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Running Head: PATTERNING IN CHILDREN’S CALCULATION
MacKay, K.J., & De Smedt, B. (2019) Patterning counts: Individual differences in children’s
calculation are uniquely predicted by sequence patterning. Journal of Experimental Child
Psychology, 177, 152-165. doi: 10.1016/j.jecp.2018.07.016
Patterning Counts: Individual Differences in Children’s Calculation are Uniquely Predicted by
Sequence Patterning
Kelsey J. MacKay & Bert De Smedt
Faculty of Psychology and Educational Sciences
KU Leuven, University of Leuven, Belgium
Word Count: 7790
Corresponding author:
Prof. Dr. Bert De Smedt
Parenting & Special Education Research Unit, University of Leuven
Leopold Vanderkelenstraat 32, box 3765, B-3000 Leuven, BELGIUM
Tel. +32 16 32 57 05; Fax. +32 16 32 59 33
e-mail: Bert.DeSmedt@kuleuven.be
Running Head: PATTERNING IN CHILDREN’S CALCULATION
Patterning Counts: Individual Differences in Children’s Calculation are Uniquely Predicted by
Sequence Patterning
THE ROLE OF PATTERNING IN CHILDREN’S CALCULATION 3
Abstract
Many studies have examined the cognitive determinants of children’s calculation, yet the
specific contribution of children’s patterning abilities to calculation remains relatively
unexplored. This study investigated if children’s ability to complete sequence patterns (i.e. add
the missing element into: 2–4–?– 8) uniquely predicted individual differences in calculation and
whether these associations differed depending on the type of stimuli in these sequence patterns,
i.e. number, letter, time or rotation.
Participants were 65 children in first and second grade (Mage=7.40, SDage=0.44). All
children completed four tasks of sequence patterning (number, letter, time, and rotation).
Calculation was measured via addition and subtraction tasks. We also measured cognitive
determinants of individual differences in calculation, namely symbolic number comparison,
motor processing speed, visuospatial working memory, and non-verbal IQ, to verify if patterning
predicted calculation when these additional measures were controlled for.
We observed significant relationships between the patterning dimensions and calculation,
except for the rotation dimension. Follow-up regressions, controlling for the aforementioned
cognitive determinants of calculation, revealed that the number and time dimensions were strong
predictors of calculation while the evidence for the letter dimension was only anecdotal and for
the rotation dimension was non-existent, suggesting some degree of specificity of different types
of sequence patterning in predicting calculation. Symbolic magnitude processing remained a
powerful unique correlate of calculation performance.
These findings add to our understanding of individual differences in calculation ability,
such that sequence patterning could begin to be considered as one of the cognitive skills
underlying calculation ability in young children.
THE ROLE OF PATTERNING IN CHILDREN’S CALCULATION 4
Introduction
Many studies have examined the cognitive determinants of children’s calculation, yet the
specific role of children’s (sequence) patterning remains relatively unexplored. Even though
there have been suggestions of the links between patterns and mathematics (Papic, Mulligan, &
Mitchelmore, 2011; Starkey, Klein, & Wakeley, 2004; Warren, Cooper, & Lamb, 2006), and
patterning remains a component of children’s education (Clements, Sarama, & Liu, 2008),
research on patterning as one of the potential components of individual differences in
mathematics development is lacking, with the existing literature on the origins of individual
differences in mathematics being mainly focused on working memory or numerical processing
(Peng, Namkung, Barnes, & Sun, 2016; Schneider et al., 2017; Vanbinst & De Smedt, 2016).
While the existing body of patterning studies have almost exclusively examined the association
between repeating patterns and calculation (Miller, Rittle-Johnson, Loehr, & Fyfe, 2016; Papic et
al., 2011), sequence patterning is a relatively understudied type of patterning, in comparison to
repeating patterns. Therefore, the goal of the current study was to address this gap in the
literature and examine the specific role of sequence patterning in elementary school children’s
calculation. In the remainder of the introduction, we summarize the existing literature on
relational thinking, patterning, and present the aims of the current study.
Sequence patterns, compared to other types of patterning (e.g., repeating patterns), might
be more critical to children’s mathematical development, when referring to counting sequences
and number representation (e.g., counting; Thomas, Mulligan, & Goldin, 2002). The reason for
this could be due, for example, to sequence patterns being related to relational thinking.
