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Follow-up questions influence the measured number knowledge in the Give-a-number task

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The Give-a-number task is one of the most frequently used tests to measure the number knowledge of preschoolers at the time they acquire the meaning of symbolic numbers. In the task, an experimenter asks for a specific number of objects from a child. The literature utilizes several versions of this task, and usually it is assumed that the different versions are equivalent and that they do not have an effect on the measured number knowledge. In the present study, the specific potential effect of the follow-up questions posed after a trial on the measured number knowledge is investigated. Three versions of follow-up questions are compared. The results demonstrate that different versions affect the measured number knowledge of children. These results highlight that follow-up questions should be considered in studies using the Give-a-number task, and more generally, various versions of the Give-a-number task may have an essential effect on the measured number knowledge, thereby partly accounting for conflicting findings in the literature.
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Cognitive Development 57 (2021) 100968
Available online 29 January 2021
0885-2014/© 2021 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/).
Follow-up questions inuence the measured number knowledge in
the Give-a-number task
Attila Krajcsi
ELTE, E¨
otv¨
os Lor´
and University, Institute of Psychology, Department of Cognitive Psychology, 1064, Budapest, Izabella utca 46, Hungary
ARTICLE INFO
Keywords:
Give-a-number task
Cardinality principle
Subset-knowers
Preschooler number knowledge
ABSTRACT
The Give-a-number task is one of the most frequently used tests to measure the number knowl-
edge of preschoolers at the time they acquire the meaning of symbolic numbers. In the task, an
experimenter asks for a specic number of objects from a child. The literature utilizes several
versions of this task, and usually it is assumed that the different versions are equivalent and that
they do not have an effect on the measured number knowledge. In the present study, the specic
potential effect of the follow-up questions posed after a trial on the measured number knowledge
is investigated. Three versions of follow-up questions are compared. The results demonstrate that
different versions affect the measured number knowledge of children. These results highlight that
follow-up questions should be considered in studies using the Give-a-number task, and more
generally, various versions of the Give-a-number task may have an essential effect on the
measured number knowledge, thereby partly accounting for conicting ndings in the literature.
1. Introduction
Understanding numbers and math is one of the pillars of our culture. Research on how symbolic numbers are initially understood
and what representations are utilized is key to understanding how humans understand symbolic numbers. Human-specic, symbolic
number understanding is acquired at approximately the age of 4 years. Currently, the most important tool with which the phases of this
initial symbolic number learning is investigated among preschoolers is the Give-a-number (GaN) task (Wynn, 1990, 1992). In the GaN
task, children are asked to provide a specic number of objects. This number starts at 1 and upon a successful response, the next larger
number is asked, while in the case of an incorrect response, the preceding smaller number is asked again. The experimenter may go
back and forth between the largest known number and the smallest unknown number three times to establish the limit of the childs
number knowledge.
Using the GaN task, Karen Wynn (1990, 1992) described the sequential phases of symbolic number acquisition. Initially, while
children know the series of number words (i.e., they can recite the one-two-three etc. list), they are unable to give any number of
objects reliably, which children are termed pre-knowers (or non-knowers). At approximately the age of 3 years, children can give 1
object, but not more, termed one-knowers. A few months after this, they are also able to give 2 objects, termed two-knowers, then after
another few months, they become three-knowers, and subsequently, four-knowers. One-, two-, three-, and occasionally four-knowers
are termed subset-knowers, because they can utilize only a subset of their counting list. In other words, even if children know the
counting word series up to an appropriately high value (e.g., up to 10), in the GaN task, they are unable to give all of these values
correctly. Typically at approximately the age of 4 years, children start to give any amount that is in their counting list, termed
E-mail address: krajcsi@gmail.com.
Contents lists available at ScienceDirect
Cognitive Development
journal homepage: www.elsevier.com/locate/cogdev
https://doi.org/10.1016/j.cogdev.2020.100968
Received 6 April 2020; Received in revised form 14 October 2020; Accepted 15 October 2020
Cognitive Development 57 (2021) 100968
2
cardinality-principle-knowers (CP-knowers). It is assumed that at this point their symbolic number knowledge becomes general,
because they understand the cardinality principle, i.e., the principle that after counting a set of objects, the last number word denotes
the amount of objects in that set.
The description of this development is essential because this is the data that models attempt to account for (see a selection of the
most prominent theories of symbolic number acquisition in Carey, 2004, 2009; Carey & Barner, 2019; Piazza, 2010; vanMarle et al.,
2018). Furthermore, in many studies, other developmental numerical cognition phenomena are contrasted against the development
described using the GaN task (for example, see Davidson, Eng, & Barner, 2012; Le Corre, 2014; Le Corre & Carey, 2007; Sella &
Lucangeli, 2020), thus, the GaN task is considered the gold standard for measuring changes in initial symbolic number understanding.
While the GaN task and the concept of cardinality-principle knowledge are key components in the cognitive description of initial
symbolic number acquisition in preschoolers, there are both conceptual and methodological issues related to these descriptions. On a
conceptual level, while the concept of cardinality-principle knowledge supposes a meaningful understanding of the counting pro-
cedure, together with general knowledge about numbers, several phenomena seemingly contradict this idea. For example, not all CP-
knowers are able to estimate the cardinality of a set that includes more than 4 items (Le Corre & Carey, 2007); not all CP-knowers are
able to compare symbolic numbers (Davidson et al., 2012; Le Corre, 2014); not all CP-knowers know the cardinality of a set when an
item is added to a set with known cardinality (Davidson et al., 2012); or not all CP-knowers know the cardinality of a set when an item
is removed from a set with known cardinality (Sella & Lucangeli, 2020). Relatedly, it is also possible that the GaN task can be solved
mechanically without a rich understanding of numbers (Davidson et al., 2012). On a methodological level, several versions of the GaN
task are used in the literature without ensuring that the different versions are equivalent in the sense that they give the same mea-
surement result. For example, while 4-knowers are usually not considered CP-knowers, in other cases they are categorized as
CP-knowers, e.g., in Wynn (1992), or as an opposite example, even 5-knowers are not considered as CP-knowers (Le Corre, 2014). As
another example, when specifying the threshold of knowing a specic number, most research uses the criterion that the child must give
2 correct answers out of 3 responses (67 %), however, others use the criterion of 2-out-of-2 (100 %) or 1-out-of-2 responses (50 %) (J.
B. Wagner & Johnson, 2011), or may use an entirely different approach, e.g., employing Bayesian modeling (Negen, Sarnecka, & Lee,
2011). (See additional examples about the follow-up question below.) The methodological and conceptual issues in this regard may be
related. In its simplest form, if the various versions of the tasks are not equivalent, then some versions must be imprecise or even
invalid. Obviously, using partially invalid methods may lead to seemingly contradicting evidence (see similar examples of how
methodological variations can profoundly inuence conclusions in Gunderson, Spaepen, & Levine, 2015; K. Wagner, Chu, & Barner,
2019).
Because many different versions of the GaN task are considered equivalent without empirical conrmation of these assumptions, it
is not known whether these different versions truly offer the same measurement results. The aim of the present study is to empirically
verify the supposed equivalence of one of these variations, i.e., the use of various follow-up questions in the GaN task. The broader aim
of this study is to partly understand how variations in the GaN task can produce different observed number knowledge and how these
differences can contribute to seemingly contradictory phenomena in the development of symbolic number knowledge.
