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This study investigated the developmental dynamics of mathematical performance during children's transition from preschool to Grade 2 and the cognitive antecedents of this development. 194 Finnish children were examined 6 times according to their math performance, twice during each year across a 3-year period. Cognitive antecedents, that is, counting ability, visual attention, metacognitive knowledge, and listening comprehension, were tested at the first measurement point. The results indicated that math performance showed high stability and increasing variance over time. Moreover, the growth of math competence was faster among those who entered preschool with an already high level of mathematical skills. The initial level of math performance, as well as its growth, was best predicted by counting ability. Although quite a large body of research has been carried out on the learning of math, most studies have focused on specific math disabilities (Geary & Hoard,. However, little is known about the development of chil-dren's overall math performance (Ai, 2002), particularly during their early years of formal education. The aim of the present study was to examine the developmental dynamics of children's math performance during the transition from preschool to Grade 2 and the roles that a variety of cognitive antecedents play in this development. At the same time, heterogeneity in the developmen-tal trajectories that children show in their math performance was also investigated.
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Developmental Dynamics of Math Performance From Preschool to Grade 2
Kaisa Aunola, Esko Leskinen, Marja-Kristiina Lerkkanen, and Jari-Erik Nurmi
University of Jyva¨skyla¨
This study investigated the developmental dynamics of mathematical performance during children’s
transition from preschool to Grade 2 and the cognitive antecedents of this development. 194 Finnish
children were examined 6 times according to their math performance, twice during each year across a
3-year period. Cognitive antecedents, that is, counting ability, visual attention, metacognitive knowledge,
and listening comprehension, were tested at the first measurement point. The results indicated that math
performance showed high stability and increasing variance over time. Moreover, the growth of math
competence was faster among those who entered preschool with an already high level of mathematical
skills. The initial level of math performance, as well as its growth, was best predicted by counting ability.
Although quite a large body of research has been carried out on
the learning of math, most studies have focused on specific math
disabilities (Geary & Hoard, 2001; Rourke, 1993), on clinical
samples (Temple, 1991), and on specific subcomponents of math-
ematics (Geary, 1993; Geary, Hoard, & Hamson, 1999; Jordan,
Hanich, & Kaplan, 2003; Kulak, 1993; Share, Moffitt, & Silva,
1988). However, little is known about the development of chil-
dren’s overall math performance (Ai, 2002), particularly during
their early years of formal education. The aim of the present study
was to examine the developmental dynamics of children’s math
performance during the transition from preschool to Grade 2 and
the roles that a variety of cognitive antecedents play in this
development. At the same time, heterogeneity in the developmen-
tal trajectories that children show in their math performance was
also investigated.
Development of Math Performance
Mathematical performance is made up of a number of compo-
nents, such as basic knowledge of numbers, memory for arithmet-
ical facts, understanding of mathematical concepts, and the ability
to follow procedures (Dowker, 1998). It has also been suggested
that the development of mathematical skills progresses in a hier-
archical manner: Learning basic concepts and skills provides the
basis for mastering more complex skills and procedures (Entwisle
& Alexander, 1990; Gelman & Gallistel, 1978; Nesher, 1986). For
example, when the ability to retrieve basic number facts has
become automatic, attentional resources can be devoted to more
complex problem solving (Gersten & Chard, 1999). It has also
been shown that mathematical knowledge begins to develop well
before formal schooling starts (Resnick, 1989). For example, dur-
ing early childhood, children intuitively start to construct certain
fundamental concepts, such as those describing absolute size (e.g.,
big, small), part–whole relations, and protoquantitative reasoning
schemas. These then provide the basis for their further mathemat-
ical development (Ginsburg, 1997; Resnick, 1986, 1989).
