TKQ score and total Zareki-R score was high (.84) and remained high when age was partialed out (.60). All 11 subtests in the Zareki-R were positively correlated with the total TKQ score, with Pearson's r values ranging from .33 (perceptual estimation) to .77 (reading numbers). These correlations remained significant when age was taken into account except for perceptual estimation and written comparison.

In summary, the greatest improvement in time knowledge was observed between the 6-and 7year age groups and, to a lesser extent, between the 7-and 8-year age groups, with only a slight improvement after 8 years. However, the developmental pattern was different according to the subtest. For instance, orientation (present-time awareness) exhibited a ceiling effect in children from 7 years of age; in contrast, estimation of interview duration continued to improve between the 9and 10-year age groups. The TKQ showed a unidimensional structure, justifying the computation of only one score summing the subscores; however, this was partly due to the correlation of all the sub- p Orientation (OR) 5 3.6 (1.2) 4.7 (0.6) 4.9 (0.3) 4.9 (0.4) 5.0 (0.0) 4.6 (0.8) 16.4 <.001 Sequences (SEQ) 3 0.9 (0.8) 2.1 (1.0) 2.4 (0.7) 2.5 (0.6) 2.9 (0.3) 2.2 (0.9) 18.3 <.001 Time units (TU) 4 1.7 (0.9) 3.1 (0.8) 3.7 (0.7) 3.8 (0.5) 3.9 (0.2) 3. Overall observed correlations between the TKQ score and the Zareki-R score

A linear regression with TKQ score as the dependent variable and the 11 Zareki-R subtests as explanatory variables showed that four subtests remained significant at p < .05: reading numbers, number line estimation, oral comparison, and contextual estimation. A backward selection confirmed that these four subtests explained 75% of the variance of the total TKQ score, which was almost the same as the percentage of variance explained by the 11 subtests (77%) (see Table 4).

The same regressions with backward selection (p < .01 for a variable to be retained) were performed with dependent variables in the form of three partial scores on the TKQ: a ''conventional time score" (CTS, the sum of the four first subtests in the TKQ), a ''long durations" score (LDS, the sum of the birthday and life span subtests), and the interview duration score (IDS). The subtests in the Zareki-R selected were only one subtest for IDS (number line estimation), two subtests for LDS (reading numbers and contextual estimation), and three subtests for CTS (mental calculation, reading numbers, and contextual estimation).

In sum, these results show a strong association between time knowledge and number processing in typically developing children. Furthermore, the results also suggest that different cognitive processes involved in time processing influence TK (number knowledge, perception of magnitude, contextual estimation, and verbal working memory involved in number comparisons). Correlations between the Zareki subscores and the total TKQ score according to children's age

Correlations between subtests in the Zareki-R and total TKQ score varied according to age (Table 5).

No clear pattern appeared in the younger children (6 years), who showed floor effects in many subtests. In the older children (10 years), contextual estimation was the only subtest showing a strong positive link with TKS. Reading numbers was linked with TKS at 7 years, oral comparison at 7 and 8 years, and positioning a number on a scale at 7 and 9 years. In summary, these results suggest that associations of TK with specific number processing subtests change with age.

Discussion

The current study in typically developing school children aged 6 to 11 years describes the development of TK and analyzes the relationships between overall TK development and numerical skills. The proposed definition of TK (i.e., the correct representation and use of the various time units) implies numbers. We hypothesized that TK is linked to at least three aspects of numerical abilities: the knowledge of numbers, analogical estimations, and contextual estimations.

Development of time knowledge

The TKQ used for the evaluation of TK included subtests widely investigated in previous studies in typically developing children (for orientation or present-time awareness, see [START_REF] Friedman | Image and verbal processes in reasoning about the months of the year[END_REF][START_REF] Friedman | Analog and semantic models of judgments about the months of the year[END_REF][START_REF] Friedman | Children's representations of the pattern of daily activities[END_REF]; for temporal sequences, see [START_REF] Friedman | The development of children's understanding of cyclic aspects of time[END_REF][START_REF] Friedman | The development of children's knowledge of temporal structure[END_REF][START_REF] Friedman | Children's representations of the pattern of daily activities[END_REF]; for telling the time on a clock, see [START_REF] Burny | Towards an agenda for studying learning and instruction focusing on time-related competences in children[END_REF][START_REF] Cohen | Developmental progression of clock face drawing in children[END_REF][START_REF] Friedman | Children's analog and digital clock knowledge[END_REF][START_REF] Vakali | Clock time in seven-to ten-year-old children[END_REF] and also new subtests such as those exploring children's representation of long durations (life span and birthday).

Despite this seemingly heterogeneous composition, the TKQ showed a relatively unidimensional structure, justifying the generation of a total score. However, this one-factor solution was partially attributable to age differences, as suggested by the lower age-adjusted partial correlations between the subtests and especially the partial correlations close to 0 between some conventional TK subtests (telling the time on a clock) and the subtests concerning long durations (birthday and life span). This observation led to the generation of three partial scores in addition to the total TKQ score: a conventional time score (sum of the first four subtests), a long duration score (sum of birthday and life span), and the interview duration score.

A strong age effect on all subtests in the TKQ was observed as well as many developmental patterns already reported in previous studies. For example, [START_REF] Friedman | The development of children's understanding of cyclic aspects of time[END_REF][START_REF] Friedman | Children's representations of the pattern of daily activities[END_REF] provided evidence that (the baby, the child, the young man, and the old man) have not been studied in children's biological conceptions and might depend on their experience of the real world.

The results about birthdays showed that the children were capable of accurately identifying their birthdays only during the primary school years. This progression in children's ability to position their birthdays in time, based on past and future events, corroborates [START_REF] Friedman | The development of children's knowledge of the times of future events[END_REF] findings regarding children's knowledge of the time to future events as occurring between 7 and 10 years.

Relationships between time knowledge and numerical skills

The total TKQ score was strongly related to numerical skills, explaining 70.6% of the variance (r = .84) when only the total Zareki-R score was considered and 77% of the variance when performances on the 11 numerical subtests were considered. Regression analyses showed that four numerical subtests explained an almost equivalent proportion of the variance (75%): reading numbers, number line estimation, contextual estimation, and comparison of numbers presented orally.

When the three partial TKQ total scores (rather than the total TKQ score) were the dependent variables, the same subtests in the Zareki-R were selected. Number line estimation was the only subtest retained for IDS; mental calculation, reading numbers, and contextual estimation were retained for CTS; and reading numbers and contextual estimation were retained for LDS. In other words, as hypothesized, TK was found to be predicted by different numerical skills.

Each of these numerical predictors of TK represents a different domain of numerical skills. For the conventional time score, reading numbers and mental calculation are clearly school dependent, as is knowledge of the relationships between time units (e.g., ''One hour is 60 minutes"). To master time sequences or units, and to tell the time on a clock correctly, children need to compute with numbers.

The regression analysis predicted the interview duration score only from number line estimation, which requires a quantity to be associated with a number [START_REF] Dehaene | Varieties of numerical abilities[END_REF]. Indeed, to solve this task, children need to master the analogical representation of the ''triple code" model [START_REF] Cohen | Cerebral networks for number processing: Evidence from a case of posterior callosal lesion[END_REF]. In other words, children encountering difficulties in evaluating the passage of time need to be trained in the analogy between quantity and number.