1.1. Follow-up questions in the Give-a-number task trials
When administering the GaN task, it is critical to differentiate between competence- and performance-based errors. Children may
give incorrect responses either because they cannot identify and use the numerical meaning of a number word, or because, although
they are able to identify and use the value, they are unable to execute the task correctly, for example, they may accidentally skip a
number word or miss an object, even if, in most instances they solve the task correctly. To avoid these performance errors, Wynn (1990,
1992) asked children to check the given set at the end of the trials; if they had not counted the set, they were asked explicitly to recount
it. If the result of the checking or the result of the recount was different from the required number, the experimenter reminded the
children of the required number and asked the children to x the error.
In the literature, several variations of these follow-up questions are utilized. In some versions, the children are always asked to
recount (e.g., Le Corre & Carey, 2007) or to recount the set only if they have not counted the given set (e.g., Le Corre, Van de Walle,
Brannon, & Carey, 2006), in other versions, children are only asked to check their responses but do not have to recount their given set
(e.g., Sarnecka & Lee, 2009), and there are variations in which follow-up questions are not utilized at all (e.g., Mussolin, Nys, Leybaert,
& Content, 2012; Posid & Cordes, 2018). Other combinations of subtle details regarding the follow-up question are also observable in
the literature, for example, xing a response is asked only if the result of the childs counting differed from the required amount (Le
Corre et al., 2006).
While the GaN task is used in various forms in the literature, the differences are not justied or explained explicitly in those works.
Importantly, it is possible that various types of follow-up questions might have different effects. Since checking and recounting in-
structions are used to avoid performance errors (as supposed in Wynn, 1990, 1992), if a lack of these instructions or different variations
of instructions truly introduce more performance errors, then different versions of the task should reveal different performance. This
also means that various versions of the task may categorize the same child into different number-knowledge categories, thus, intro-
ducing a systematic bias and questioning the validity of some of the GaN task versions.
One source of information on whether the follow-up questions at the end of the GaN task trial inuence performance is how
frequently children correct their original response after the question. Unfortunately, the data available in the literature are not
conclusive on whether follow-up questions inuence the identied number knowledge level. According to Sarnecka and Lee (2009), at
the end of the trials, it was asked whether the given set is N, and the correction was extremely low (0.2 %). Le Corre et al. (2006) asked
children to recount the set if counting was not employed in their original response. They reported the proportion of correct xes and
A. Krajcsi
Cognitive Development 57 (2021) 100968
3
the proportion of xes in the correct direction (i.e., in contrast to a lack of corrections or corrections that made the response even more
erroneous) and found an important difference between subset-knowers and CP-knowers: While subset-knowers usually left the
incorrect response unxed or made the response even worse (approximately only 20 % of the trials in which children were asked to x
were corrected), CP-knowers were more successful (approximately 80 % of the to-be-xed trials were corrected). Note that the pro-
portion values of the two studies cannot be compared directly because, in the Sarnecka and Lee (2009) study, the base number for the
reported proportion is the number of all trials, while in the Le Corre et al. (2006) study, it is only the number of trials where a x was
asked for (i.e., the number of trials when the result of the childs counting differed from the required amount); additionally, the latter
study considered a response correct even if the difference of the correct and given response was 1 (e.g., 4 was accepted as correct
response for 3). In summary, it is not straightforward whether children frequently correct their responses, and whether the follow-up
questions have an important effect on measured performance.
Note that another potential mechanism may inuence the effect of follow-up questions at the end of GaN task trials. Although
follow-up questions were intended to avoid performance errors, an explicit request to recount the set and the comparison of the given
set with the required amount may have a training effect. First, in a more subtle form, even a simple question as to whether the given
response is correct may hint to children that they may have made a mistake and may encourage them to look for other solutions or
strategies. Conversely, it is also possible that after a recount or other follow-up instruction, children may suppose that their response
had been incorrect, even if the response was correct, consequently, because of the correction, they may modify an otherwise correct
response to be incorrect. In these latter cases, the follow-up question may cause worse performance than the task versions without
follow-up questions. Second, in line with the training hypothesis, in a training study, Kelly Mix and her colleagues demonstrated that
counting and labeling a set at the same time (e.g., Look, this page has three crackers. Can you say it with me? Three crackers. Now lets
count them 1, 2, 3!) results in larger number knowledge measured with the GaN task in three-and-half-year old children (Mix,
Sandhofer, Moore, & Russell, 2012). In the same study, it was found that using counting alone or using labeling alone did not improve
number knowledge when compared with a control group. Therefore, the follow-up questions may induce training and children may
show better number knowledge with those questions compared with a more passive version of the GaN task. In another study, a similar
counting and labeling instruction induced a training effect even after a 5-minute-long practice (Posid & Cordes, 2018). (Note that there
may be important differences between the recounting version of the GaN task and these training studies. For example, the training task
has a different context compared with the GaN measurement task. Therefore, while the recounting version of the GaN task may have a
training effect, this effect cannot be taken for granted.) Similar feedback-based performance change in sequence learning has been
described in various conditions (Lange-Küttner, Averbeck, Hirsch, Wießner, & Lamba, 2012). To the current authors knowledge, the
literature has to date not considered the potential effect of this type of feedback at the end of the GaN task trials, and the consequences
of this potential effect remain unknown. It is not clear how much the follow-up questions could help children to avoid performance
errors or how much they might inuence responses, either by teaching more effective strategies or conceptual knowledge, or by
worsening initially correct responses.
To summarize, while various follow-up questions are utilized at the end of the GaN task trials, it is not known whether these
versions measure different level of number knowledge among children, potentially leading to validity issues in some versions of the
GaN task. Relatedly, while these follow-up questions were originally applied to avoid performance errors, it is not known whether
these questions have a training effect.
1.2. Aim of the study
The aim of the present study is to compare the effect of various follow-up questions on the performance of children in the GaN task.
More specically, three GaN task versions are contrasted here: the task without any follow-up questions; the task with a question about
whether the response is correct; and the task with the request to recount the given set. It is investigated whether the three conditions
inuence the measured number knowledge of children and the number of corrections after their original response in the task. Notice
that it is assumed throughout the entire study that only the observed number knowledge is available in the present data, but not the
realability of children. Note that these task versions do not cover all the variations that can be found in the literature; nonetheless,
they cover some of the main versions, where it is reasonable to suppose that differences may arise. Additionally, no combinations of
these follow-up questions (such as in Wynn, 1990, 1992) are tested here because the primary interest is whether the follow-up
questions have an effect in their purest forms. Relatedly, the present work does not use the classic task version as described in
Wynn (1990, 1992) as a starting point. Rather, similar to many versions in the literature, the present work uses a version that is
modied in several ways (see specic details and references in the Task and procedure section). Still, the present version serves as an
appropriate test case for whether equality of various GaN task versions could be granted. Finally, note that the main aim of the present
study was not to investigate whether the follow-up questions help to avoid performance error or whether they induce training or both;
rather, the aim was to reveal whether the follow-up question versions have an effect on the childrens performance in any way.