Most of the research in the field so far has focused on specific
learning disabilities (Geary, 1993; Geary et al., 1999; Ginsburg,
1997; Rourke, 1993; Rourke & Conway, 1997) or on the role of
certain subcomponents of mathematics, such as word-problem
solving or arithmetic computations (Geary, 1993; Geary et al.,
1999; Kulak, 1993; Share et al., 1988). Although this research has
increased the understanding of math disabilities and of different
components of math performance, less is known about how overall
math performance develops among unselected populations. More-
over, understanding of the heterogeneity of the developmental
trajectories children show in math development is limited. To
address these issues, the present study focused on examining both
the developmental dynamics of overall math performance and the
heterogeneity in math developmental trajectories during the period
when children transferred from preschool to Grade 2.
The development of mathematics may proceed over time in two
ways. The first is that children’s knowledge and skills gradually
accumulate over time. Children who start with good skills and
sophisticated knowledge in math increase their performance over
time more than those who start with poorer skills. Such a cumu-
lative pattern should be manifested in high stability, increasing
interindividual differences in performance over time (“fan-out”),
and the amplifying nature of the developmental process (i.e., the
initial level of performance predicts positive subsequent growth in
performance; Aunola, Leskinen, Onatsu-Arvilommi, & Nurmi,
2002; Bast & Reitsma, 1997). The alternative possibility is that
individual differences in performance decrease rather than increase
with time: Children who originally start with a low level of skills
and related knowledge increase the speed of their development and
catch up with those who originally have higher levels of these
skills. Such a decrease in individual differences may be due, for
example, to the systematic instruction children receive at school or
Kaisa Aunola and Jari-Erik Nurmi, Department of Psychology, Univer-
sity of Jyva¨skyla¨, Jyva¨skyla¨, Finland; Esko Leskinen, Department of Math-
ematics and Statistics, University of Jyva¨skyla¨, Jyva¨skyla¨, Finland; Marja-
Kristiina Lerkkanen, Department of Teacher Education, University of
Jyva¨skyla¨, Jyva¨skyla¨, Finland.
This study is part of the ongoing Jyva¨skyla¨ Entrance Into Primary
School (JEPS) Study and was funded by Finnish Academy Grants 63099
and 778230. We express our gratitude to all the children and teachers
participating in this study. Special thanks to Hannele Ika¨heimo for valuable
comments on our article.
Correspondence concerning this article should be addressed to Kaisa
Aunola, Department of Psychology, P.O. Box 35, 40014 University of
Jyva¨skyla¨, Finland. E-mail: aunola@psyka.jyu.fi
Journal of Educational Psychology Copyright 2004 by the American Psychological Association
2004, Vol. 96, No. 4, 699–713 0022-0663/04/$12.00 DOI: 10.1037/0022-0663.96.4.699
699
to the increasing difficulty of the target operations to be learned, as
has been found for the development of reading skills (Leppa¨nen,
Niemi, Aunola, & Nurmi, 2004; Parrila, Aunola, Leskinen, Nurmi,
& Kirby, 2003; Phillips, Norris, Osmond, & Maynard, 2002).
Although a number of studies have recently been carried out on
the development of reading skills (Aunola et al., 2002; Bast &
Reitsma, 1997, 1998; Leppa¨nen et al., 2004; Parrila et al., 2003;
Phillips et al., 2002), little is known about the developmental
dynamics of mathematics. In one study, Jordan, Kaplan, and
Hanich (2002) followed children’s progress in math four times
every half year from the second grade to the third grade. They
found that those children who had disabilities in math but not in
reading developed at a faster rate in overall math performance than
those who had difficulties in both math and reading. Children who
had disabilities in math but not in reading also showed a slightly
faster rate of growth than normally achieving children. Jordan et
al. concluded from their findings that mathematical difficulties are
not necessarily stable over time. One limitation of the Jordan et al.
study, however, is that the possibility of a technical ceiling effect
was not taken into account in the measurement of math perfor-
mance. This may have caused the growth of children with high
mathematical achievement to level off somewhat, a possibility that
was also suggested by the authors themselves. In two other studies
carried out thus far, the results have shown a partly different
pattern from those reported by Jordan et al. Williamson, Appel-
baum, and Epanchin (1991) followed children’s math performance
from Grade 1 to Grade 8. They found that individual differences
in mathematical achievement increased (fanned out) over time
and that there was a modest positive correlation between the
initial level of math achievement and the rate at which this level
improved. A similar result was reported by B. O. Muthe´n and
Khoo (1998) among two cohorts of adolescents followed up
yearly from Grade 7 to Grade 12. One limitation of all these
previous studies is that they have focused on examining the
development of math performance during a period when the
children had already acquired some basic knowledge of math.