Relational thinking, in the context of mathematical cognition, refers to the idea of inspecting the
connections between two or more objects to solve a problem (Molina, Castro, & Ambrose,
THE ROLE OF PATTERNING IN CHILDREN’S CALCULATION 5
2005). While repeating patterns would also activate this type of thinking, sequence patterns are a
more complex method by which to examine children’s relational thinking. Taking relational
thinking into account, and the complexity of sequence patterns compared to repeating patterns,
sequence patterns are a potentially more beneficial type of patterning to examine compared to
repeating patterning. Moreover, the idea of relational thinking has been applied in the domain of
algebra (Carpenter, Levi, Franke, & Zeringue, 2005) as well as fraction learning (Empson, Levi,
& Carpenter, 2011), and has been shown to be an important step in the development of arithmetic
to algebraic learning (Napaphun, 2012). Sequence patterning can therefore be investigated with
regard to its importance in mathematics competency, potentially through the mechanism of
relational thinking.
Although relational thinking can be conceived as a potential general mechanism in
sequence patterning, it is also possible that relational thinking is only involved in more specific
associations with sequence patterning. For instance, a sequence pattern can be in numbers (i.e.,
2-4-6-8) but can also be in letters (i.e., a-c-e-g). The question regarding relational thinking in
sequence patterns can therefore also include the distinction between domain-specific (i.e.,
sequence patterns being related to mathematics only within some of these dimensions) and
domain-general (i.e., every dimension of sequence patterning is related to mathematics) sequence
patterns. In examining distinct dimensions of sequence patterning and their relations to
mathematics, the potential mechanism of relational thinking in the association between sequence
patterning and mathematics can be examined as being either domain-specific or domain-general.
A pattern is any replicable regularity in which the items have relations to one another and
are predictable (Economopoulos, 1998; Fyfe, McNeil, & Rittle-Johnson, 2015; Hendricks,
Trueblood, & Pasnak, 2006; Papic et al., 2011). Patterns can be found in a variety of domains
THE ROLE OF PATTERNING IN CHILDREN’S CALCULATION 6
including, but not limited to, poetry, music, as well as mathematics (Bjorklund & Pramling,
2014), showing it is important to understand the placement and use of patterning skills.
The type of patterning that is usually the focus of the classroom as well as the literature
on children’s mathematical development, is that of repeating patterns (i.e., a-b-a-b-a-b; Clements
et al., 2008; Rittle-Johnson, Fyfe, McLean & McEldoon, 2013), and these repeating patterns are
deemed important for later mathematical development (Economopoulos, 1998; Ford & Crew,
1991; Rittle-Johnson, Fyfe, Loehr, & Miller, 2015). However, much less focus is given to
sequence patterning. Piaget (1952) theorized that a milestone in children’s cognitive
development lies in seriation and transitivity, in other words, sequences or patterns, showing it is
a relevant area for research on children’s mathematical development. Sequence patterns can also
be linked to an understanding of ordinality, such that it involves participants providing yes/no
answers regarding a set of numbers appearing in a row. Importantly, ordinality has been found to
be linked to various aspects of mathematical ability, specifically the processing of symbolic
numbers (Goffin & Ansari, 2016; Lyons, et al., 2016). Further, both repeating and sequence
patterns offer a method by which children can exercise their relational thinking, as mentioned
above, to understand how items in a pattern relate to one another (Blanton & Kaput, 2005;
Pasnak et al., 2015). Taken together, these arguments motivated the use of sequence patterning in
the current study.
A sequence pattern is defined as a type of pattern in which the items follow a certain
sequence, that is not repeating, but rather is increasing or decreasing at a constant rate and is
sometimes referred to as a growing pattern (Economopoulous, 1998; Papic et al., 2011). As an
example, a sequence pattern involving numbers would look like: 2 - 4 - 6 - 8 - 10. In some tasks
using sequence patterning, a sequence is presented with one of the numbers missing (i.e., 2 - 4 - ?
THE ROLE OF PATTERNING IN CHILDREN’S CALCULATION 7
- 8 - 10) and children are then presented with four options to choose from, of which one is the
correct answer (Gadzichowski, 2012).
Previous intervention studies (Hendricks et al., 2006; Kidd et al., 2013, 2014) have
shown that patterning instruction involving a variety of patterns (e.g., repeating, symmetrical
increasing sequence patterns) in many dimensions (e.g., number, letter, shapes, symbols, objects)
improves academic achievement in mathematics as compared to control groups receiving other
types of instruction (e.g., math, reading, social studies) or business-as-usual instruction. These
studies provide evidence that relationships between patterning and mathematical ability exist.
However, these findings were based on the combination of all types of patterns, including
sequence patterns, in all dimensions, leaving the disentanglement of which type(s) of patterning
and in which dimensions contribute to these findings.
Lee, Ng, Bull, Pe, and Ho (2011), examined the role of sequence patterning in 10-year
old children’s mathematical abilities via a number series task (sequences of whole or rational
numbers following a rule) and a function machine task (pairs of numbers with a specific
relationship). They observed unique associations between the patterning tasks and different
measures of mathematical ability (i.e., computational fluency and algebraic proficiency).