If there are differences between the effects of the follow-up question versions, then it is essential not only to describe the exact
method a study follows, but also to consider these differences in reviews, meta-analyses and validation works. Also, the difference
between the effect of follow-up questions is important because it only makes sense to investigate whether there are performance errors
or training effects if differences between the follow-up questions truly exist (unless we suppose that an equally strong performance
error and training effects annul the observable effect). On the other hand, if no differences between the present versions of the follow-
up questions (including the no-question version) could be observed, then follow-up questions may be omitted from the GaN task,
leading to a much shorter version of the task which could be easier to administer and could be less demanding for the children.
A. Krajcsi
Cognitive Development 57 (2021) 100968
4
2. Methods
Three- and four-year-old preschoolers solved different follow-up question versions of the GaN task. The study was approved by the
ethics committee of the Faculty of Education and Psychology, E¨
otv¨
os Lor´
and University, Hungary.
2.1. Participants
The study initially included 210 preschool participants. Among them, children who could not complete the tasks (N=9) or whose
counting list was not longer than their measured number knowledge (N=4) were excluded from the analysis. The nal sample
comprised 197 preschoolers, 99 girls and 98 boys, with a mean age of 4;1 years, SD of 0.5;, ranging between 3;1 and 5; years. Seventeen
Hungarian preschools were involved in the measurement, 8 of them in the capital, and 9 of them in a country town, all of them mostly
receiving children of middle-class families.
2.2. Tasks and procedure
2.2.1. Give-a-number task
The children were presented with a pile of sponge balls (approximately 30 balls, with a diameter of approximately 2 cm) and the
experimenter asked them to provide a specic number of balls. In this task, I want to see if you could pick these balls appropriately.
Could you give me x balls? Just place them in front of me.The numbers were asked in the following pseudorandom order: 1, 3, 2, 6, 4,
5, 7, 9, and 8. This order was used to overcome the predictive nature of an increasing series. The entire series of these values was
repeated four times, resulting in 36 trials. The numbers provided by children were recorded. (Further details of this task version are
given below.)
Three versions of the GaN task with three follow-up question conditions at the end of the trials were used. (1) In the No follow-up
version, no follow-up question was asked after a trial, and the children were not encouraged to recount the set when an incorrect
answer was given. (2) In the Is it Nversion, after the rst response, the experimenter asked the child, Is this N balls?, using the
required number in the question. If the child answered no, then the child was asked again to give N balls. (3) In the Recounttask
version at the end of the trials, the child was asked to recount the given set. If the result was different from the required amount, the
required amount was asked for again.
Participants were randomly assigned to the three follow-up question groups. The nal sample included 65, 67 and 65 children in
the No follow-up, Is it Nand Recountgroups, respectively. The age of the three groups did not differ (age as an interval variable
was compared across the three follow-up question groups; all three group means were 4;1 years,
ω
2
=-0.008, KruskalWallis test:
χ
2
(2,
N=197) =0.92, p=0.631).
The GaN task is employed in various forms in the literature. Since it has not been formerly investigated whether different versions
of the task measure number knowledge differently or whether various details interact (i.e., the presence or strength of a version-based
effect may inuence another version-based effect), it is not possible to tell whether specic details of the current task inuence the
result. The minimal supposition of the present work is that if, in any version used in the literature, an effect of different versions can be
observed, then it should be taken as a cautionary sign that the various versions must be considered and further investigated. In the
present paragraph, the specic properties of the task are summarized, for which properties different versions can be observed in the
literature. (1) In the present version, 30 balls were used as an initial set. Various set sizes are used in the literature (e.g., 15 objects in
Gunderson et al., 2015; 10 objects in K. Wagner et al., 2019) and this set size may inuence the response when children give incorrectly
all the available objects (Figs. 1 and 2 in Gunderson et al., 2015; Figs. 1 and 6 in. K. Wagner et al., 2019). However, the latter in-
formation is not used in the present study. (2) The classic GaN task uses the titration method: After a successful trial, the next larger
number is asked, and after an unsuccessful trial, the preceding smaller number is asked, until the rst number that is given incorrectly 2
out of 3 times is found (Wynn, 1990, 1992). Instead of this titration procedure, in the present work, all numbers between 1 and 9 are
asked in the order described above (see similar methods e.g., in Marchand & Barner, 2020; Mussolin et al., 2012; Sarnecka & Gelman,
2004; Sella & Lucangeli, 2020; Wagner & Johnson, 2011). The non-titration version was chosen to investigate additional properties of
the GaN task, which results are reported elsewhere. Importantly, the number knowledge calculation method used here mimics the
titration method (see the Analysis section below for more details). (3) While, in the titration method, numbers are initially asked in
increasing order, and subsequently, the order depends on the responses of the children, the present version uses pseudorandom order,
where randomization makes it impossible to rely on the order of the numbers in the task (see similar methods in Marchand & Barner,
2020). (4) Because it is assumed that once they understand 5, preschoolers are able to give any numbers from their counting list,
usually numbers larger than 5 are not tested, although several studies ask for numbers beyond 5 (e.g., see Almoammer et al., 2013;
Barner, Libenson, Cheung, & Takasaki, 2009; Cheung, Slusser, & Shusterman, 2016; Marchand & Barner, 2020; Mussolin et al., 2012;
Wagner & Johnson, 2011). The present dataset includes numbers larger than 5 to investigate other specic properties of the GaN task,
which results are reported elsewhere. Nonetheless, in the present study, in line with this widespread assumption, children who know at
least number 5 are categorized as CP-knowers (see more details in the Analysis section below) and data on larger numbers are not
analyzed here. (5) In the original GaN task version, all investigated numbers were asked at least once or twice, and the numbers at the
edge of participantsnumber knowledge were asked 3 times (Wynn, 1990, 1992). In the present version, all numbers were asked 4
times, according to which a more precise measurement is expected.
A. Krajcsi
Cognitive Development 57 (2021) 100968
5
2.2.2. Counting list knowledge
Counting list knowledge was measured to ensure that the number knowledge measured using the GaN task is not constrained by the
childrens counting list: Only children who had larger counting list than their number knowledge (for CP-knowers the counting list
should go beyond 6) were included in the analysis. In the counting list task, the child was asked to continue the verbal list: Could you
continue this series? I will start it, and you continue! One, two, three…” The task was given twice and the highest numbers in the
correctly recited series were recorded.
The two tasks were measured in a single session. First, the GaN task and then the Counting list knowledge task was given. A session
lasted approximately 1520 min. Data were collected by nine experimenters, all of whom were blind to the aim of the study.
2.3. Analysis
2.3.1. Calculating number knowledge in the Give-a-number task
The average performance (proportion of the correctly given trials) was calculated for each child and each number. For a child, a
number was considered known according to two criteria. First, the proportion of correct responses must be higher than 50 % (i.e., 75 %
or 100 %). This calculation method is comparable with the routinely used titration method, where it was either tested whether a child
succeeded [] reliably(Wynn, 1992), or a number was evaluated as not known when the number was not given at least two out of
three times (Le Corre & Carey, 2007), and when the previous number was given at least two out of three times (Cheung et al., 2016).