However, no studies have been carried out that have examined
the developmental dynamic of math performance starting from
before the beginning of formal instruction. The present study
made an effort to add to previous research by examining the
development of math performance during the children’s first
efforts to learn basic math skills, that is, during their transition
from preschool to Grade 2. Because we assumed that skill
development may be rapid during this period, the children’s
math performance was measured every half year.
It has been suggested recently that to understand development,
it may be necessary to identify heterogeneity in individuals’ de-
velopmental trajectories rather than examining the development
purely on a sample level (Bergman, Magnusson, & El-Khouri,
2003; B. Muthe´n & Muthe´n, 2000; Nagin, 1999). It is possible, for
example, that different subgroups of children follow different
developmental trajectories (see also Jordan et al., 2002, 2003).
Consequently, to complement the variable-oriented approach, the
present study also used a person-oriented approach to examine
whether different kinds of developmental trajectories of math
performance could be identified among children moving from
preschool to primary school.
Cognitive Antecedents of Math Performance
Several cognitive antecedents have been suggested as factors
that play a role in the development of math performance. One such
variable is counting ability, namely, children’s understanding of
how to count objects and their knowledge of the order of numbers
(Geary, 1993; Gelman & Gallistel, 1978). It has been shown, for
example, that children with math-related disabilities are deficient
in their basic counting abilities regardless of their IQ or reading
status (Geary, 1990; Geary, Bow-Thomas, & Yao, 1992; Geary et
al., 1999; Geary, Hamson, & Hoard, 2000; Ohlsson & Rees, 1991).
One possible reason why counting ability is an important anteced-
ent of future math skills is that it leads to the automatic use of
math-related information and thus allows attentional resources to
be devoted to more complex problem solving and more elaborate
procedures (Gersten & Chard, 1999; Resnick, 1989). Another
possibility, as suggested by Siegler (1986; see also Geary, Brown,
& Samaranayake, 1991; Lemaire & Siegler, 1995), is that counting
is a backup strategy in the acquisition of arithmetic knowledge,
being reflected not only in subsequent level of automatization of
fact retrieval but also in the accuracy of fact retrieval. After many
successful counts, re-presentation of the problem leads to a stron-
ger activation of the candidate answers, leading children to rely
increasingly on the retrieval of solutions instead of counting. The
associative strength between a specific problem and a possible
answer increases directly as a function of the frequency with which
an individual gives that answer in response to the problem. If the
child repeatedly makes counting errors, then incorrect solutions
may become associated with a specific problem (Siegler, 1986; see
also Geary et al., 1991; Temple & Sherwood, 2002).
Another cognitive antecedent that has been assumed to be
related to math performance is metacognitive knowledge (Butter-
field & Ferretti, 1987), which refers to a child’s knowledge and
understanding of cognitive processes (Flavell, 1976). For example,
Lucangeli, Coi, and Bosco (1997), studying fifth graders, found
that those who had low metacognitive awareness were less effi-
cient in solving math problems than those who showed high
metacognitive awareness. Similar findings have recently been re-
ported by Desoete, Roeyers, and Buysse (2001) and Borkowski
(1992). There are two possible explanations for these results. First,
metacognitive knowledge may reflect overall cognitive ability
(Alexander, Carr, & Schwanenflugel, 1995), which has been
shown to be highly correlated with academic achievement in
mathematics (Alarcon, Knopik, & DeFries, 2000). Second, meta-
cognitions may enable learners to adjust appropriately to varying
problem-solving tasks, demands, and contexts, thereby demon-
strating their importance in all learning processes (Boekaerts,
1999; Desoete et al., 2001). For example, Swanson (1990) found
that high-metacognitive pupils outperformed lower metacognitive
pupils in problem solving regardless of their overall aptitude level.