The predictors of long duration score were reading numbers (necessary to identify ages of life span and birthday) and contextual estimation. This numerical ability involves an accurate judgment about quantities, whereby a large number may be ''a few" (e.g., 200 fans at a football game) and a small number may be ''a lot" (e.g., 10 children in a family). The ability to make such judgments requires general knowledge of the world. This knowledge is needed to answer some items in the TKQ correctly. For example, when children are asked to evaluate the duration of developmental periods (from infancy to childhood, from childhood to adulthood, and from adulthood to old age), they need to know how time is involved in human growth and also that the change in appearance is not proportional to the time intervals. According to [START_REF] Shallice | The involvement of the frontal lobes in cognitive estimation[END_REF], contextual estimation is based on executive functions. These functions are also involved in conventional time knowledge (e.g., to answer questions such as ''Can you tell me the seasons of the year in order?" and ''Is a month shorter or longer than a year?"), and they are also needed to estimate the time required to plan daily activities [START_REF] Harel | It takes nine days to iron a shirt: The development of cognitive estimation skills in school age children[END_REF].

Lastly, the results about the relationships between TK development and numerical skills according to children's age have shown that, as predicted, the significant numerical subtests were not the same according to children's age. First, reading numbers, number line estimation, and especially oral comparison were the subtests that were most correlated with the total TKQ score for the youngest children

Fig. 1 .

 1 Fig. 1. Material for the life-span questions.

10F.

  Labrell et al. / Journal of Experimental Child Psychology xxx (2015) xxx-xxx

  'theory of magnitude" and the second relating to contextual estimation. These two components are thought to rely on two different brain areas: the inferior parietal cortex and the frontal lobe, respectively.

	F. Labrell et al. / Journal of Experimental Child Psychology xxx (2015) xxx-xxx

Please cite this article in press as: Labrell, F., et al. Time knowledge acquisition in children aged 6 to 11 years and its relationship with numerical skills. Journal of Experimental Child Psychology (2015), http://dx.doi.org/10.1016/j. jecp.2015.10.005 '

Table 1

 1 Participants' school grade, gender, and age.

	School grade	Age group (years)	n	Girls [n (%)]	Age in years [mean (SD), range]
	1	6	17	8 (47)	6.41 (0.14), 6.2-6.7
	2	7	19	12 (63)	7.52 (0.13), 7.2-7.7
	3	8	26	13 (50)	8.71 (0.26), 8.1-9.0
	4	9	24	11 (46)	9.51 (0.29), 9.0-10.0
	5	10	19	13 (68)	10.58 (0.29), 10.1-11.1
	Total		105	57 (54)	8.65 (1.40), 6.2-11.1

Table 2

 2 Means (and standard deviations) of the subtests in the TKQ according to children's age.

	Age group: (n):	Max 6-7	7 (19)	8 (26)	9 (24)	10 (19)	Total	F
		(17)					(105)	(4,100)

  Within each age group, it remains questionable whether the TKQ can be considered as unidimensional.

								3 (1.0)	35.9	<.001
	Telling the time on a clock	5	2.1 (1.6) 3.7 (1.0) 3.7 (0.9) 4.3 (0.6) 4.9 (0.3) 3.8 (1.3)	21.6	<.001
	(CL)
	Life span (LS)	6	2.1 (2.0) 3.7 (2.2) 4.8 (1.5) 4.6 (1.5) 5.2 (0.8) 4.2 (1.9)	10.2	<.001
	Birthday (BIR)	6	1.6 (2.0) 3.8 (2.0) 4.7 (1.5) 5.2 (1.1) 5.5 (0.7) 4.3 (2.0)	19.7	<.001
	Interview duration (ID)	10	4.0 (3.6) 5.7 (3.6) 8.2 (1.5) 7.10	8.90	6.94 (3.1) 10.4	<.001
						(2.7)	(1.2)
	Total score (TKS)	39	15.9	26.9	32.3	32.3	36.3	29.38	50.5	<.001
			(6.0)	(7.0)	(4.1)	(3.1)	(2.1)	(8.1)

Note. Significant age effect at p < .0001 for all six subscores. Post hoc comparison of the means (Scheffé test): OR, SEQ, TU, CL, BIR, ID, TKS: 6 < 7, 8, 9, 10; LS: 6 < 8, 9, 10; TU, ID, and TKS: 7 < 8, 9, 10; BIR: 7 < 10; CL: 7, 8 < 10; ID and TKS: 8, 9 < 10. F. Labrell et al. / Journal of Experimental Child Psychology xxx (2015) xxx-xxx Please cite this article in press as: Labrell, F., et al. Time knowledge acquisition in children aged 6 to 11 years and its relationship with numerical skills. Journal of Experimental Child Psychology (2015), http://dx.doi.org/10.1016/j. jecp.2015.10.005 tests with age.

Table 3

 3 Correlations and age-adjusted partial correlations between the ''time" subtests.Note. p < .05 if r > .20; p < .01 if r > .25; p < .001 if r > .32; alpha = .86; age-adjusted alpha = .66. OR, orientation; SEQ, sequence; TU, time units; CL, telling the time on a clock; LS, life span; BIR, birthday; ID, interview duration. Relationships of the total time score with the subtests and the total score of the Zareki-R number processing and calculation battery.

		SEQ	TU	CL	LS	BIR	ID
	OR	.50 (.29)	.52 (.28)	.49 (.26)	.38 (.19)	.47 (.23)	.32 (.10)
	SEQ		.59 (.32)	.52 (.25)	.38 (.13)	.64 (.43)	.43 (.21)
	TU			.56 (.23)	.51 (.27)	.59 (.29)	.45 (.19)
	CL				.29 (.02)	.38 (.003)	.36 (.09)
	LS					.57 (.39)	.33 (.14)
	BIR						.44 (.21)

Please cite this article in press as: Labrell, F., et al. Time knowledge acquisition in children aged 6 to 11 years and its relationship with numerical skills. Journal of Experimental Child Psychology (2015), http://dx.doi.org/10.1016/j. jecp.2015.10.005

Table 5

 5 Correlations (Pearson r) of the Zareki-R subtests with TKS by age. time orientation at around 7 years of age. Indeed, knowledge of numbers on which time orientation depends is largely school dependent.In contrast, the children's precise representation of life-span intervals (LS subtest) scored lower than in adults even at 10 years of age. Accurate estimation of the three time intervals (baby-child, child-young man, and young man-old man) appeared to be a very difficult task for young children.It can be considered that, in order to succeed in the LS subtest, respondents need to attribute an age to each stage and also to perform a subtraction (e.g., age of the adult minus age of the child) in the same manner as adults do. The age for mastering subtraction in French children is approximately 8 years[START_REF] Von Aster | Time knowledge acquisition in children aged 6 to 11 years and its relationship with numerical skills[END_REF]. Correct estimations of the probable age of each human figure