However, their patterning tasks only included numerical stimuli, leaving it unresolved as to
whether only numerical patterning abilities or more general patterning abilities are predictive of
mathematical performance.
In a follow-up study, Lee et al. (2012) investigated the same relationships but in younger
six-year old children. They administered executive functioning tasks, numerical and arithmetic
proficiency measures and three types of patterning tasks. They administered a similar function
machine task, as described above, with both geometric shapes and numbers. They also included a
THE ROLE OF PATTERNING IN CHILDREN’S CALCULATION 8
shapes-within-shapes patterning measure, which involved manipulating the size, shape and
position of the shapes (e.g., ; Lee et al., 2012). The authors observed that the
patterning tasks explained individual differences in numerical and arithmetical proficiency, and
that the numerical patterning task explained higher variance compared to the geometrical task
and shapes-within-shapes task. While Lee et al. (2012) made an important step in understanding
the role of sequence patterns, they did not consider other specific determinants of individual
differences in mathematics, such as number processing, as we will do in the current study.
More recently, Pasnak et al. (2016) examined the specific relationship between sequence
patterning, specifically, and mathematical abilities in first-grade children. The results showed
that children’s fall scores on number sequences were significantly correlated with their spring
scores on the standardized math assessment, while the fall scores on the standardized math
assessment were not significantly correlated with spring scores on number sequences. One
limitation of this study, however, much like many other patterning studies (e.g., Hendricks et al.,
2006; Kidd et al., 2013, 2014), is the limited specificity regarding mathematical abilities. In other
words, these studies used general, standardized mathematical measures that examined large
components of mathematical ability (e.g., quantitative concepts, mathematical concepts), leaving
it unresolved as to which type of mathematical ability sequence patterning is specifically related.
Because predictors of individual differences are dependent on the math ability under study (De
Smedt, 2016; Peng et al., 2016; Schneider et al., 2017), the current study restricted the focus to
calculation (i.e., addition and subtraction) as it plays a quintessential role in mathematics and
given that the existing body largely has focused on this part of mathematics.
The present study
As hinted briefly in the literature summary of patterning studies, there are two major
THE ROLE OF PATTERNING IN CHILDREN’S CALCULATION 9
shortcomings in previous patterning research which the present study will aim to resolve. Firstly,
previous patterning research has not investigated different dimensions of sequence patterning,
leaving it unresolved whether sequence patterning generally relates to mathematics or if this
association is domain-specific (i.e., number patterns relate to mathematics, while letter patterns
do not). Secondly, previous research examining the major determinants of individual differences
in calculation have not considered sequence patterning as a potential major determinant,
therefore it is unknown whether sequence patterning uniquely contributes to individual
differences in calculation, beyond other known cognitive determinants of calculation ability, such
as numerical magnitude processing or visuospatial working memory (De Smedt et al., 2009;
Peng et al., 2016; Schneider et al., 2017; Vanbinst & De Smedt, 2016).
Based on these two shortcomings in patterning research, two research questions were
derived. The first was to determine the associations between dimensions of the sequence
patterning task and calculation. If the association is domain-specific, we expected to see
correlations between the number and time dimensions (given their proximity to number) and
calculation, whereas, if the association is general, we expected to see correlations between each
of the sequence patterning dimensions and calculation. The second question was to determine if
sequence patterning plays a unique role in explaining the individual differences in children’s
calculation when symbolic number comparison, motor processing speed, visuospatial working
memory and non-verbal IQ were taken into account. To follow-up previous patterning literature
(Hendricks et al., 2006; Kidd et al., 2013, 2014; Pasnak et al., 2016), we focused on children in
early elementary school.
To answer these two questions, we administered a sequence patterning task in four
dimensions (number, letter, time and rotation) as well as a calculation task assessing both
THE ROLE OF PATTERNING IN CHILDREN’S CALCULATION 10
addition and subtraction. Also, to assess how patterning uniquely contributes to calculation skills,
various cognitive measures were added, as the literature clearly shows that individual differences
in arithmetic can be explained by a variety of cognitive skills (Chu, vanMarle, & Geary, 2015;
Desoete, 2015; Xenidou-Dervou, De Smedt, Van Der Schoot, & van Lieshout, 2013). In terms of
cognitive measures, we included numerical magnitude processing, as it is known to be crucial in
explaining the individual differences found in children’s calculation performance (Vanbinst & De
Smedt, 2016), but restricted our focus to symbolic measures, in view of the meta-analysis by
Schneider et al. (2017) showing that these are significantly more correlated to individual
differences in mathematics than non-symbolic ones. We also focused on visual measures for
working memory and IQ, seeing as the patterning task is also visual in nature. Visuospatial
working memory has also been found to be related to calculation, further motivating its use in
the current study (e.g., De Smedt et al., 2009; Schmerold, 2017). Against this background, we
assessed children’s symbolic number comparison, motor processing speed, visuospatial working
memory, and non-verbal IQ in addition to sequence patterning as potential variables that could
explain variance in children’s calculation ability.