According to Swanson, high metacognitive knowledge can even
compensate for overall ability because, for any task at hand, it
leads to more efficient information processing than is possible for
those with a low level of metacognitive skills.
The third possible antecedent that may be detrimental to math-
ematical performance is attentional deficit (Badian, 1983). For
example, difficulties with executive control and attentional allo-
cation have been shown to be negatively associated with arithmetic
skills (see, e.g., McLean & Hitch, 1999). Similarly, children with
700 AUNOLA, LESKINEN, LERKKANEN, AND NURMI
attention-deficit/hyperactivity disorder have been found, early in
their school careers, to show delayed automatization of number
facts (Ackerman, Anhalt, & Dykman, 2001). It has been suggested
that attentional resources are important because they affect how
children initiate and direct their processing of information in
different tasks, how they comprehend those tasks, and how they
retrieve representations of to-be-remembered information when
doing tasks (Geary et al., 1999).
Because mathematical tasks also often include auditory cues as
well as visual ones, the fourth possible antecedent of math perfor-
mance is the ability to understand instructions, namely, listening
comprehension. For example, listening comprehension and the
ability to handle verbal information are, it has been argued, im-
portant for solving verbally presented mathematical problems (Jor-
dan et al., 2003).
Although many studies have been carried out on the anteced-
ents of math performance, this research has several limitations.
First, most studies have been cross-sectional. Consequently,
they have not provided the possibility of examining the extent
to which the variables under study may predict subsequent math
development (Hecht, Torgesen, Wagner, & Rashotte, 2001). In
the present study, we were able to examine the extent to which
predictor variables were associated with the level of math
performance, as well as their impact on the progress of math
performance. Second, most of the research on the antecedents
of math performance has been carried out with samples repre-
senting individuals with mathematical disabilities or else it has
compared children with and without learning disabilities (Bull
& Johnston, 1997; Geary, 1993; Jordan et al., 2002; McLean &
Hitch, 1999). Less is known about the antecedents of math
performance among nonselected populations. Third, previous
research on the cognitive antecedents of math performance has
focused on factors that are specific to mathematics, such as
counting ability and the retrieval of arithmetical facts from
semantic memory (Geary, 1993). Less is known about the role
that a broader spectrum of cognitive processes, not only those
specific to mathematics, plays in the development of math
performance (Kail & Hall, 1999; see also Share et al., 1988).
The present study examined the role of a broad range of
cognitive antecedents, such as counting ability, metacognitive
knowledge, attentional resources, and listening comprehension,
in the development of children’s math performance using lon-
gitudinal data on children in the course of their transfer from
preschool to Grade 2.
The Role of Sex
The findings on the sex differences in children’s math per-
formance are contradictory (for a review, see Geary, 1996). In
some studies, no sex differences have been found in the level of
math performance (Hyde, Fennema, & Lamon, 1990; Randhawa
& Randhawa, 1993; Skaalvik & Rankin, 1994), particularly
during the elementary school years (Friedman, 1989). However,
other studies have shown that the growth rate in math is steeper
among male students than female students. For example, B. O.
Muthe´n and Khoo (1998) reported that among 7th–13th graders,
boys showed more growth in math skills than girls. Similarly,
Leahey and Guo (2001) showed that boys had a faster rate of
acceleration in math than girls despite an equal initial level of
skills. Some studies have also suggested that, by the end of high
school, boys do outperform girls in mathematics (Hyde et al.,
1990; Mills, Ablard, & Stumpf, 1993; Skaalvik & Rankin,
1994). These differences have particularly been found among
talented students (Benbow, 1988; Friedman, 1989; Mills et al.,
1993). Overall, it has been argued that sex differences exist
(Geary, 1996) but that their timing has not been clearly estab-
lished (Leahey & Guo, 2001). Consequently, in the present
study, the role of sex in children’s level and growth of math
performance was also investigated.