		6 years (n = 17)	7 years (n = 19)	8 years (n = 26)	9 years (n = 24)	10 years (n = 19)
	Counting dots	.41	.44	-.005	.08	.30
	Counting backward	.28	.11	.05	.21	-.46 *
	Number dictation	-51 *	.49 *	.09	.16	-.22
	Mental calculation	.32	.40	.34	-.07	.24
	Reading numbers	.05	.75 ***	.20	.03	-.05
	Number line estimation	.22	.84 ***	.24	.61 **	-.06
	Oral comparison	.22	.84 ***	.53 **	.30	.03
	Perceptual estimation	.28	-.11	.21	-.11	-.08
	Contextual estimation	.08	.36	.34	.41 *	.82 ***
	Problem solving	.57 *	.34	.30	-.12	.35
	Written comparison	-.39	.26	.15	.04	-
	Total Zareki-R	.26	.69 ***	.43 *	.26	.31

Please cite this article in press as: Labrell, F., et al. Time knowledge acquisition in children aged 6 to 11 years and its relationship with numerical skills. Journal of Experimental Child Psychology (2015), http://dx.doi.org/10.1016/j. jecp.2015.10.005 children master

F.Labrell et al. / Journal of Experimental Child Psychology xxx (2015) xxx-xxx

Please cite this article in press as: Labrell, F., et al. Time knowledge acquisition in children aged 6 to 11 years and its relationship with numerical skills. Journal of Experimental Child Psychology (2015), http://dx.doi.org/10.1016/j. jecp.2015.10.005 (i.e., 7 and 8 years). Indeed, working memory, which is involved in oral comparison, is the executive function young children need to answer all of the subcomponents in the TKQ (i.e., present-time awareness, telling the time on a clock, birthday, life span, and duration of the interview). Second, number line estimation was the most strongly correlated subtest with the total TKQ score among children from 9 years. For these children, time knowledge appeared to be mostly linked to the ability to associate a quantity with a number, as for the birthday, life span, and duration of the interview subtests.

Third, contextual estimation, thought to rely on children's general knowledge of the world, was the most strongly correlated subtest with the total TKQ score in 10-year-old children, the oldest group.

Limitations and perspectives

Due to ceiling effects for some subtests in the TKQ (e.g., orientation) and to floor effects for others (e.g., life span), interpretation of the results of the correlation analyses according to children's age should be made with caution. Further studies are needed with more participants by age group in order to (a) propose new versions of the TKQ better adapted to each age group and (b) examine whether the structure of the TKQ is unidimensional or not within each age group.

Future studies should investigate the role of cognitive factors, especially verbal skills, in the relationship between time knowledge and numerical skills. The above-mentioned four components of the Zareki-R involved in time knowledge may be differentially impaired in children with developmental disorders such as children with developmental dyslexia, specific language impairment [START_REF] Gooch | Time perception, phonological skills and executive function in children with dyslexia and/or ADHD symptoms[END_REF][START_REF] Helland | Visual-sequential and visuo-spatial skills in dyslexia: Variations according to language comprehension and mathematics skills[END_REF], or specific nonverbal deficits. Studies in these special populations of children could clarify the roles of language, reading, and/or visuospatial skills in TK acquisition. Life span (LS) (four pictures: baby, child, young man, and old man): How long does it take: (1) for a baby (here) to become a child (here)? (2) for a child (here) to become a young man (here)? (3) for a young man to become an old man? Birthday (BIR): (1) ⁄ How old are you? (2) ⁄ When is your birthday? (3) How long ago was your birthday? (4) How long is it to your next birthday?

Interview duration (ID): (1) For how long have we been here together? ⁄ Questions not considered in the calculation of subscores and total score.

⁄⁄

Considered as correct if the child's answer is in the same half day (if it is 10 o'clock and the answer is ''it is 8 o'clock" or ''it is midday," the child's answer is considered as correct, but if the child's answer is ''it is 4 p.m.," it is considered as incorrect).

Appendix B. The 11 subtests of the Zareki-R Counting dots (CoDo): Presented on six cards with 13, 5, 15, 10, 8, and 18 dots, respectively. Score from 0 to 6.

Counting backward (CoBa): From 23 to 1 and from 67 to 54. Score from 0 to 4.


Florence Labrell
email: florence.labrell@gmail.com

Hugo Câmara-Costa

Virginie Kieffer

Georges Dellatolas

TIME KNOWLEDGE IN MEDULLOBLASTOMA 1 Time knowledge difficulties in children treated for medulloblastoma

Keywords: time processing, action-perception coupling, number, medulloblastoma, child Time-processing, action-perception coupling, number, medulloblastoma, child

.

The present study compares TK in 38 children treated for MCT (mean age 11.6 years) with 105 typically-developing children (TDC).

Performances on all TK subtests were significantly lower in the MCT group. The results also confirmed lower mean IQ in children treated for MCT, related to slow processing speed and low performance on working memory and non-verbal tasks. However, low IQ did not explain the considerable difficulties in the acquisition of TK.

Introduction

In typically developing children, there are numerous studies and theories exploring the development of temporal perception and temporal thinking. In children treated for medulloblastoma, a malignant tumor of the cerebellum, there are only a few studies investigating temporal skills, despite a generally admitted role of the cerebellum in time processing.

Time-related investigations in typically developing children

How do children acquire a mature conception of time? According to a recent account [START_REF] Mccormack | Three types of temporal perspective: characterizing developmental changes in temporal thought: Development of temporal thought[END_REF], very young children have a script-based, event-dependent temporal perspective, based on representations of repeated event sequences, such as bedtime routines, being able to accurately order the events in these sequences (e.g. story time comes after brushing the teeth). Around 3 years, children are able to make a categorical distinction between events that have already happened (i.e. past "unalterable" events), and events that have yet to happen (i.e. future potentially "alterable" events). A mature event-independent temporal perspective based on a linear temporal framework (from age 5-6 years) requires hypothetical thinking about the past and the future, possibly related to the capacity for "mental time travel" [START_REF] Tulving | Episodic memory: from mind to brain[END_REF]. For instance, if A, B, and C are successive past events, when B occurred, A was in the past and C in the future.

The development of time perception during childhood has been investigated for short time intervals (from sub-seconds to a few seconds) with tasks of temporal reproduction, temporal generalization or temporal discrimination, under the "internal clock" model of temporal processing [START_REF] Allman | Properties of the internal clock: firstand second-order principles of subjective time[END_REF][START_REF] Gibbon | Scalar timing in memory[END_REF]. These tasks do not use any verbal coding of time (i.e. numbers and time units), and are difficult to apply to long durations (e.g. hours). They show improvement in time sensitivity during childhood, associated with an improvement in other cognitive functions such as attention, working memory or processing speed [START_REF] Droit-Volet | Time perception in children: A neurodevelopmental approach[END_REF]. These tasks are clearly different from time-related tasks using conventional time units in their instructions, such as time production tasks (e.g. producing a duration of a number of seconds) or in their required answers (i.e. giving a verbal estimation of the duration using numbers and time units). These are based on the correct mental representation of the duration of a given time unit (e.g. how long is one second, one minute, one hour, etc) and can be considered as "cognitive estimation" tasks [START_REF] Shallice | The involvement of the frontal lobes in cognitive estimation[END_REF]. In Friedman's work with children from preschool to late adolescence, the use of conventional time units with longer durations (e.g. days, weeks, months) was partially investigated. For instance, to "how long" questions (e.g. how long is it until your next birthday?), responses in conventional units (e.g. a number of days, weeks, or months) were rare and very inaccurate in 4-year-olds, not systematic and with variable accuracy in 7-year-olds, and systematic and generally accurate in 10-year-olds [START_REF] Friedman | The Development of Children's Knowledge of the Times of Future Events[END_REF]. In another series of experiments exploring the representation of the days of the week and the months of the year, from second grade to college, Friedman proposed a 2-stage model, where the younger subjects use a verbal-list-based representational processing system and the older subjects an additional "image system" based on temporal distances [START_REF] Friedman | The Development of Children's Knowledge of Temporal Structure[END_REF].