Second, a number was considered as known only if the same set size was not given for other number words at least twice (e.g., if 3 was
given correctly when asking for 3, and 3 was also given at least twice when asking for 4, then 3 is not considered as known). (If the
criterion is set to a much stricter and somewhat unrealistic expectation of 100 % performance, and if giving a number to other number
words even once would make the number to be unknown, then the analysis still provides the same pattern of results as presented in the
Results section below.)
To mimic the classic titration method, number knowledge was specied as the largest known number after which the rst unknown
number follows. In other words, number knowledge is the smallest unknown number minus 1. For example, if the smallest unknown
number is 4, then the number knowledge of that child is 3. This calculation method imitates the titration method because in the
titration method, when an unknown number is reached, no further numbers are asked, and the previous number is considered as the
limit of the childs knowledge. In other words, even if the present method measured the entire range between 1 and 9, the number
knowledge calculation method imitated the titration method, in which data collection is interrupted after reaching the rst unknown
number.
If childrens measured number knowledge was at least 5, they were considered a CP-knower.
2.3.2. Calculating counting list knowledge
Counting list knowledge was calculated as the mean of the highest numbers recited correctly from the two counting list repetitions.
The mean was considered a compromise between minimum reliable performance (the minimum of the two values, i.e., that number
was reached consistently) and the maximum peak performance (the maximum of the two values, i.e., a value that is not always
reached). Because numbers should be recited starting from 4, the smallest upper limit of the counting list knowledge could be 4.
Analyses were performed using LibreOfce Calc spreadsheet, the CogStat (version 2.0) data analysis software (Krajcsi, 2020), and
the Jamovi statistical software (The jamovi project, 2020) packages. In CogStat, after specifying the analysis task with the appropriate
variables, analyses are run automatically based on the properties of the data. In that software, the selection of the hypothesis tests is
based on the common textbook suggestions extended with additional considerations found in the methodological literature. For the
specic decision tree in CogStat, see the online documentation of the software.
2.3.3. Measurement level of the number knowledge
In most of the following analyses, number knowledge was handled as an interval variable. While number knowledge in the subset-
knowersrange may be considered an interval variable, pre-knowers and CP-knowers may violate the equal intervals precondition, and
it may be more appropriate to handle number knowledge as an ordinal number. However, the mean could be a more useful descriptive
statistic for the expected value than the median (e.g., see Fig. 1 and 2). Several further details ensure that handling number knowledge
as an interval variable does not invalidate the present ndings. (1) Because, in all relevant hypothesis tests, the assumptions for the
parametric tests were violated, the same hypothesis test results would be expected if the number knowledge was handled as an ordinal
variable. (2) The full distributions of the data are also presented (see Fig. 1 and 2) where measurement level is irrelevant. (3) Some of
the analyses comparing the three follow-up question groups rely on nominal variables (e.g., see the left panel of Fig. 2 and the related
results), which analyses reveal the same pattern as observed in the interval variable-based analyses.
3. Results and discussion
The raw data are available at https://osf.io/584y3/.
Because the counting list knowledge may limit the number knowledge measured using the GaN task, children whose counting list
knowledge was not higher than their measured number knowledge were excluded from further analysis (N=4). Note again that the
counting list knowledge task was not a pre-test, but it was given after the GaN task.
A. Krajcsi
Cognitive Development 57 (2021) 100968
6
3.1. Observed number knowledge
The measured number knowledge differed between the three follow-up question groups (see Fig. 1; measured number knowledge
as an interval variable was compared across the three follow-up question groups,
ω
2
=0.035, KruskalWallis test (normality
assumption for ANOVA was violated):
χ
2
(2, N=197) =9.42, p=0.009). Specically, number knowledge of the Recount condition
was signicantly higher than the number knowledge in the other two conditions (post hoc Dunns test, ps <0.04). If pre-knowers are
coded as 0, and CP-knowers are coded as 5-knowers, the average number knowledge is 3.0 for the No follow-up condition, 2.7 for the Is
it N condition and 3.6 for the Recount condition, therefore, based on the descriptives, the recounting procedure causes an approxi-
mately 0.51.0 higher number knowledge index.
This difference in the observed number knowledge could be caused by at least two sources. First, it is possible that, in the Recount
condition, the average number knowledge is higher because there are more CP-knowers than in the other two conditions (e.g., because
the recount instruction may have a training effect, causing a qualitative change in number knowledge). Second, independent of the
previous possibility, average number knowledge in the Recount condition may be higher than in the other conditions because subset-
knowers (i.e., children who are not yet CP-knowers) know larger numbers (e.g., more 3- or 4-knowers) compared with the subset-
knowers in the other two conditions, while they do not become CP-knowers. These two possibilities are different both mathemati-
cally (i.e., whether mean is increased by more 5 values or by more values below 5) and conceptually (i.e., whether seeming change
occurs within subset-knower category or not). It is also possible that both of these factors contribute to the observed difference
simultaneously.
To investigate the rst possibility, i.e., whether there are more CP-knowers in the Recount condition compared with the other two
conditions, the proportion of CP-knowers and subset-knowers were compared across the three follow-up question conditions (see
Fig. 2, left panel). The results revealed that there were signicant differences in the proportion of the CP-knowers in the three groups
(number knowledge as a dichotomous nominal variable (i.e., CP-knower or subset-knower, where subset-knower category also
included the pre-knowers) was compared across the three follow-up question groups, Cram´
ers V measure of association: φ
c
=0.212;
Pearsons chi-squared test:
χ
2
(2, N=183) =8.196, p=0.017), and the descriptive data suggested that in the Recount condition there
were more CP-knowers than in the other two conditions (while there were 19 (32 %) and 17 (27 %) CP-knowers in the N follow-up and
in the Is it N conditions, respectively, there were 31 (51 %) CP-knowers in the Recount condition).
To investigate the second possibility, i.e., whether there were differences between the conditions in the subset-knowing range
(including the pre-knowers), the average number knowledge of the subset-knowers (i.e., excluding the CP-knowers from the sample)
between the three conditions was compared (Fig. 2, middle and right panels). The results showed no signicant differences between
the three follow-up question groups in the observed number knowledge of subset-knowers (among subset-knowers the number
knowledge as an interval variable was compared across the three follow-up question groups,
ω
2
=0.011, KruskalWallis test
(normality assumption for ANOVA was violated):
χ
2
(2, N=116) =3.33, p=0.189). The same non-signicant result was found if the
number of the specic number-knowers were compared between the three conditions (among subset-knowers the number knowledge
as a nominal variable was compared across the three follow-up question groups, Cram´
ers V measure of association: φ
c
=0.214;
Pearsons chi-squared test:
χ
2
(6, N=116) =10.580, p=0.102). (Note that, in the latter analysis, the Cram´
ers V measure of asso-
ciation was in the same order of magnitude as in the previous analysis for the proportion of subset- and CP-knowers. While the former
association proved to be signicant, it was not signicant in the latter analysis. It is possible that the follow-up questions also had an
effect on the number of specic subset-knowers, but because of the smaller sample size, the analysis lacked sufcient statistical power.
Nonetheless, the present data lacks evidence for different proportions of specic subset-knowers between the follow-up conditions).