The Finnish school system differs from schooling in the United
States and many European countries. Primary school with formal
teaching begins in Finland in the year children become 7 years old,
that is, one year later than in the United States. Before primary
school, Finnish children typically participate in preschool for one
year. Finnish preschool, therefore, is equivalent to kindergarten in
the United States. Preschool in Finland is not compulsory, but
almost all children attend it. In the Finnish preschool, there is no
formal teaching in mathematics, but children are encouraged to
play with numbers and mathematical concepts.
Aims
This study examined the following research questions:
1. How does children’s math performance develop from pre-
school to Grade 2? It was assumed that the development of
mathematical performance would follow one of two alternative
patterns. According to the first hypothesis, the development of
math performance is a cumulative process that is characterized by
high stability, widening individual differences over time, and a
positive association between the initial level and the rate of growth
in performance (B. O. Muthe´n & Khoo, 1998; Williamson et al.,
1991). As an alternative hypothesis, it was assumed that the
development of math performance would follow a reversed pat-
tern: There would be a decrease in individual differences across
time, and those who were originally doing worse would catch up
with those who used to show a superior performance.
2. What are the major cognitive antecedents of math perfor-
mance during this period? Four possible antecedents were inves-
tigated: counting ability, visual attention, listening comprehension,
and metacognitive knowledge. It was hypothesized that initial high
counting ability would be associated with the level of math per-
formance and would also increase its subsequent growth rate
(Geary, 1993). Because metacognitive knowledge, listening com-
prehension, and visual attention may reflect children’s overall
level of cognitive competence and their adoption of effective
learning strategies, they too were assumed to contribute to the level
and growth of math performance.
3. To what extent do boys and girls differ in the level and growth
of their math performance? We hypothesized that, although both
sexes start equal, boys would have a faster rate of growth than girls
(Leahey & Guo, 2001; B. O. Muthe´n & Khoo, 1998).
4. What kinds of developmental trajectories can be identified in
the development of math performance by applying a person-
oriented approach; that is, to what extent is there heterogeneity in
the growth trajectories of math performance, and what are the
major antecedents of math development within each type of
trajectory?
701
DEVELOPMENTAL DYNAMICS OF MATH PERFORMANCE
Method
Participants and Procedure
A total of 194 (103 boys, 91 girls) 5- to 6-year-old children (M75.0
months, SD 3.3 months) participating in the Jyva¨skyla¨ Entrance Into
Primary School Study (Nurmi & Aunola, 1999) were examined during
their transition from preschool to primary school. The original sample
consisted of all the children (n210) from two medium-size municipal-
ities in central Finland who were born in 1993. Parental permission to
gather data from the children was obtained from the parents of 207
children. Of these 207 children, 194 participated at each measurement
point during the study.
The sample was homogeneous in terms of race and cultural background,
as is typical in Finnish schools. Moreover, all the children spoke Finnish as
their native tongue. Four pupils were enrolled in special education classes
from the beginning of the first grade. In addition to these four children,
seven children in normal classes followed a special curriculum for at least
one school subject. Three of the children started the second grade straight-
away after preschool (i.e., they skipped the first grade).
Background information was gathered from the parents of 191 children.
A total of 159 of the children were from families with two parents, 19 of
the children were from families consisting of the mother or the father living
with her or his new spouse and their children, and 13 of the children were
living with their single mother. The number of children per family ranged
from1to11(M2.80, SD 1.50). A total of 18% of the mothers and
14% of the fathers had a degree from an institution of university standing,
68% of the mothers and 75% of the fathers had a qualification from an
institution of professional or vocational education, and 14% of the mothers
and 11% of the fathers had no occupational education.
A total of 17% of the mothers and 24% of the fathers were working in
the higher white-collar professions, 53% of the mothers and 25% of the
fathers were working in lower level white-collar professions, 17% of the
mothers and 34% of the fathers were working in blue-collar professions,
6% of the mothers and 12% of the fathers were in private business, and 7%
of the mothers and 5% of the fathers were students, housewives/-husbands,
or pensioners.
Preliminary analyses comparing the math performance of children from
different socioeconomic backgrounds, as defined by the mothers’ and
fathers’ socioeconomic status (SES), showed that children with parents in
blue-collar professions performed less well in math and also showed a
lower level of initial counting skills than those with parents in the higher
white-collar professions. Moreover, children with fathers in blue-collar
professions performed less well in math and showed a lower level of initial
counting skills than those with fathers in lower white-collar professions.