In a previous report [START_REF] Labrell | Time knowledge acquisition in children aged 6 to 11 years and its relationship with numerical skills[END_REF], we defined Time Knowledge (TK) as the correct representation and use of familiar time units (i.e. from seconds to years). A Time Knowledge Questionnaire (TKQ) was proposed for the evaluation of TK in elementary school children (6 to 11 years). The TKQ assesses conventional time knowledge (temporal orientation, temporal sequences, relationships between time units, and telling the time on a clock), estimation of longer durations (months, years) related to life span or birthday, and estimation of the duration of the interview. All the items in the TKQ involve time units, and consequently numbers. In typically developing children, TK increased dramatically with age, especially between 6 and 8 years, and correlated strongly with the acquisition of numerical skills and working memory. In older children, a numerical cognitive estimation task (e.g. the number 10 is few for "leaves on a tree, but many for "children in a family"), was the only numerical factor associated with TK. Another tool previously proposed for the assessment of time-processing abilities in non-typically developing children [START_REF] Janeslätt | Patterns of Time Processing Ability in Children with and without Developmental Disabilities[END_REF]) also contains many "cognitive estimation" questions, such as saying which of two proposed activities takes a shorter time, or saying which activity can be completed in a given time. There is strong evidence of the role of the cerebellum in cognition [START_REF] Schmahmann | The cerebellar cognitive affective syndrome[END_REF] and also continuing debate about the description and understanding of the cerebellar functions [START_REF] Baumann | Consensus paper: the role of the cerebellum in perceptual processes[END_REF][START_REF] Koziol | Consensus paper: the cerebellum's role in movement and cognition[END_REF]. Some authors suggest that the cerebellum contributes to cognitive processing when temporal demands are critical [START_REF] De Zeeuw | Motor Learning and the Cerebellum[END_REF]Rahmati et al., 2014). For instance, De Zeeuw and[START_REF] De Zeeuw | Motor Learning and the Cerebellum[END_REF], after describing the mechanisms of motor learning in the cerebellum, specifically the adaptation of the vestibulo-ocular reflex and eye blink conditioning, concluded that the most characteristic feature among all the functions of the cerebellum is its ability to control timing at a high resolution, and suggested that this cerebellar temporal function might be also critical in complex timing-sensitive processes involved in cognition and episodic memory. According to a recent review [START_REF] Baumann | Consensus paper: the role of the cerebellum in perceptual processes[END_REF], there is general consensus that cerebellar contributions to timing are more pronounced with relatively short intervals (under 1 s), such as those involved, for instance, in eye blink conditioning, interception of a moving object, velocity perception or temporal prediction violations. Impressive evidence for the role of the cerebellum in sub-second duration processing comes from a study reporting that a 4-day training in a temporal discrimination task in the millisecond range (around 200 ms) induces structural changes in the right cerebellum in humans [START_REF] Bueti | Learning about Time: Plastic Changes and Interindividual Brain Differences[END_REF]. Neuro-imaging studies exploring time processing suggest that sub-second timing tasks are processed in the motor system and recruit sub-cortical networks, such as the basal ganglia and the cerebellum, whereas suprasecond durations activate cortical structures and utilize attention capacity and working memory to keep track of time [START_REF] Hayashi | Dissociable neuroanatomical correlates of subsecond and suprasecond time perception[END_REF][START_REF] Wiener | The image of time: a voxel-wise meta-analysis[END_REF].

Processing time and the cerebellum

However, there is also some evidence that the cerebellum is activated by temporal tasks involving suprasecond durations. For instance, in functional neuroimagery studies of "mental time travel" [START_REF] Tulving | Episodic memory: from mind to brain[END_REF], the cerebellum is activated when subjects vividly envision a personal future event [START_REF] Szpunar | Neural substrates of envisioning the future[END_REF] or repeatedly imagine taking a short walk in a familiar environment, either in the past or in the future [START_REF] Nyberg | Consciousness of subjective time in the brain[END_REF]. In patients with cerebellar lesions, many investigations report time processing deficits, as well as the difficulty in separating "pure" time processing deficits from other motor or cognitive deficits induced by the lesion of the cerebellum [START_REF] Keren-Happuch | A meta-analysis of cerebellar contributions to higher cognition from PET and fMRI studies[END_REF]. In a seminal study [START_REF] Ivry | Timing functions of the cerebellum[END_REF], 27 adults with various cerebellar pathologies (stroke, tumor, cerebellar degeneration) had difficulty maintaining a simple rhythm and discriminating between small differences in the duration of two intervals in the 160 to 640ms range, whereas patients with other neurological deficits (Parkinson disease, cerebro-cortical or peripheral pathologies) did not show deficits in timing functions. More recently [START_REF] Gooch | Interval timing disruptions in subjects with cerebellar lesions[END_REF] eleven adults with cerebellar lesions (mainly stroke) were compared to 17 controls on temporal estimation, temporal production and temporal reproduction tasks, in the suprasecond range (2-12s). Patients with cerebellar damage showed significant impairments on these tasks, more pronounced for short intervals (near 2 s) than for longer intervals (near 12 s). The authors suggest a disruption of a left cerebellar-right frontal network implicated in working memory. The role of working memory in time reproduction has been also suggested by another study [START_REF] Harrington | Does the representation of time depend on the cerebellum?: Effect of cerebellar stroke[END_REF], which compared perception and reproduction of short time intervals (under 1s) between 21 patients with cerebellar damage after stroke and 30 controls. The authors found impairments only in time reproduction and hypothesized that cerebellar damage could disrupt the maintenance of time intervals in working memory during time reproduction. Furthermore, evidence of an involvement of working memory in time reproduction of "short" time intervals (up to a few seconds) has been reported in other studies, showing, for instance, that the continuation phase of time reproduction (e.g. continuing tapping every 300ms or every 600ms) activates a covert auditory-rehearsal network in the prefrontal cortex [START_REF] Rao | Distributed neural systems underlying the timing of movements[END_REF]. Likewise, other investigations indicated selective deficits of verbal working memory in patients with cerebellar lesions [START_REF] Ravizza | Cerebellar damage produces selective deficits in verbal working memory[END_REF], and provided evidence for a cerebellar contribution to verbal workingmemory in healthy subjects [START_REF] Marvel | Functional Topography of the Cerebellum in Verbal Working Memory[END_REF]. However, similar studies performed with children are scarce. [START_REF] Bartlett | Medulloblastoma[END_REF]. Children treated for malignant cerebellar tumors (MCT), especially medulloblastoma, often present a Full Scale IQ (FSIQ) decline, at least in the first 7 years after the end of treatment [START_REF] Grill | Long-term intellectual outcome in children with posterior fossa tumors according to radiation doses and volumes[END_REF][START_REF] Palmer | Patterns of intellectual development among survivors of pediatric medulloblastoma: a longitudinal analysis[END_REF], possibly related to core cognitive deficits such as low processing speed and deficient attention and working memory (Mabbott, Penkman, Witol, Strother, & Bouffet, 2008a;[START_REF] Palmer | Processing Speed, Attention, and Working Memory After Treatment for Medulloblastoma: An International, Prospective, and Longitudinal Study[END_REF]. FSIQ decline and core cognitive deficits have been associated with radiotherapy-induced white matter loss [START_REF] Mulhern | Neurocognitive deficits in medulloblastoma survivors and white matter loss[END_REF], and also with the tumor-and surgery-induced cerebellar damage [START_REF] Grill | Critical risk factors for intellectual impairment in children with posterior fossa tumors: the role of cerebellar damage[END_REF][START_REF] Puget | Injuries to inferior vermis and dentate nuclei predict poor neurological and neuropsychological outcome in children with malignant posterior fossa tumors[END_REF].