Fig. 1. Number knowledge as a function of the three follow-up conditions, showing individual data and boxplots (left) and point and interval
estimations of the mean (right). In the charts, 0 number knowledge means that the child is a pre-knower, and 5 number knowledge means the child
is a CP-knower. Note that the condence intervals were calculated with the assumption of normal distribution, which was not the case in the present
data (e.g., see the individual data on the left chart), therefore, these interval estimations can only be considered as approximations. The dotted lines
of the x-axes denote that those variables are nominal.
A. Krajcsi
Cognitive Development 57 (2021) 100968
7
In summary, the follow-up questions used at the end of the GaN task inuenced the observed number knowledge, displaying higher
number knowledge for the Recount follow-up condition than for the Is it N follow-up or without a follow-up question. Importantly, the
difference was mainly rooted in the higher proportion of CP-knowers with the Recount follow-up, and not in the number knowledge of
subset-knowers. In other words, the Recount version mainly had an effect among the CP-knowers.
3.2. Number of corrections
The follow-up questions had a signicant effect on the number of corrections, i.e., the number of trials in which the children
modied their original response (Fig. 3; number of corrections as an interval variable was compared across the follow-up question
groups,
ω
2=0.302, KruskalWallis test (normality and homogeneity of variance assumptions for ANOVA were violated):
χ
2(2,
N=197) =84.4, p <0.001). When no follow-up question was asked, hardly any corrections were observed (mean number of
correction was 0.2, 0.6 %); for the Is it N follow-up the mean number of corrections (3.2, 8.9 %) was signicantly higher; and the
Recount follow-up question caused the highest mean number of corrections (9.1, 25.3 %; all pairwise post hoc comparisons (Dunns
test) were signicant, ps<.001). In parallel with these results, the proportion of children that corrected their responses at least once
differed between the three groups: 6% in the No follow-up question group, 45 % in the Is it N group, and 77 % in the Recount group (the
presence of at least one correction as a dichotomous nominal variable was compared across the follow-up question groups, Pearsons
chi-squared test:
χ
2(2, N =197) =66.739, p <0.001).
When the number of corrections was compared between children with different observed number knowledge (Fig. 4), the number-
knowledge had a signicant effect on the number of corrections (number of corrections as an interval variable was compared across the
various number knowledge groups,
ω
2=0.0751, KruskalWallis test (normality and homogeneity of variance assumptions were
violated):
χ
2 (5, N =197) =19.3, p =0.002). According to the post hoc pairwise comparisons (Dunns test), pairs pre-knower vs 2,
pre-knower4, 12, 14, 1CP, 23, and 34 differed signicantly, suggesting that 2- and 4-knowers primarily made the most
corrections.
Comparing the present results with existing reports on the proportion of corrections, we nd that Sarnecka and Lee (2009) who
used a condition similar to the present Is it N condition, found a low 0.2 % proportion of corrections, closer to what we observed in the
No follow-up condition (Fig. 3). It is possible that as-of-yet non-recognized factors, or factors that are not documented at all, caused this
difference. Le Corre et al. (2006) used a task version similar to the present Recount version (although they did not ask for recounting if
Fig. 2. Mosaic plot of the proportion of subset-knowers and CP-knowers as a function of the three follow-up conditions (left panel). Number
knowledge as a function of the three follow-up question conditions among subset-knowers (middle and right panels). See additional notes in Fig. 1.
Fig. 3. The number of corrections as a function of the three follow-up question conditions. See additional notes in Fig. 1.
A. Krajcsi
Cognitive Development 57 (2021) 100968
8
children already used counting spontaneously in their responses). On the one hand, they found a similarly large proportion of cor-
rections (20 % for subset-knowers and 80 % for CP-knowers) as in the present study, but the difference between the subset-knowers
and CP-knowers was not observed in the present results (Fig. 4). Note again that there are other differences in the Le Corre et al. (2006)
study that make a direct comparison difcult. For example, the basis for the proportion is the number of to-be-xed trials; corrections
that were made in the wrong number direction were not considered corrections; incorrect responses with a difference of 1 were
accepted as correct responses. These difculties regarding a comparison of the studies and the difculties concerning the differences in
the methods used highlights again the need for a more detailed description of how different versions of the GaN task can inuence
performance.
3.3. Performance changes during the session
The present GaN task version took longer to complete than the classic version. Several properties of the present task contributed to
the increased length: Instead of the titration method, all numbers between 1 and 9 are asked; numbers beyond 5 are also asked; all
numbers are asked four times. It is possible that this relatively long task may have been more tiresome for children and affected their
performance negatively. On the contrary, the supposed training or practice effect may improve performance further during a longer
task. Any of those two opposing effects may inuence the performance. To investigate whether the longer task impacted the results, the
mean performance (proportion of the correctly solved trials) was calculated for each child for each of the four series in the GaN task.
The proportion of correctly solved trials did not change over time (the mean proportions of the four series were 60 %, 58 %, 59 %, and
57 %, respectively, Friedman test (normality assumption for ANOVA was violated):
χ
2(3, N=197) =6.87, p=0.076; and while
performance slightly decreased over time, the change was not signicant despite the relatively large sample size). In a special case, it is
possible that both the tiredness and the training effect work, and the two effects have similar effect sizes, then the two effects may
extinguish each other, leaving no visible effect on performance.
To check whether the performance change differed between the follow-up question groups, an additional mixed ANOVA was
conducted with the proportion of correct responses in the four series as the within-subject factor and using the follow-up question
groups as the between-subject factor. For the two main effects the results replicated the previous analyses above: i.e., the series effect
was not signicant (F(3, 582) =2.395, p=0.067), while the follow-up question group effect was signicant (F(2, 194) =3.90, p=
0.022). Critically, the interaction of the two factors was not signicant (F(6, 582) =0.952, p=0.457), demonstrating no difference in
performance change between the follow-up question groups. However, descriptive results of the groups suggested that, in the Recount
group, the performance decreased. The change was monotonic across all 4 series, hinting that the change may not have been due to
noise, but there may be an existing effect with small effect size for which the present sample size did not provide sufcient statistical
power. By checking the effect of the series separately in the three groups, a signicant performance change was observed across the
series in the Recount group, but not in the other two groups: Running separate analyses for the three groups, the mean performance of
the 4 series as repeated measures interval variables was compared, Friedman test (normality assumption was violated):
χ
2(3,
N=65) =9.75, p=0.021 for the Recount group,
χ
2(3, N=65) =2.83, p=0.418 for the no No follow-up group, and
χ
2(3,
N=67) =2.11, p=0.549 for the Is it N group (Fig. 5).
Even if a performance change was present in the Recount group (note that the interaction of the primary ANOVA above was not
signicant), the decrease was moderate: The performance changed from 69 % (series 1) to 63 % (series 4). This was similar to the
results of Marchand and Barner (2020) where, in two studies, using the recount version of the GaN task, there were more children for
whom performance decreased between two GaN administrations compared with children whose performance increased; however, the
difference in change direction was not signicant. This pattern is counter to a training effect, the outcome of which predicts an
increased performance over time; or at least the training effect is weaker than the tiredness effect. Contrastingly, one may infer that the
Fig. 4. Number of corrections as a function of number knowledge. See additional notes in Fig. 1. Note that number knowledge is denoted as nominal
variable on the x-axis, even if this variable can be considered at least an ordinal variable, because the current analysis handles that variable as a
nominal variable (i.e., comparing the groups of children with different number knowledge).