However, parents with different socioeconomic backgrounds did not differ
in the amount of arithmetic they had taught to their children.
For the preschool assessments, the children came from 21 preschool
groups. In the first and second grades, the children came from 17 classes
(13 schools). The preliminary analyses showed no classroom variation
either in the initial level or in the growth of children’s math performance
(Aunola, Leskinen, & Nurmi, 2004), suggesting that children were not
selected into particular classes, for example, because of SES.
The children were examined six times: twice during their preschool year,
that is, (a) in October 1999 (N207) and (b) in April 2000 (N200);
twice during their first primary school year, that is, (c) in October 2000
(N196) and (d) in April 2001 (N196); and twice during their second
primary school year, that is, (e) in October 2001 (N197) and (f) in
March 2002 (N196).
Cognitive antecedents of math performance, that is, counting ability,
visual attention, listening comprehension, and metacognitive knowledge,
were assessed at Time 1 in an individual testing situation. Math perfor-
mance was assessed at each measurement point. At Time 1 and Time 2, the
test was performed in an individual testing situation. At Times 3– 6, it was
performed in a group situation in the classroom. All the tests were carried
out by trained investigators and took place in a suitable room on either the
preschool or primary school premises. Children who were absent from
school on the day of testing (e.g., because of illness), were tested as soon
as they were back at school again. The attrition of 11 children was due to
the fact that the families of these children had moved to other districts and
were not able to participate in the study later on. The attrition of 2 children
was due to missing data.
Measures
Mathematical Performance
In the present study, we set out to examine the developmental dynamics
of math performance across six measurement points, using latent growth
curve modeling. This procedure set at least two requirements for the
measurement of math performance: First, we had to be able to assume that
the measurements were identical across the measurement points. Second,
there had to be no floor or ceiling effects at any of the measurement points
(Williamson et al., 1991). These are challenging requirements in the case
of a rapidly developing academic skill, such as math performance.
In the present study, the children’s mathematical performance was
assessed by means of the Diagnostic Test for Basic Mathematical Concepts
(Ika¨heimo, 1996). This test was designed to assess overall math compe-
tence in the central domains of mathematical instruction in the elementary
school curriculum. The test consists of five subtests assessing different
subskills, parts of which are easy enough to be sensitive to differences in
the early stages of math competence and parts of which are difficult enough
to avoid a ceiling effect. Because this stage of skills development is so
rapid, it was not possible to focus on specific components of math. The
structure of the test was identical at all measurement points. However, to
avoid a ceiling effect across the measurement points, we added more
difficult items to the test as the children became more skilled in
mathematics.
Knowledge of ordinal numbers. The children’s knowledge of ordinal
numbers was assessed by means of two tasks. The children were first
shown a picture of a sequence of boy figures and then asked to circle a
particular one (“The boys are in a line. Circle the third boy from the
beginning”; “The boys are in a line. Circle the seventh boy from the
beginning”).
Knowledge of cardinal numbers and basic mathematical concepts.
The children’s knowledge of cardinal numbers and basic mathematical
concepts, such as equal, more, and less, was measured by tasks (7 tasks at
the first measurement point and 12 at the other measurement points) that
became progressively more difficult. In each task, the children were shown
a picture of a set of balls and asked to draw a specific number of balls in
the space given (e.g., “Draw as many balls as there are in the model,”
“Draw five balls fewer than there are in the model,” “Draw four balls more
than there are in the model”).
Number identification. The children’s ability to perceive the corre-
spondence between a particular number and the number of objects in a
figure was assessed by two kinds of task (four tasks at the first measure-
ment point and six tasks at the other measurement points). In the first set
of tasks, the children were shown a picture that included a set of balls and
four different numbers written below them. They were then asked to circle
the number that corresponded to the number of balls in the figure (e.g.,
“How many balls are there in the picture? Circle the right number”). In the
other set of tasks, the children were shown a picture that included a specific
number and were asked to draw as many balls as were shown in the picture
(e.g., “Draw as many balls as are shown in the picture” [e.g., 8]).