Time-related investigations in patients with cerebellar lesions

Cognitive and time-related investigations in children treated for medulloblastoma

Proposed core cerebellum-dependent cognitive deficits, possibly explaining the FSIQ decline in children treated for cerebellar tumors, (see [START_REF] Chevignard | Core Deficits and Quality of Survival after childhood medulloblastoma: a review[END_REF], for a recent review), include deficits in sequence learning [START_REF] Leggio | Cognitive sequencing impairment in patients with focal or atrophic cerebellar damage[END_REF], verbal working memory [START_REF] Ravizza | Cerebellar damage produces selective deficits in verbal working memory[END_REF], attention [START_REF] Palmer | Processing Speed, Attention, and Working Memory After Treatment for Medulloblastoma: An International, Prospective, and Longitudinal Study[END_REF], executive function (Koustenis, Hernáiz Driever, de Sonneveille, & Rueckriegel, 2016;[START_REF] Law | Executive function in paediatric medulloblastoma: The role of cerebrocerebellar connections[END_REF], and processing speed [START_REF] Mabbott | Core neurocognitive functions in children treated for posterior fossa tumors[END_REF][START_REF] Palmer | Processing Speed, Attention, and Working Memory After Treatment for Medulloblastoma: An International, Prospective, and Longitudinal Study[END_REF][START_REF] Wolfe | Executive Dysfunction in Pediatric Posterior Fossa Tumor Survivors: A Systematic Literature Review of Neurocognitive Deficits and Interventions[END_REF]. At least some of these deficits may be related to a specific role of the cerebellum in motor and cognitive learning through the establishment of automatisms after repeated execution of the same motor or cognitive task [START_REF] Ramnani | Automatic and Controlled Processing in the Corticocerebellar System[END_REF]. A cerebellar injury-induced automation deficit could account for a lower processing speed and excessive demands on attention (Ait [START_REF] Khelifa-Gallois | Clinical Evidence of the Role of the Cerebellum in the Suppression of Overt Articulatory Movements During Reading. A Study of Reading in Children and Adolescents Treated for Cerebellar Pilocytic Astrocytoma[END_REF].

In a study with subjects aged 11-36 years treated for medulloblastoma or benign cerebellar tumors (astrocytomas), short (in the 400-ms range) and long (30-60 min) duration perception was investigated, as well as estimation of the time needed to walk 20 meters, before and after actual performance of this walk [START_REF] Hetherington | Perception and estimation of time in long-term survivors of childhood posterior fossa tumors[END_REF]. The tumor group showed deficits in short duration perception compared to the controls, whereas long-duration estimation was overall spared. However, the medulloblastoma group overestimated retrospectively the time actually required to walk 20 meters (e.g. answer 36 s, actual 17 s) and there was no correlation in this group between actual time and retrospective estimation. The authors concluded that the medulloblastoma group had This seems in accordance with another recent investigation using a virtual-reality paradigm, reporting that, compared to healthy controls, cerebellar patients showed no beneficial influence of action execution on perception [START_REF] Christensen | An intact action-perception coupling depends on the integrity of the cerebellum[END_REF].

Children treated for medulloblastoma, showed preserved temporal discrimination for durations between 0.5 and 8 seconds, but difficulty reproducing short intervals (0.5 to 1 second), associated with low processing speed [START_REF] Droit-Volet | Time perception in children: A neurodevelopmental approach[END_REF]. Another study in the same patients, exploring rhythm perception and production, showed intact rhythm discrimination but slow and variable spontaneous motor tempo and disrupted sensorimotor synchronization [START_REF] Provasi | Disrupted sensorimotor synchronization, but intact rhythm discrimination, in children treated for a cerebellar medulloblastoma[END_REF]. "Conceptions of time" were compared between 20 children treated for malignant cerebellar tumors, mainly medulloblastoma, aged 6-12 years, and 20 controls, using multiple choice questions on the duration of daily activities (e.g. brushing your teeth for 30 minutes, is it long, short, or average?), planning (e.g. if you have 5 minutes left, is it possible to eat a sandwich?), and "time units" (e.g. during the night you are likely to sleep 4 hours, 18 hours, or 10 hours?) [START_REF] Labrell | Conceptions of time in children treated for malignant cerebellar tumours[END_REF]. The tumor group showed more difficulty correctly judging the duration of daily activities. All these questions are cognitive estimate questions.

However, cognitive estimation is not about time exclusively [START_REF] Bullard | The Biber Cognitive Estimation Test*1[END_REF]. A recent version of the cognitive estimation task [START_REF] Macpherson | Bringing the Cognitive Estimation Task into the 21st Century: Normative Data on Two New Parallel Forms[END_REF], initially proposed by [START_REF] Shallice | The involvement of the frontal lobes in cognitive estimation[END_REF], favored the "speed" questions (e.g. what is the maximum speed of a Harley-Davidson motorbike?) in order to bring the cognitive estimation task into the 21 st century. In these questionnaires, cognitive estimation is generally considered to be a uni-dimensional trait, justifying the computation of a total score, where time, duration or speed questions are mixed with questions about length, distance, weight or quantity.

The present study

Overall, there is converging evidence for the role of the cerebellum in time processing of sub-second durations, as well as some evidence for cognitive estimation difficulties in patients with cerebellar lesions for longer durations, possibly related to deficient action-perception coupling. The present study We discuss our findings using two models of temporal processing as a framework, the dominant internal clock model (Coull, Cheng, & Meck, 2011a;[START_REF] Droit-Volet | Time perception in children: A neurodevelopmental approach[END_REF][START_REF] Gibbon | Scalar timing in memory[END_REF] and the more recently-proposed state-dependent network model of temporal processing [START_REF] Buonomano | State-dependent computations: spatiotemporal processing in cortical networks[END_REF][START_REF] Goel | Timing as an intrinsic property of neural networks: evidence from in vivo and in vitro experiments[END_REF][START_REF] Karmarkar | Timing in the Absence of Clocks: Encoding Time in Neural Network States[END_REF]. The first model postulates that estimating duration depends on the accumulation of temporal information in the working memory [START_REF] Coull | SMA Selectively Codes the Active Accumulation of Temporal, Not Spatial, Magnitude[END_REF], whereas the second proposes a qualitative, rather than quantitative, nonlinear and action-dependent coding of time.