A. Krajcsi
Cognitive Development 57 (2021) 100968
9
recount instruction may prevent performance error, although this prevention effect may decrease over time. The supposed tiredness
effect is also in line with the observed number of corrections: The tiredness effect was observable in the group where the children made
a substantial number of corrections, whereas in the other two groups, where corrections were much rarer, the supposed tiredness effect
was not observable. Importantly, the observed potential change in performance over time showed that even if longer administration of
the task may have slightly inuenced performance, the Recount follow-up question still demonstrated higher observed number
knowledge.
4. General discussion
The Give-a-number task is likely the most important task for measuring the initial acquisition of symbolic numbers among pre-
schoolers. At the end of the trials follow-up questions aim to x potential performance errors; however, various versions of these
follow-up questions are used in the literature, and it is not known whether the different questions can have a different effect on
performance. The present study compared the effect of three different follow-up questions, and found that the different questions have
effects on the measured performance. First, it was found that the Recounting instruction caused better performance in preschoolers
than the No follow-up question and the Is it N versions. This result is in line with an existing training study in which children showed
better performance in the GaN task after counting-and-labeling training (similar to the present Recounting condition) compared with
counting training, labeling training or control group (similar to the present Is it N or and the No follow-up question conditions) (Mix
et al., 2012). This result is also consistent with the explicit intention of including the recount instruction to prevent performance errors
in the GaN task (Wynn, 1990, 1992). Second, detailed analyses revealed that improved performance in the Recount follow-up con-
dition had been caused mainly by a higher proportion of CP-knowers and not by better number knowledge of the subset-knowers.
Third, and unsurprisingly, the follow-up question inuenced the number of corrections: There were hardly any spontaneous correc-
tions when no follow-up questions were used, there were more corrections in the Is it N condition and the most corrections were
observed for the Recount condition. Fourth, some aspects of the results hinted that performance in the Recount condition decreased
over time, supporting the notion that the recounting instruction prevented performance error through additional verication.
The results provided herein are essential because they demonstrate that different versions of the follow-up questions may cate-
gorize children differently in terms of whether they are subset- or CP-knowers (there were almost twice as many CP-knowers in the
Recount condition than in the Is it N and No follow-up conditions), a miscategorization that may have led to invalid data in several
studies. These results also highlight the need for the precise reporting of follow-up questions and the need to consider the type of
questions in reviews and meta-analyses. More generally, if there are further differences in the measured number knowledge caused by
additional variations in the GaN task, studies using different versions are incomparable, because preschoolers are categorized
inconsistently. One may argue that the present study did not use the GaN task version proposed by Wynn (1990, 1992), but a modied
version; therefore, the present results cannot be used to warn about GaN task versions in general. However, even if some of the version
related effects depend on other parameters of the task (i.e., even if different task version dimensions interact), it does not mean that the
GaN task version proposed by Wynn (1990, 1992) cannot work differently if only a single parameter of the task was modied. Rather,
the present result serve as a warning that various GaN task versions should not automatically be assumed as equivalent, but this
equality must be veried empirically.
The specic differences that were demonstrated here raise a next question: While the Recount follow-up question increased the
proportion of CP-knowers, it may not be clear whether recounting improved performance by avoiding performance error (as was
originally intended with the introduction of these follow-up questions), or whether recounting trained children, or whether both
effects worked in parallel. Some aspects of the results indicate that the recounting instruction may activate verication processes,
which may help to avoid performance errors. A former training study by Mix et al. (2012) hints that training at least plays a role in this
follow-up question effect. The possibility that performance may be altered by the follow-up instruction may also be related to chil-
drens partial number knowledge around their number knowledge limitation (Barner & Bachrach, 2010; Gunderson et al., 2015;
ORear, McNeil, & Kirkland, 2021; Wagner et al., 2019). The effect may also be in line and related to the fact that the reliability of
specic number knowledge is moderate (Marchand & Barner, 2020). Future studies may establish more specic sources of this
follow-up question effect.
In summary, while various versions of the Give-a-number task are used in the literature, and while they are supposed to be
equivalent, follow-up questions inuence the observed performance, which makes the results of the literature incomparable and
Fig. 5. The proportion of correct trials in the GaN task as a function of the series of numbers (i.e., time) and follow-up question groups.
A. Krajcsi
Cognitive Development 57 (2021) 100968
10
renders some of the task versions invalid or unreliable. Further validation studies may establish an optimal version for measuring the
number knowledge of preschoolers. The present result serves as an important caution that supposedly equivalent versions of the GaN
task may give rise to essential differences in observed performance, and further studies are needed to discover whether additional
variations of the GaN task may inuence the measured number knowledge. These clarications are essential because the conceptual
issues regarding initial symbolic numerical understanding can be resolved only if the GaN task, which is one of the most important
tests, measures the numerical ability validly.
Acknowledgments
I thank Marta Fedele, Edina Fintor, Petia Kojouharova and Tam´
as Sz˝
ucs for their comments on the manuscript. The work was
supported by the National Research, Development and Innovation Fund (NKFI 132165), CELSA Research Fund (CELSA/19/011), ELTE
E¨
otv¨
os Lor´
and University, Faculty of Education and Psychology, and Central European University, Department of Cognitive Sciences.
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90008-P
A. Krajcsi
... Knowledge of counting principles is typically assessed with the Give-a-number ("GaN" or "Give-THE PLURAL COUNTS N") task (Wynn, 1992;Patro & Haman, 2012;Krajcsi, 2019;Krajcsi, Fintor, & Hodossy, 2019). ...
... Then they generalise the successor rule on all other number-words within their count list (at least up to 10). The GaN procedure allows to distinguish between the "subset-knowers" and "CP-knowers", with some level of uncertainty in the case of "four-knowers" and "five-knowers" (c.f. Lee & Sarnecka, 2011;Krajcsi et al., 2019;see also O'Rear, McNeil, & Kirklans, 2020). The GaN task is, however, not sufficient to discriminate between CP-knowers-non-mappers and CP-knowers-mappers. ...
... This may be important also in a developmental context. Over the past several years, more and more work has shown that in learning the meanings of numerals, not only the numerical magnitude plays an important role, but also the ordinal aspect, which can be related to mapping numbers to spatial dimensions (Reynvoet & Sasanguie, 2016;Lyons & Beilock, 2011, 2013, Sella, Lucangeli, & Zorzi, 2018, 2019Sella, Lucangeli, Cohen Kadosh & Zorzi, 2020;but cf. Brannon & Van De Walle, 2002;Libertus, Feigenson, Halberda, & Landau, 2014;Spaepen et al., 2018). ...