Method
Participants
Participants were 65 typically developing children, ranging in age from six to nine years
old (28 girls, 37 boys, Mage = 7.40, SDage = 0.44) from three different international schools in
Belgium. Twenty-five children were recruited from two of the schools in the spring, while 40
children were recruited from the third school the following fall. Grouping all the children
together, 15 were in Grade 1 (23%) and 50 were in Grade 2 (77%). All children were taught in
THE ROLE OF PATTERNING IN CHILDREN’S CALCULATION 11
English and followed similar curricula. All students were able to communicate well in English,
follow the task instructions and ask questions clearly if help was needed.
Materials
Patterning. This task was the same as in Kidd et al. (2014) and consisted of 12 problems
in each of the four dimensions of patterns (i.e., number, letter, time and rotation), for a total of 48
items. The items were all combined, regardless of dimension, as decided by a Latin Square table.
There were four practice items (one for each dimension). The patterns consisted of five stimuli,
with one item missing, as shown by a question mark. The items varied such that the first, third or
fifth item of the pattern could be missing. Depending on whether the item was vertically or
horizontally presented, four possible answers were presented beside or below the problem,
respectively. Children received a score out of 12 for each of the four dimensions of patterns. For
an example of each of the dimensions of patterns, see Figure 1.
Calculation. The task comprised six sections, each increasing in difficulty. There were
two versions; one for addition and one for subtraction, for a total of 12 subsections. The sections
were lettered and were categorized based on the sum and minuend of the problems for the
addition and subtraction versions, respectively. The sections were made to increase in difficulty
with two assumptions: that smaller sums and minuends were easier and that problems in which
children were not required to carry were easier than those that required the child to carry. The
items increased from single- to double- to triple-digit with each set presenting first no-carry-
problems followed by carry-problems. Each section contained six problems and had
approximately an equal number of problems in each decade. Problems that contained a decade, 0
or 1 in either the operands or solution were excluded. A list of the specific problems is provided
in the Appendix. The children moved through the sections at their own pace and were instructed
THE ROLE OF PATTERNING IN CHILDREN’S CALCULATION 12
to continue through as many sections as possible. There was no time limit. The total number of
problems answered correctly, per operation, was used as the score.
Symbolic number comparison. We administered the SYMP Test (Brankaer, Ghesquière,
& De Smedt, 2017) to assess children’s symbolic magnitude comparison skills. The task of the
child was to cross out as fast and accuracy as possible the largest number when given a pair of
numbers. The task existed in two versions, single-digit and double-digit, each comprising of four
practice problems followed by 60 problems, of which children completed as many as possible in
45 seconds. Children received a score for the total number of problems correct for each version
of the task. Performance on this group administered test is known to correlate very highly with
classic computerized versions of the comparison task in children of this age (rs > .58; Brankaer
et al., 2017).
Motor processing speed. This measure was taken from the SYMP Test (Brankaer et al.,
2017) and was administered as a control task to account for the speed in which children
completed the symbolic number comparison tasks. It had exactly the same design as the SYMP
Test (four practice items followed by 60 items) , except that the stimuli were pairs of shapes that
were either solid black or white with a black outline. Children were asked to cross out the black
shape. Children received one score for the total number of problems correct within the time limit
(20 seconds).
Visuospatial working memory. The Corsi Span measure was used to assess visuospatial
working memory (De Smedt et al., 2009). The experimenter placed a wooden plate to which 9
blocks at random positions were glued. The experimenter indicated to the child that she would
tap a sequence of blocks and the child’s task was to repeat the tapped sequence, in the same
order. After two practice problems, the test stimuli then began, starting with three sequences of
THE ROLE OF PATTERNING IN CHILDREN’S CALCULATION 13
two blocks. The sequence increased with one block after three items of the same list length. The
task stopped when the child could not complete all three problems of the same length. The
child’s score was calculated to be the total number of items solved correctly.
Non-verbal IQ. We used the block design task from the Wechsler Intelligence Scale for
Children – III (WISC-III; Wechsler, 1991) as a proxy for non-verbal IQ.
General Procedure
The testing order was the same for every child. Children first completed the group
administered tests in the following order: patterning task, calculation task, single-digit number
comparison, motor processing speed, double-digit number comparison. After that, visuospatial
working memory task and non-verbal IQ were administered individually in a quiet room.