Methods

Patients and controls

The participants were 38 children (24 boys, 14 girls) aged 6.1-20.4 years who had been treated for a malignant cerebellar tumor (MCT) and 105 typically-developing children (TDC) (48 boys, 57 girls).

The characteristics of children previously treated for MCT are shown in Table 1. The tumor treatment consisted of surgery, radiotherapy and chemotherapy. The inclusion criteria were having had a malignant cerebellar tumor treated in the Pediatric Oncology Department in the Gustave Roussy Institute (Villejuif, France), being older than 6 years, out of treatment and with no relapse history. The study was approved by the Institutional Review Board (IRB) as a non-interventional study and the parents gave their informed consent for their children's participation. The children came from middleand upper-middle-class backgrounds and were living in urban areas in Metropolitan France and French overseas territories.

The control children, who participated in previous research [START_REF] Labrell | Time knowledge acquisition in children aged 6 to 11 years and its relationship with numerical skills[END_REF], were in general education schools and had experienced no schooling delay. They came from middle-and upper-middle-class backgrounds and were living in an urban area in France.

Informed consent was obtained from all parents in order to conduct this research.

Insert years. Nineteen children were younger than 11.25 years (which was the age limit for available TK data in the typically-developing children) and 19 were older (11.5 to 20.4 years). There were 24 boys and 14 girls. Age at diagnosis was under 3 years for 13 children, between 3 and 8 years for 13, and 8 to 18 years for 12 children. The mean time-lapse since diagnosis was 5.6 years and the mean time-lapse since the end of treatment was 4.6 years. One patient was examined at the end of treatment one month after diagnosis; all the other patients were examined at least 5 months after the end of treatments.

Thirty-four patients were treated for a medulloblastoma and 4 for an ependymoma. All patients received chemotherapy; all except one underwent surgery; and 35 (92%) radiotherapy. Eight patients (21%) were in special school.

FSIQ scores ranged from 48 to 130 (mean 83); nine patients (24%) obtained FSIQ<70. The Verbal Comprehension Index (mean=93.6) was significantly higher than the Perceptual Reasoning Index (mean 83.7), the Processing Speed Index (mean 80.4) and the Working Memory Index (mean=81.5), (p<.001 in all three cases, paired t-test).

Material

The Time Knowledge Questionnaire (TKQ, Labrell, Mikaeloff, Perdry, & Dellatolas, 2016) was used (see Appendix A). This questionnaire comprises seven subtests. The first four subtests investigate conventional time knowledge: orientation (OR), sequences (SEQ), time units (TU), and telling the time on a clock (CL), and comprise 5, 3, 4, and 5 questions respectively. The questions about telling the time on a clock use pictures of five analogue clocks with five different positions of the minute and hour hands. Each answer is coded as correct (1 point) or incorrect (0 points), for a total conventional time score from 0 to 17. The fifth subtest, life span (LS), uses four pictures (a baby, a child, a young man, and an old man) illustrating the biological process of ageing at four different stages in development from infancy to old age. The child is asked three questions about the time required to go from one age to the next (i.e. from baby to child, from child to young man, and from young man to old man). Each answer is coded as 0, 1, or 2 points based on the answers of 20 adults: 2 points if the answer is in the adult range, 1 point if it is borderline and 0 points otherwise. The sixth subtest, birthday (BIR), comprises three questions (''When is your birthday?", ''How long ago was your birthday?", and ''How long is it to your next birthday?") and correct answers are taken into account in the calculation of the score. For these three questions, the correct answer is available (for each child, date of birth and date of interview are recorded), which makes it possible to use a coding for error.

Zero points are given if the child does not know the date of his or her birthday. Otherwise, for both of the last two questions (i.e. time intervals to the previous and next birthdays), 0 points are given for ''don't know" and errors exceeding 12 months, and 1, 2, and 3 points are given for errors (i.e. absolute difference between the answer and the correct interval) greater than 3 months, between 3 months and 1 month, and less than 1 month respectively (i.e. total BIR score from 0 to 6). The last subtest concerns interview duration (ID). At the end of the interview, the examiner asks the child ''For how long have we been here together?" The median duration of the interview is15 min (inter-quartile range = 14-21 min, range = 5-42). Zero points are given for ''don't know" answers. For all other answers, 0 to 10 points are given according to the absolute difference between the log of the real duration of the interview and the log of the answer (in minutes).

Procedure

The children previously treated for MCT were interviewed from January 2014 to May 2015 at the Gustave Roussy Institute, when they had medical appointments scheduled. The TDC were interviewed individually at their schools, in a quiet room, from February 2015 to June 2015.The participants' answers were fully transcribed.

Statistical analysis

Statistical analysis used SAS version 9 software. To compare TK acquisition between children treated for malignant cerebellar tumor (MCT group) and typically developing children (TDC group) the General Linear Model (GLM) procedure was used, with TK sub scores and TK total score as dependent variables, and age-group (4 levels) and group (2 levels: MCT and TDC) as between-subject factors. Among children older than 10 years, where the age distribution differed between the MCT group (mean age 13.6 years, SD=2.5, range=10.1-20.4) and the TDC group (mean=10.6 years, Based on age-group data among the TDC, TK z-scores were calculated for each subject in the MCT group. Only the z-score of the total TK score (TKZ) will be presented. Pearson-correlation coefficients were used for the relationship between IQ and TKZ, and the relationship between these scores and age, age at diagnosis and time-lapse since diagnosis. T-tests were used to ascertain whether subjects in special education institutions obtained lower IQ and TKZ scores.

F

Results

TK in MCT and TDC groups

The mean total TK z-score was -2.5. Twenty children (53%) obtained TKZ<-2 and 27 (71%) TKS<-1.

Positive TKZ was obtained by 6 children (16%), the highest score being 0.81.

Table 2 compares the time knowledge questionnaire (TKQ) between TDC and children treated for MCT. Performance on all TKQ subtests and TK total score (TKS) were significantly higher in the TDC group than in the MCT group when age was taken into account. Subjects older than 10 years were all aged less than 11.25 years in the TDC group, but between 10 and 20 years in the MCT group.

However, despite this age difference between the groups of children older than 10 years, TKS was significantly lower in the MCT group (p<.001), together with most of the TKQ subtests, in particular telling the time on a clock, long durations (life span and birthday) and temporal sequences.

Insert Table 2 about here

TK and neuropsychological factors

Table 3 shows the factors associated with FSIQ, IQ components and TKZ score, as well as the relationship between TKZ and IQ. Older age and long time-lapse since diagnosis (or long time-lapse since the end of treatments) had a negative impact on FSIQ scores, the processing speed index (PSI), the perceptual reasoning index (PRI), and (for older age) the working memory index (WMI), but not on the verbal comprehension index (VCI) or TKZ. Enrollment in a special educational institution was (r=0.30,p=0.07) and WMI (r=0.33,p=0.05).