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Already in toddlerhood, children begin to master the system of number word meanings. The role of grammar, and in particular grammatical number inflection, in early stage of this process has been well documented. It is not clear, however, whether the influence of the grammatical language structure also extends to more complex later stages. In the current study, we have addressed this problem by using differences in the grammatical number paradigms between Polish and German, in particular, the inconsistency of the grammatical number of the verb and the noun for numbers above four. One-hundred-fifty-three Polish-speaking children and 124 German-speaking three-to-six-year-old children took part in the study. Their main task was to compare symbolic numbers (Arabic numerals and spoken number-words) in the range of small numbers (2-4) large numbers (5-9) and between ranges. In addition, counting skills (Give-a-number and count-list) and mapping between non-symbolic (dot sets) and symbolic representations of numbers were checked. The children also performed working memory tests (Corsi-blocks and digit span). Based on Give-a-number and mapping tasks, participants were divided into subset-knowers, CP-knowers-non-mappers and CP-knowers-mappers (cf. LeCorre, 2014). As expected, grammatical number structure influenced performance: Polish-speaking children, later than the German ones, achieved the CP-knowers stage and, after it was achieved, they fared worse in the numerical comparison task, which was further mediated by response side. Importantly, however, there were no significant differences in the mapping task between non-symbolic and symbolic representations of numbers between Polish and German groups. We conclude that cross-linguistic differences in the grammatical number paradigms can significantly affect the development of representations and processing of numbers not only at the stage of acquiring the meaning of the first number-words, but also at later stages, when dealing with symbolic numbers.
... How many squares? bases of symbolic number (Davidson et al., 2012;Le Corre & Carey, 2007;, whether different components of their knowledge emerge concurrently (Slusser et al., 2013;Slusser & Sarnecka, 2011), and whether the GaN task is used optimally to monitor their number knowledge (Krajcsi, 2021). In this article, we review some persistent and some recent uncertainties about this classic description of the development of symbolic number knowledge. ...
... This titration method defines the limit of the children's cardinal knowledge. There are alternative versions of GaN task, but it is assumed that different versions yield equivalent results (Krajcsi, 2021). ...
... Third, while it is assumed that various versions of the GaN task yield similar results, this is not necessarily the case. For example, omitting recounting instruction may increase performance errors (Krajcsi, 2021), which leads to a biased assessment of cardinal knowledge. ...
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Learning the meaning of number words is a lengthy and error‐prone process. In this review, we highlight outstanding issues related to current accounts of children’s acquisition of symbolic number knowledge. We maintain that, despite the ability to identify and label small numerical quantities, children do not understand initially that number words refer only to sets of discrete countable items, not to other nonnumerical dimensions. We question the presence of a sudden change in children’s understanding of cardinality, and we report the limits of the give‐a‐number task. We also highlight that children are still learning the directional property of the counting list, even after acquiring the cardinality principle. Finally, we discuss the role that the Approximate Number System may have in supporting the acquisition of symbolic numbers. We call for improvements in methodological tools and refinement in theoretical understanding of how children learn natural numbers.
... In terms of methodological implications, the present results are consistent with the growing concern that the give-n task may underestimate (pre-counting and) counting-based cardinality knowledge (Barner & Bachrach, 2010;Krajcsi, 2021;Mou et al., 2021;Sella et al., 2021;Wagner et al., 2019). If future results confirm that the give-n task requires the more advanced cardinal-count concept, assessing the CP-knower level might better involve a combination of tasks such as the how-many task and a how-many application task (e.g., the cardinal-identity task) and using a composite cardinality score as in the present study. ...
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The give-n task is widely used in developmental psychology to indicate young children’s knowledge or use of the cardinality principle (CP): the last number word used in the counting process indicates the total number of items in a collection. Fuson (1988) distinguished between the CP, which she called the count-cardinal concept, and the cardinal-count concept, which she argued is a more advanced cardinality concept that underlies the counting-out process required by the give-n task with larger numbers. One aim of the present research was to evaluate Fuson’s disputed hypothesis that these two cardinality concepts are distinct and that the count-cardinal concept serves as a developmental prerequisite for constructing the cardinal-count concept. Consistent with Fuson’s hypothesis, the present study with twenty-four 3- and 4-year-olds revealed that success on a battery of tests assessing understanding of the count-cardinal concept was significantly and substantially better than that on the give-n task, which she presumed assessed the cardinal-count concept. Specifically, the results indicated that understanding the count-cardinal concept is a necessary condition for understanding the cardinal-count concept. The key methodological implication is that the widely used give-n task may significantly underestimate children’s understanding of the CP or count-cardinal concept. The results were also consistent with a second aim, which was to confirm that number constancy concepts develop after the count-cardinal concept but before the cardinal-count concept.
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Studies on children's understanding of counting examine when and how children acquire the cardinal principle: the idea that the last word in a counted set reflects the cardinal value of the set. Using Wynn's (1990) Give-N Task, researchers classify children who can count to generate large sets as having acquired the cardinal principle (cardinal-principle-knowers) and those who cannot as lacking knowledge of it (subset-knowers). However, recent studies have provided a more nuanced view of number word acquisition. Here, we explore this view by examining the developmental progression of the counting principles with an aim to elucidate the gradual elements that lead to children successfully generating sets and being classified as CP-knowers on the Give-N Task. Specifically, we test the claim that subset-knowers lack cardinal principle knowledge by separating children's understanding of the cardinal principle from their ability to apply and implement counting procedures. We also ask when knowledge of Gelman & Gallistel's (1978) other how-to-count principles emerge in development. We analyzed how often children violated the three how-to-count principles in a secondary analysis of Give-N data (N = 86). We found that children already have knowledge of the cardinal principle prior to becoming CP-knowers, and that understanding of the stable-order and word-object correspondence principles likely emerged earlier. These results suggest that gradual development may best characterize children's acquisition of the counting principles, and that learning to coordinate all three principles represents an additional step beyond learning them individually. This article is protected by copyright. All rights reserved
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Understanding the way in which counting represent numerosities was shown to be a long-lasting process. As shown in the Give-a-number task, acquiring the meanings of verbal number words goes through successive developmental stages in which children first learn the cardinal meanings of small number words one at a time before generalizing the cardinal principle they have induced from the first three number words to all number words within their counting range. This acquisition would take about a year, and would be completed by the age of 3 ½ years. The aim of the present study was to provide a conceptual replication of the developmental sequence described in Wynn’s study nearly 30 years ago using the Give-a-number task. A first cross-sectional study was conducted on 213 Belgian children aged between 39 and 74 months using the Give-a-number task to examine the developmental pattern and the influence of age on this acquisition. The time span of acquisition was examined in a second study in which 34 children were tested five times every months between the age of 36 to 52 months. Results showed that acquiring the cardinal meanings of number words spread out over a protracted period, far more extended than assumed by Wynn. Furthermore, children do not generalize all-at-once to large number words, the cardinal knowledge they learned on small number words. Rather, number words were found to be learned one at a time in a really progressive manner. Results were discussed with regard to their implications for the existing theories and in relation with other tasks assessing the acquisition of verbal number symbols.
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How does cross-linguistic variation in linguistic structure affect children's acquisition of early number word meanings? We tested this question by investigating number word learning in two unrelated languages that feature a tripartite singular-dual-plural distinction: Slovenian and Saudi Arabic. We found that learning dual morphology affects children's acquisition of the number word two in both languages, relative to English. Children who knew the meaning of two were surprisingly frequent in the dual languages, relative to English. Furthermore, Slovenian children were faster to learn two than children learning English, despite being less-competent counters. Finally, in both Slovenian and Saudi Arabic, comprehension of the dual was correlated with knowledge of two and higher number words.