Results
Descriptive Statistics
Descriptive statistics for all the tasks and their reliabilities can be found in Table 1. The
addition and subtraction items of the calculation task were highly positively correlated (r = .64, p
< .001), as were the single- and double-digit number comparison scores (r = .70, p < .001), and
were therefore combined into a total calculation score and a total number comparison score,
respectively, to reduce the number of correlations. In addition, the original patterning task from
Kidd et al. (2014) had 48 items; however, we discovered an error in one rotation item after
testing and therefore, this item was removed from the analysis, resulting in 11 instead of 12 items
for this dimension. Results show no ceiling or floor effects, with each task having high
reliability, with the exception of the rotation dimension which was found to have a lower
reliability.
THE ROLE OF PATTERNING IN CHILDREN’S CALCULATION 14
Correlational analyses
A summary of the correlations between each of the measures can be found in Table 2. All
patterning dimensions except rotation were positively significantly correlated with children’s
calculation. These relationships can be seen graphically in Figure 2. The number and time
dimensions significantly positively correlated with children’s total number comparison score.
The letter and time dimensions significantly positively correlated with children’s visuospatial
working memory. None of the patterning dimensions correlated with motor processing speed.
Only the rotation dimension of the patterning test correlated significantly with children’s non-
verbal IQ. Age correlated only significantly positively with the letter dimension of the patterning
task as well as motor speed and visuospatial working memory while sex only significantly
correlated with number comparison. Finally, it should be noted that all the patterning dimensions
significantly correlated with each other, with the exception of the correlation between the letter
and rotation dimensions.
Linear regressions
Four linear regressions were executed to examine how each of the dimensions of the
patterning tasks uniquely predicted children’s calculation scores. Other cognitive determinants of
children’s calculation were added to determine if the dimensions contributed additional
explained variance in children’s calculation, namely symbolic number comparison, motor
processing speed, visuospatial working memory, non-verbal IQ. Age was added in the linear
regressions as it was significantly correlated with the letter dimension of the patterning task. Sex
was not added as a predictor as it showed no significant correlations with any of the dependent
variables (Table 2). The children in this study came from three different schools, and therefore
school was first entered as a dummy variable to account for school-to-school differences. The
THE ROLE OF PATTERNING IN CHILDREN’S CALCULATION 15
results of the regression analyses were similar whether or not school was accounted for and
therefore only the results without the dummy variables are reported.
The outcome of the regression analyses is depicted in Table 3. It is important to note that
the regression diagnostics (variance inflation factor) indicated that there was no evidence for
multicollinearity of the predictors. The regression analyses indicated that the number, time, and
letter dimensions were significant unique predictors of children’s calculation, controlling for age,
symbolic number comparison, motor processing speed, visuospatial working memory and non-
verbal IQ, while the rotation dimension was not. Symbolic number comparison emerged as a
powerful unique predictor, even when patterning abilities (in the different dimensions) were
controlled for.
We also ran Bayesian analyses, which allowed us to test the degree of support for a given
predictor. We calculated the Bayes factor, which is the ratio of the evidence for the alternative
hypothesis compared to the null (see Andraszewicz et al., 2015 for an excellent elaboration).
Adding the Bayes factors to our analyses (Table 3) allowed us to infer which hypothesis for a
given predictor is the most plausible (i.e. alternative vs. null) and to identify for which
predictors, the evidence in the data is the strongest. The Bayesian analyses were generally in line
with the frequentist data, supporting the observation that the number and time dimensions were
strong predictors of the variability in children’s calculation. The evidence for the contribution of
the letter dimension was only anecdotal. For the rotation dimension, there was more evidence in
the data for the null hypothesis of no association, although this evidence was also only anecdotal.
Symbolic number comparison was an even stronger predictor (as indicated by the Bayes factors)
than any of the patterning measures. Interestingly, the Bayes factors for age, motor processing
speed, visuospatial working memory, and non-verbal IQ and were all below 1, indicating that
THE ROLE OF PATTERNING IN CHILDREN’S CALCULATION 16
there is more evidence in the data for the null hypothesis (i.e. that this variable is not correlating
with calculation) than for the alternative hypothesis.
Discussion
The goal of the present research was to determine the specific role of sequence patterning
in elementary school children’s calculation. Two research questions were posed for the current
study: 1) Are associations between dimensions of the sequence patterning task and calculation
ability domain-specific (i.e., only some dimensions of sequence patterning are associated with
calculation) or domain-general (i.e., every dimension of sequence patterning is associated with
calculation)?, and 2) Does sequence patterning play a unique role in explaining the individual
differences in children’s calculation when symbolic number comparison, motor processing
speed, visuospatial working memory and non-verbal IQ are added? The current data revealed
that number, letter and time patterning but not rotation patterning correlated with calculation
ability. These associations remained, once symbolic number comparison, motor processing
speed, visuospatial working memory and non-verbal IQ were controlled for. This suggests an
important and unique role for sequence patterning in predicting individual differences in
calculation, as we elaborate in more detail below.