Insert Table 3 about here Among subjects with FSIQ scores under 70 (n=9), between 70 and 90 (n=16) and higher than 90 (n=13), mean TKZ was -4.7 (4.1), -1.6 (2.0), and -2.1 (2.0) respectively. These three means differed significantly (F (2,35) = 4.3, p=0.02) and all differed significantly from 0 (p<0.01 in all three cases).

Figure 1 shows the distribution of TKZ according to FSIQ group. Six among the 13 subjects with FSIQ in the normal range (>90) obtained a total TKZ score under 2.

Discussion

The present results confirm the lower mean IQ in children treated for malignant cerebellar tumors (MCT), mostly medulloblastoma, related to slow processing speed and low performance on working memory and non-verbal tasks. In our findings the most original results indicate that these children and young adults show considerable difficulties in the acquisition of time knowledge (TK), which cannot be completely explained by low IQ. Among subjects with IQ in the normal range (>90), the mean TK z-score (TKZ) was -2. Correlations were modest between TKZ and FSIQ or Working Memory Index, and low between TKZ and the other IQ components.

The definition of TK adopted, i.e. correct use and representation of time units, logically links TK to numerical skills: the correct use of time units implies the use of numbers. Numerical skills and semantic memory, such as knowledge of the relationships between time units and understanding of the principles of division, contribute to the representation of a given time unit (e.g. knowing that there are 24 hours in a day, 60 minutes in an hour, understanding what one quarter means, etc.). According to a previous study in typically developing children [START_REF] Labrell | Time knowledge acquisition in children aged 6 to 11 years and its relationship with numerical skills[END_REF], investigating the relationship between TK and numerical skills, TK depends on four numerical factors: academic knowledge of numbers and number facts, number line estimation (e.g. correspondence between a number and a distance), verbal working memory, and contextual estimation (e.g. the number 10 is few for "leaves on a tree", but many for "children in a family"). Most of the above four Academic knowledge of numbers and number facts could partly depend on sequence learning and automation, and could contribute to explaining difficulties in learning temporal sequences or telling the time on a clock. Sequence learning deficit may be difficult to distinguish from a working memory deficit. In patients with lesions of the cerebellum, some studies have pointed to sequencing difficulties [START_REF] Leggio | Cognitive sequencing impairment in patients with focal or atrophic cerebellar damage[END_REF], and others to verbal working memory difficulties [START_REF] Ravizza | Cerebellar damage produces selective deficits in verbal working memory[END_REF]. Verbal working memory strategies, such as subvocal rehearsal, are likely to be adopted in time reproduction tasks for relatively short intervals (up to a few minutes). The "theory of magnitude" [START_REF] Walsh | A theory of magnitude: common cortical metrics of time, space and quantity[END_REF], postulating that the same cognitive system, thought to be located in the inferior parietal cortex, is devoted to common processing mechanisms for the evaluation of time, number and space, and could account for the association of number line estimation with time. In the same vein, [START_REF] Friedman | The Development of Children's Knowledge of the Times of Future Events[END_REF] adopted a spatial coding of time (i.e. distances) to investigate the development of children's knowledge of future events. However, to our knowledge, there is no clear evidence for general "magnitude" deficits in patient with cerebellar pathologies. Life span and duration of the interview are clearly cognitive estimations according to the definition proposed by [START_REF] Shallice | The involvement of the frontal lobes in cognitive estimation[END_REF]: answers to cognitive estimation questions are based on "general knowledge available to almost all subjects". This contrasts with the reproduction of a given duration after auditory or visual presentation, where a working memory strategy can be applied (e.g. rehearsal). Cognitive estimation may be deficient in patients with lesions of the cerebellum. Executive dysfunction has been described in patients with cerebellar pathologies as a consequence of disconnections between the cerebellum and the frontal cortex [START_REF] Law | Executive function in paediatric medulloblastoma: The role of cerebrocerebellar connections[END_REF]. Executive functions are classically attributed to the frontal lobes, as is cognitive estimation. Patients with frontal lesions and planning dysfunctions gave bizarre answers to cognitive estimate questions [START_REF] Shallice | The involvement of the frontal lobes in cognitive estimation[END_REF]. Furthermore, impaired action-perception coupling has been described in patients with lesions of the cerebellum [START_REF] Christensen | An intact action-perception coupling depends on the integrity of the cerebellum[END_REF]. If we consider that the representation of a given time unit (e.g. one hour) is based on sets of actions that it is possible to perform within this time unit, intact action-perception coupling is required to questions about time requiring cognitive estimation. In the study by [START_REF] Hetherington | Perception and estimation of time in long-term survivors of childhood posterior fossa tumors[END_REF], time estimation was not improved by action in patients treated for medulloblastoma.

Our results can be interpreted by way of two different models of temporal processing that have been proposed (among others): (i) the very successful internal clock model [START_REF] Allman | Properties of the internal clock: firstand second-order principles of subjective time[END_REF]Coull, Cheng, & Meck, 2011b;[START_REF] Droit-Volet | Time perception in children: A neurodevelopmental approach[END_REF][START_REF] Gibbon | Scalar timing in memory[END_REF], in which time is quantitative (number-and accumulation-based) and linear; (ii) the neural network state model [START_REF] Buonomano | State-dependent computations: spatiotemporal processing in cortical networks[END_REF][START_REF] Goel | Timing as an intrinsic property of neural networks: evidence from in vivo and in vitro experiments[END_REF][START_REF] Karmarkar | Timing in the Absence of Clocks: Encoding Time in Neural Network States[END_REF], in which time is qualitative and not linear. The strong correlation between TK and numerical skills, described above, can be considered as an argument in favor of the first model. Nonetheless, the possible relationship of our results with the neural network state model merits some comments.

"Timing in the absence of clocks" has been proposed as a model of time processing by neural networks in the millisecond range [START_REF] Karmarkar | Timing in the Absence of Clocks: Encoding Time in Neural Network States[END_REF]. Using computer simulations, the authors conclude that cortical networks are inherently able to tell time as a result of time-dependent changes in network state. In this model, there is no linear metric of time, and a given interval is encoded in the context of preceding events and "temporal objects". According to these authors, the model applies in the millisecond range, but timing in the second range is likely to be more consistent with a linear metric of time. It is important to note, however, that "temporal objects" can be generated not only by neural networks in the millisecond range, but also by behavior in the second, minute, or even longer-range. Traditionally, in humans, during temporal reproduction, temporal discrimination or temporal generalization task, using intervals in the supra-second range, subjects have to be attentive, but they are also often asked to avoid counting [START_REF] Rattat | What is the best and easiest method of preventing counting in different temporal tasks?[END_REF], which could interfere with the internal clock model. Indeed, counting is not the only possibility to code a duration of a certain number of seconds, or longer. Coupling the duration of the stimulus with an action could be an alternative strategy. For instance, it is possible to start rehearsing a favorite poem, song or piano sonata at a constant speed at the onset of the stimulus, and code the duration by the stopping-point (the "state") in the poem, the song or the piano sonata. Action-based coding of time of this sort is qualitative, non-numerical and not based on the accumulation of units. Rather it is script-based and The above-mentioned dichotomy concerning the concept of time has its philosophical correspondence.