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There is an ongoing debate concerning how numbers acquire numerical meaning. On the one hand, it has been argued that symbols acquire meaning via a mapping to external numerosities as represented by the approximate number system (ANS). On the other hand, it has been proposed that the initial mapping of small numerosities to the corresponding number words and the knowledge of the properties of counting list, especially the order relation between symbols, lead to the understanding of the exact numerical magnitude associated with numerical symbols. In the present study, we directly compared these two hypotheses in a group of preschool children who could proficiently count (most of the children were cardinal principle knowers). We used a numerosity estimation task to assess whether children have created a mapping between the ANS and the counting list (i.e., ANS-to-word mapping). Children also completed a direction task to assess their knowledge of the directional property of the counting list. That is, adding one item to a set leads to he next number word in the sequence (i.e., successor knowledge) whereas removing one item leads to the preceding number word (i.e., predecessor knowledge). Similarly, we used a visual order task to assess the knowledge that successive and preceding numbers occupy specific spatial positions on the visual number line (i.e., preceding: [?], [13], [14]; successive: [12], [13], [?]). Finally, children's performance in comparing the magnitude of number words and Arabic numbers indexed the knowledge of exact symbolic numerical magnitude. Approximately half of the children in our sample have created a mapping between the ANS and the counting list. Most of the children mastered the successor knowledge whereas few of them could master the predecessor knowledge. Children revealed a strong tendency to respond with the successive number in the counting list even when an item was removed from a set or the name of the preceding number on the number line was asked. Crucially, we found evidence that both the mastering of the predecessor knowledge and the ability to name the preceding number in the number line relate to the performance in number comparison tasks. Conversely, there was moderate/anecdotal evidence for a relation between the ANS-to-word mapping and number comparison skills. Non-rote access to the number sequence relates to knowledge of the exact magnitude associated with numerical symbols, beyond the mastering of the cardinality principle and domain-general factors.
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Do children learn number words by associating them with perceptual magnitudes? Recent studies argue that approximate numerical magnitudes play a foundational role in the development of integer concepts. Against this, we argue that approximate number representations fail both empirically and in principle to provide the content required of integer concepts. Instead, we suggest that children's understanding of integer concepts proceeds in two phases. In the first phase, children learn small exact number word meanings by associating words with small sets. In the second phase, children learn the meanings of larger number words by mastering the logic of exact counting algorithms, which implement the successor function and Hume's principle (that one-to-one correspondence guarantees exact equality). In neither phase do approximate number representations play a foundational role.
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How do children acquire exact meanings for number words like three or forty‐seven? In recent years, a lively debate has probed the cognitive systems that support learning, with some arguing that an evolutionarily ancient “approximate number system” drives early number word meanings, and others arguing that learning is supported chiefly by representations of small sets of discrete individuals. This debate has centered around findings generated by Wynn's (1990, 1992) Give‐a‐Number task, which she used to categorize children into discrete “knower level” stages. Early reports confirmed Wynn's analysis, and took these stages to support the “small sets” hypothesis. However, more recent studies have disputed this analysis, and have argued that Give‐a‐Number data reveal a strong role for approximate number representations. In the present study, we use previously collected Give‐a‐Number data to replicate the analyses of these past studies, and to show that differences between past studies are due to assumptions made in analyses, rather than to differences in data themselves. We also show how Give‐a‐Number data violate the assumptions of parametric tests used in past studies. Based on simple non‐parametric tests and model simulations, we conclude that (1) before children learn exact meanings for words like one, two, three, and four, they first acquire noisy preliminary meanings for these words, (2) there is no reliable evidence of preliminary meanings for larger meanings, and (3) Give‐a‐Number cannot be used to readily identify signatures of the approximate number system. This article is protected by copyright. All rights reserved.
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While much research has focused on understanding the process by which young children learn to count, little work has explored the effects of direct instruction on this process. In the current study, we explored the impacts of training children in an explicit counting procedure on two distinct cardinality tasks. Two- to 5-year-old children first participated in a Give-N task in which counting proficiency was assessed, and then participated in a short instruction session where explicit counting was modeled and encouraged. Following training, children were significantly better at identifying which of two cards contained a set size outside of their range of counting mastery (Huang, Spelke, & Snedeker, 2010) and were more likely to improve on a secondary numerical production task (Give-N; Wynn, 1990, 1992) compared with children in the control group. Not surprisingly, a greater proportion of children in the count training condition overtly counted during the cardinality task, a strategy that was found to be the strongest indicator of performance. Together, results reveal that even 5 min of counting instruction greatly increases the likelihood that a child will engage in counting behavior and results in improvements in cardinality judgments in two distinct numerical tasks.
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Approximate number word knowledge-understanding the relation between the count words and the approximate magnitudes of sets-is a critical piece of knowledge that predicts later math achievement. However, researchers disagree about when children first show evidence of approximate number word knowledge-before, or only after, they have learned the cardinal principle. In two studies, children who had not yet learned the cardinal principle (subset-knowers) produced sets in response to number words (verbal comprehension task) and produced number words in response to set sizes (verbal production task). As evidence of approximate number word knowledge, we examined whether children's numerical responses increased with increasing numerosity of the stimulus. In Study 1, subset-knowers (ages 3.0-4.2years) showed approximate number word knowledge above their knower-level on both tasks, but this effect did not extend to numbers above 4. In Study 2, we collected data from a broader age range of subset-knowers (ages 3.1-5.6years). In this sample, children showed approximate number word knowledge on the verbal production task even when only examining set sizes above 4. Across studies, children's age predicted approximate number word knowledge (above 4) on the verbal production task when controlling for their knower-level, study (1 or 2), and parents' education, none of which predicted approximation ability. Thus, children can develop approximate knowledge of number words up to 10 before learning the cardinal principle. Furthermore, approximate number word knowledge increases with age and might not be closely related to the development of exact number word knowledge. Copyright © 2014 Elsevier Inc. All rights reserved.
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Many have proposed that the acquisition of the cardinal principle (CP) is a result of the discovery of the numerical significance of the order of the number words in the count list. However, this need not be the case. Indeed, the CP does not state anything about the numerical significance of the order of the number words. It only states that the last word of a correct count denotes the numerosity of the counted set. Here, we test whether the acquisition of the CP involves the discovery of the later-greater principle - that is, that the order of the number words corresponds to the relative size of the numerosities they denote. Specifically, we tested knowledge of verbal numerical comparisons (e.g., Is 'ten' more than 'six'?) in children who had recently learned the CP. We find that these children can compare number words between 'six' and 'ten' only if they have mapped them onto non-verbal representations of numerosity. We suggest that this means that the acquisition of the CP does not involve the discovery of the correspondence between the order of the number words and the relative size of the numerosities they denote.
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We investigated whether specific input helps 3-1/2-year-olds discover that the last word in a count represents its cardinal value (i.e., the cardinal word principle). In Study 1, we contrasted four training approaches. The only approach to yield significant improvement was to label a set's cardinality and then immediately count it. This training is consistent with previously hypothesized mechanisms based on juxtaposing a set's cardinal label with its count in close temporal contiguity (e.g., Klahr & Wallace, 1976; Schaeffer, Eggleston, & Scott, 1974), as well as general theories of comparison and categorization (e.g., Gentner, 2005). In Study 2, we asked parents to read picture books to their preschool children and found that they rarely provide cardinal labels immediately followed by counting, even when asked to read a book about number.