With regard to the first research question, correlations determined that sequence
patterning is significantly positively related to addition and subtraction skills, such that only
number, letter, and time patterns were associated with calculation. Firstly, our results extend the
literature that has studied the association between repeating patterning and mathematics
(Economopoulos, 1998; Ford & Crew, 1991; Papic et al., 2011; Rittle-Johnson et al., 2015;
Starkey et al., 2004) and show that sequence patterning, in certain dimensions (i.e., number,
letter, and time), is another type of patterning related to calculation abilities. Our study also
THE ROLE OF PATTERNING IN CHILDREN’S CALCULATION 17
therefore replicated and extended Lee et al. (2012), who found that number patterns were
associated with arithmetic proficiency. Further, the results provide evidence that sequence
patterning is related to a specific type of mathematical ability, i.e., calculation, which extends the
findings from previous sequence patterning literature examining broad mathematical measures
(Hendricks et al., 2006; Kidd et al., 2013, 2014; Lee et al., 2011, 2012; Pasnak et al., 2016). We
focused on calculation as the dependent variable as it plays a quintessential role in mathematics
and given that prior research has largely focused on this part of mathematics. With regard to the
strength of the correlations, number and time were found to be more strongly related to
calculation in comparison with the letter and rotation dimensions, therefore confirming some
degree of specificity within sequence patterning.
As hinted above, the rotation dimension of the patterning task did not correlate with
individual differences in calculation. Lee et al. (2012) found that numerical patterning explained
more variance in numerical proficiency compared to the geometrical task, showing that perhaps
geometrical tasks, such as the rotation sequence patterns used in the current study, explain less
variance with regard to mathematical abilities. On the other hand, the rotation task was found to
have low reliability in the current study, showing that it is possible that the task was too difficult
for this age group, therefore resulting in non-significant association between the rotation
sequence patterns and calculation. Taking this low reliability of the rotation dimension into
account, we should be cautious in drawing strong conclusions about domain-specificity. Further
investigation is required regarding rotation sequence patterns and their association with
calculation in this age group and older.
The similarity in the associations of both number and time with calculation, might
suggest that perhaps the cognitive representations of sequence patterns in the dimensions of
THE ROLE OF PATTERNING IN CHILDREN’S CALCULATION 18
number and time may not be so distant from one another and could even reflect basic magnitudes
in different dimensions (e.g., Bueti & Walsh, 2009; Lourenco & Bonny, 2017; Park & Cho,
2017). However, the rotation dimension did not correlate with calculation, suggesting that these
representations of space, time, and number maybe not so well connected, at least not in children.
Future studies are needed further investigate the interrelations between measures of space, time,
and number.
To further examine the possibility of domain-specificity within sequence patterning as
part of our first aim, the Bayesian regression analyses demonstrated that the association between
number and calculation was the strongest of all the dimensions on the patterning task. The
association between the time dimension and calculation was also found to be quite strong. The
evidence for the importance of letter patterns, however, was much less strong, even at the
anecdotal level, and there was no evidence for an association between the rotation measure and
calculation. In other words, if there would be a domain-general association between patterning
and calculation, one should observe similar associations for all patterning dimensions under
study, and this was not the case. Against this background, the current data suggest that the
association between patterning and calculation is more domain-specific.
Although our findings point to domain-specific relations between sequence patterning
and children’s calculation, the underlying mechanisms of this relationship are still unknown. Our
results particularly emphasize the role of the number and time dimensions of sequence patterns,
rather than the letter and rotation dimensions, in explaining children’s calculation, although the
Bayesian analyses point to number as being stronger than time. The understanding of the number
and time dimensions of sequence patterns could be related to children’s understanding of
ordinality (i.e., the ability to order numbers correctly). Recently, there has been an increased
THE ROLE OF PATTERNING IN CHILDREN’S CALCULATION 19
focus on ordinality and individual differences in mathematical achievement (Goffin & Ansari,
2016; Lyons & Beilock, 2011, 2013). However, it should be noted that while sequence patterns
and ordinality are similar, they may not tap into the same cognitive abilities. Rather, as sequence
patterning requires complex relational reasoning, while ordinality tasks do not, the underlying
mechanism could perhaps be this relational reasoning with numbers (Carpenter et al., 2005;
Empson et al., 2011; Napaphun, 2012) that is used in the sequence patterns, as opposed to the
letter and rotation sequence patterns, for which reason number and time are strong predictors of
calculation performance.