In accordance with Aristotle's claim (384-322 B.C., Physics IV, 1941) that time is a number, the dominant theoretical model for time processing is quantitative: regular pulses are produced by an "internal clock" and perception of duration is based on their number. On the other hand, a nonnumerical concept of duration was proposed by Henri Bergson (1859Bergson ( -1941)), who thought that numbers are related to space, and not to time. The duration of a piece of music, a "temporal object", could illustrate Bergson's idea. Knowledge or "consciousness" of the number of notes composing, for instance, a piano sonata is not necessary for its correct execution. This is related to the fact that notes in music are not all identical, unlike the pulses of a clock or the time units [START_REF] Bergson | Essais sur les Données Immédiates de la Conscience [Time and Free Will: An Essay on the Immediate Data of Consciousness[END_REF].

In summary, the correct representation and use of familiar time units depends both on numerical skills and on "intact action-perception coupling". Cerebellar pathologies may lead to numerical deficits related to deficient sequence learning, working memory, cognitive estimation or slow processing speed. If action-perception coupling depends on the integrity of the cerebellum, as reported recently [START_REF] Christensen | An intact action-perception coupling depends on the integrity of the cerebellum[END_REF][START_REF] Hetherington | Perception and estimation of time in long-term survivors of childhood posterior fossa tumors[END_REF], it could add to the explanation of specific timing difficulties in patients with cerebellar lesions, whatever the duration involved. Processing of subsecond timing tasks in the motor system itself suggests action-perception coupling. During processing of timing tasks in the second range, there is evidence that the cerebellum is involved in working memory, and rehearsal can be used to code time intervals by action-perception coupling. For longer durations, there is some evidence that action is less helpful for time perception among patients with lesions of the cerebellum than among healthy subject, although this needs to be confirmed by future studies.

F

  most common malignant childhood central nervous system tumor, arising infratentorially in the cerebellum or the fourth ventricle. Treatment generally includes surgery, cranial radiotherapy and chemotherapy

  -world spatiotemporal estimates with sensory and somatomotor information.

  children treated for medulloblastoma with normative data, and tries to explain the TK difficulties in relation to the proposed roles of the cerebellum in time processing and action-perception coupling.

  related to proposed cerebellar cognitive dysfunctions in patients with cerebellar lesions.

  , which is much closer to the temporal perspective of young children proposed by[START_REF] Mccormack | Three types of temporal perspective: characterizing developmental changes in temporal thought: Development of temporal thought[END_REF].

Figure 1 :

 1 Figure 1: Distribution of TKZ according to FSIQ group.

Table 1

 1 Table 1 describes the group of 38 children treated for malignant cerebellar tumors. Mean age was 11.6

	about here

  TK scores were further compared between groups within this age-group using t-tests and Wilcoxon non-parametric tests. The results of this comparison were retained because the scores were significantly lower in the MCT group, despite the significantly older age of this group.

	TIME KNOWLEDGE IN MEDULLOBLASTOMA
	SD=0.3, range=10.1-11.25),
	o
	r
	P
	e e r
	R e v i e w
	O n
	l y
	12

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  IQ scores (except for PRI) and lower TKZ. No significant relationship was observed with young age at diagnosis in this sample. TKZ was only marginally associated with FSIQ

	Child Neuropsychology
	TIME KNOWLEDGE IN MEDULLOBLASTOMA
	associated with lower
	F o
	r
	P
	e e r
	R e v i e w
	O n
	l y
	13

Table captions Table 1 .

 captions1 Characteristics of children with MCT

Table 2 .

 2 Sub scores on time knowledge questionnaire (TKQ) in typically developing children (TDC) and in children treated for malignant cerebellar tumor (MCT)

Table 3 .

 3 Factors associated with FSIQ, IQ components and TKZ score, and correlations between TKZ and IQ components

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	Child Neuropsychology
	F o
	r
	P
	e e r
	R e v i e w
	O n
	l y

Table 1 Characteristics of children with MCT N 38

 1

	Page 23 of 27	Child Neuropsychology
	Age: m (SD) [range] (years)	11.6 (3.7) [6.1 -20.4]
	Age at diagnosis	6.0 (4.4) [0.1 -18.0]
	Time-lapse since diagnosis	5.6 (4.4) [0.1 -14.9]
	Sex (boys): N (%)	24 (63)
	Tumor
	Medulloblastoma	34 (90)
	Ependymoma	4 (10)
	Treatment
	Surgery	37 (97)
	Radiotherapy	35 (92)
	Chemotherapy	38 (100)
	School
	Special institution	8 (21)
	Wechsler scales: m (SD) [range]
	Full scale IQ	83.0 (17.9) [48 -130]
	VCI	93.6 (15.8) [63 -122]
	PRI	83.7 (15.5) [54 -126]
	PSI	80.4 (18.0) [50 -118]
	WMI	81.5 (14.5) [50 -112]

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Table 2 Sub scores on time knowledge questionnaire (TKQ) in typically developing children (TDC) and in children treated for malignant cerebellar tumor (MCT)

 2 : Age group effect, p<.001 for TKS and all subtests. Group effect (TDC vs. MCT), p<.01 for OR and ID, p<.001 for TKS and all other subtests. Group by age group interaction NS (p>.05) *age group >10: TDC group: all 19 children were aged between 10.1 and 11.25. MCT group: 7 of the 26 children (27%) were aged between 10.1 and 11.25 years and 19 (73%) between 11.6 and 20.4 years. However, all means were higher among the TDC than among children with MCT. A comparison (t-test or non-parametric Wilcoxon test gave similar results) showed significant differences: p<.001 for TKS and CL, p<.01 for SEQ, LS, BIR, p<.05 for OR (Wilcoxon), but no significant difference for TU and ID.

	Typically developing children (TDC)	Malignant Cerebellar Tumors (MCT)

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Table 3 Factors associated with FSIQ, IQ components and TKZ score, and correlations between TKZ and IQ components

 3 Full Scale Intellectual Quotient; VCI: Verbal Comprehension Index; PRI: Perceptual Reasoning Index; PSI: Processing Speed Index; WMI: Working Memory Index; TKZ: Time Knowledge Z score. Figure 1. Distribution of TKZ according to FSIQ group.

	Age Age at diagnosis Time-lapse since diagnosis TKZ FSIQ: Page 25 of 27	FSIQ -0.48** -0.07 -0.36* 0.30 Child Neuropsychology VCI PRI -0.13 -0.34* 0.04 0.07 -0.13 -0.33* 0.26 0.22	PSI -0.46** 0.13 -0.47** 0.07	WMI -0.34* -0.14 -0.13 0.33*	TKZ 0.03 0.20 -0.17 -

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Funding: This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Conflict of interest: none

Life span (LS) (four pictures: baby, child, young man, and old man): How long does it take: (1) for a baby (here) to become a child (here)? (2) for a child (here) to become a young man (here)? (3) for a young man to become an old man?

Birthday (BIR):

(1) * How old are you? (2) * When is your birthday? (3) How long ago was your birthday? (4) How long is it to your next birthday?

Interview duration (ID):

(1) For how long have we been here together? * Questions not considered in the calculation of sub scores and total score. **Considered as correct if the child's answer is in the same half day (if it is 10 o'clock and the answer is ''it is 8 o'clock" or ''it is midday," the child's answer is considered as correct, but if the child's answer is ''it is 4 o'clock in the afternoon", it is considered as incorrect).