For our second aim, in order to understand the unique role of sequence patterning in
children’s calculation abilities, linear regressions as well as Bayesian analyses examined the role
of each of the dimensions alongside symbolic number comparison, motor processing speed,
visuospatial working memory, and non-verbal IQ in explaining variance in children’s calculation
abilities. These analyses further revealed that the number and time dimensions were unique as
well as stronger predictors of children’s calculation ability compared to the letter dimension
(anecdotal evidence) and rotation dimension (no evidence). These results are consistent with
those of Pasnak et al. (2016) that number sequences and not letter sequences are unique
predictors of children’s calculation abilities, although Pasnak et al. (2016) did not investigate
dimensions of time and rotation. Importantly, our findings go beyond Pasnak et al. (2016) by
showing that the association with number sequence patterns remains when other well-known
cognitive determinants of calculation are controlled for.
The current findings also replicate the observation that symbolic number processing is a
major underlying ability that predicts children’s arithmetic (Schneider et al., 2017; Vanbinst &
De Smedt, 2016). The current data extend this body of evidence by showing that this association
THE ROLE OF PATTERNING IN CHILDREN’S CALCULATION 20
also remains when sequence patterning abilities, including number patterning, are additionally
controlled for. Our data also indicate that numerical magnitude processing and number patterning
are related, yet each of them provides a unique contribution in explaining individual differences
in calculation. Both symbolic number comparison and sequence patterning tasks require children
to use relational reasoning, therefore supporting the idea that relational reasoning could be the
underlying mechanism driving both of these competencies. On the other hand, it needs to be
emphasized, as is evidenced by the differences in Bayes factors, that the evidential strength for
symbolic number comparison as a predictor of calculation was substantially higher than the
evidential strength for number or time patterns.
Future research should additionally examine the relationships between each of the
dimensions of sequence patterning and calculation (or other specific mathematical abilities) in a
longitudinal study, to verify the direction of the association, and especially to ensure that our
findings generalize beyond our sample of students in international schools. Moreover, the
addition of other known cognitive variables, such as approximation skills, could be relevant
(Xenidou-Dervou et al., 2013). Further, the current study used increasing sequence patterns,
however, decreasing sequence patterns have also been used in some patterning research
(Hendricks et al., 2006) and would be an interesting next step regarding sequence patterns, to
determine if similar associations exist for decreasing sequence patterns. Finally, a future study
could examine the same sequence patterning task in older children, to get a better sense of the
relationships with the rotation dimension, as it could have been too difficult for this age group.
As for implications of our research, Sarama and Clements (2009) discuss the importance
of including (repeating) patterns into daily environments of children. Building on this idea,
sequence patterns could also be integrated into the classroom, however, as part of a lesson plan in
THE ROLE OF PATTERNING IN CHILDREN’S CALCULATION 21
mathematics. Seeing as our research found that particularly the number and time dimensions of
sequence patterns are related to calculation, this activity in math classrooms will be an
innovative and potentially interesting pathway for children to develop their calculation skills.
In conclusion, our results revealed that there is a relationship between sequence
patterning, in particular number, letter, and time, and calculation. Further, sequence patterning in
the dimensions of number and time are strong predictors of children’s calculation abilities above
and beyond symbolic number comparison, motor processing speed, visuospatial working
memory, and non-verbal IQ, suggesting that certain types of sequence patterning make a
significant contribution to individual differences in young children’s arithmetic abilities.
Acknowledgements
We would like to thank all the students, teachers, principals, and secretaries that participated in
this research. Special thanks are due to Marinka Gadzichowski for her open communication and
sharing of the patterning task used in this study. This work was partially supported by grant
C16/16/001 of University of Leuven Research fund.
22
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Figure 1
Number Letter
Time Rotation
Figure 1. Examples of each of the dimensions of patterns used in the Patterning Task, based on
(Gadzichowski, 2012). From left to right and top to bottom, an example of number patterns,
letter patterns, time patterns and rotation patterns can be seen.
Figure 2
A B
C D
Figure 2. The linear relationships between each dimension of the patterning task and the total
calculation score.
r = .54 r = .18
r = .37r = .53
Table 1
The means, standard deviations, minimum and maximum, theoretical maximum and reliability
for all tasks completed.
M SD Min Max Theoretical
Max
Reliability
Patterning Task
Number 7.34 3.71 0 12 12 .88 a
Letter 5.95 3.31 0 11 12 .81a
Time 6.11 3.10 0 12 12 .77 a
Rotation 4.05 2.18 0 9 11 .54 a
Calculation Task Total 33.06 14.44 11 71 72 .96 a
SNC Total 43.35 9.84 19 69 120 .79b
Motor Processing Speed 25.98 6.41 12 42 60 .86b
Visuospatial working
memory
8.84 1.99 5 14 24 .77c
Non-verbal IQ 26.15 11.59 2 51 69 .87d
Note:SNC symbolic number comparison, a Cronbach alpha calculated from the current study, b
Test-retest reliability from the manual in children of the same age, c Calculated from the sample
in De Smedt et al., 2009 in children of the same age, d from Sattler, 2001, Non-verbal IQ uses
raw scores
Table 2
Zero-order correlations of all tasks used.
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
1. Number -
2. Letter .61** -
3. Time .51** .36** -
4. Rotation .36** .14 .40** -
5. Calculation .53** .37** .54** .18 -
6. Number
Comparison .31* .18 .39** .07 .56** -
7. Motor Speed .15 -.07 .22 .13 .36** .44** -
8. Visuospatial
Working Memory .20 .28* .29* .19 .33** .15 .08 -
9. Nonverbal IQ .05 .20 .23 .34** .11 .04 -.03 .23 -
10. Age .16 .26* .14 .14 .24 .10 .28* .27* .17 -
11. Sex .09 .21 -.01 -.00 -.16 -.42** -.09 -.12 -.21 .22 -
* p < .05, ** p < .01
Table 3
Linear regression models predicting calculation, estimated for the different dimensions of the
patterning task.
Predictors R2adj β p VIF Bayes
Factor
Unique
R²
Number (F (6, 54) = 9.616, p < .001) .463
Number patterning .316 .003** 1.138 24.001 .088
Age in months .027 .793 1.177 0.135 .001
Symbolic Number Comparison .439 .000** 1.366 572.690 .135
Motor Processing Speed .086 .435 1.323 0.340 .021
Visuospatial working memory .116 .263 1.176 0.620 .012
Non-verbal IQ .124 .221 1.125 0.716 .014
Letter (F (6, 54) = 8.086, p < .001) .415
Letter patterning .236 .037* 1.244 2.708 .044
Age in months -.001 .990 1.232 0.314 .000
Symbolic number comparison .470 .000** 1.355 857.067 .091
Motor Processing Speed .147 .217 1.414 0.511 .043
Visuospatial working memory .115 .290 1.188 0.623 .011
Non-verbal IQ .101 .344 1.150 0.564 .009
Time (F (6, 54) = 8.842, p < .001) .440
Time patterning .301 .010** 1.350 13.366 .067
Age in months .055 .599 1.171 0.364 .003
Symbolic Number Comparison .421 .001** 1.451 191.537 .118
Motor Processing Speed .082 .462 1.323 0.343 .018
Visuospatial working memory .106 .318 1.190 0.501 .010
Non-verbal IQ .058 .590 1.213 0.352 .003
Rotation (F (6, 54) = 7.024, p < .001) .376
Rotation patterning .107 .337 1.164 0.633 .009
Age in months .047 .670 1.172 0.368 .002
Symbolic number Comparison .535 .000** 1.271 4299.127 .217
Motor Processing Speed .067 .574 1.350 0.329 .041
Visuospatial working memory .135 .229 1.182 0.800 .015
Non-verbal IQ .106 .348 1.209 0.657 .009
Note. *p < .05, ** p < .01
Appendix – Calculation Task
ADDITION
A
2 + 4 = ______
1 + 2 = ______
5 + 4 = ______
3 + 1 = ______
2 + 6 = ______
7 + 3 = ______
B
8 + 6 = ______
5 + 11 = ______
3 + 9 = ______
13 + 6 = ______
7 + 4 = ______
12 + 3 = ______
C
11 + 27 = ______
76 + 23 = ______
35 + 13 = ______
41 + 26 = ______
32 + 54 = ______
24 + 31 = ______
D
17 + 16 = ______
35 + 26 = ______
66 + 29 = ______
49 + 22 = ______
28 + 14 = ______
27 + 58 = ______
E
113 + 105 = ______
453 + 536 = ______
324 + 161 = ______
404 + 273 = ______
211 + 384 = ______
534 + 262 = ______
F
349 + 125 = ______
108 + 167 = ______
236 + 647 = ______
513 + 178 = ______
258 + 336 = ______
568 + 424 = ______
SUBTRACTION
A
3 – 2 = _____
9 – 2 = _____
6 – 4 = _____
8 – 3 = _____
5 – 1 = _____
7 – 4 = _____
B
15 – 2 = _____
16 – 4 = _____
17 – 6 = _____
18 – 3 = _____
19 – 8 = _____
17 – 5 = _____
C
97 – 16 = _____
35 – 14 = _____
66 – 12 = _____
59 – 17 = _____
89 – 23 = _____
68 – 31 = _____
D
56 – 38 = _____
24 – 17 = _____
92 – 13 = _____
41 – 16 = _____
87 – 18 = _____
63 – 24 = _____
E
778 – 355 = _____
212 – 101 = _____
979 – 323 = _____
338 – 122 = _____
686 – 124 = _____
553 – 231 = _____
F
311 – 209 = _____
883 – 325 = _____
337 – 118 = _____
923 – 288 = _____
645 – 217 = _____
462 – 126 = _____