Word problems. The children were read aloud simple verbal mathe-
matical problems (e.g., “You have seven candies, and you get three more.
How many do you have now?”). There were 2 tasks at the first measure-
ment point, 6 problems at the second and third, 10 problems at the fourth,
and 14 problems at the fifth and sixth. After each problem, the children
were asked to write down the right solution on the answer sheet.
702 AUNOLA, LESKINEN, LERKKANEN, AND NURMI
Basic arithmetic. The children’s skill in basic arithmetic was assessed
using a set of visually presented addition (e.g., “9 3_,” “7 _
14”), subtraction (e.g., “11 2_,” “15 _9”), multiplication (e.g.,
“8 7_,” “4 700 _”), and division tasks (e.g., “48 6_,”
“240 80 _”), as well as combinations of these (e.g., “16 47
_”). There were 4 tasks at the first measurement point, 8 tasks at the
second, 18 at the third, 28 at the fourth, and 42 at the fifth and sixth. The
children were asked to do as many of them as they could.
In the diagnostic test, one point was given for each correct answer.
Consequently, the total maximum score for the test was 19 for the first
measurement (Time 1), 34 for the second (Time 2), 44 for the third (Time
3), 58 for the fourth (Time 4), and 76 for the fifth (Time 5) and sixth (Time
6). Preliminary examination of the distributions of the test scores showed
that at none of the measurement points were there any ceiling or floor
effects.
The test–retest reliability for the test was .94. The respective correlations
of the diagnostic test score with the teacher ratings of the children’s math
performance (5-point scale) at the six measurement points were, in the
present sample, .60, .63, .66, .67, .72, and .71, respectively. At Measure-
ment Points 5 and 6, the test score correlated with the Finnish arithmetic
reasoning test (Ra¨sa¨nen, 2000), with correlations of .75 and .66,
respectively.
Cognitive Antecedents
Counting ability. Children’s counting ability was tested at the begin-
ning of their preschool year (Time 1) using the following four subtests
taken from the Diagnostic Tests for Metacognitions and Mathematics
(Salonen et al., 1994):
1. Counting numbers. The child was asked to count as far as he or she
could. If the child reached 50, the test was stopped. The subtest was scored
dichotomously. One point was given if the child could count without any
mistakes from 1 to 50 and 0 points if the child made mistakes or did not
reach 50.
2. Counting forward. The child was given a starting number and asked
to count forward from it. One point was given if the child correctly counted
forward at least four numbers. The subtest consisted of four different tasks
(starting values: 3, 8, 12, and 19).
3. Counting backward. The child was given a starting number and asked
to count backward from it. One point was given if the child correctly
counted backwards at least four numbers. The subtest consisted of four
different tasks (starting values: 4, 8, 12, and 23).
4. Counting forward by number. The child was asked four questions
designed to assess knowledge of the ordinal aspects of numbers (e.g.,
“What is the number you get when you count five numbers on from two?”).
One point was given for each correct answer.
A sum score for counting ability was constructed by summing the scores
of the four subtests (maximum: 13 points). The Cronbach alpha reliability
for the test was .79. The test–retest reliability for the test was .94, and
split-half reliability was .90.
Visual attention. Children’s visual attention was assessed using a vi-
sual attention subtest of the NEPSY test battery (Korkman, Kirk, & Kemp,
1997). The NEPSY is a comprehensive instrument designed to assess
neuropsychological development in preschool and school-age children. It
has been standardized in Finland (Korkman et al., 1997), the U.S.A.
(Korkman, Kirk, & Kemp, 1998), and Sweden (Korkman, Kirk, & Kemp,
2000) for children aged 3–12 years.
The visual attention subtest from the attention/executive functions do-
main was used in this study. Its aim is to measure a child’s ability to
maintain selective visual attention by assessing the speed and accuracy
with which a child can scan an array of targets and locate them. In the test,
the child was asked to go through a set of pictures (a total of 100 pictures)
and mark the target pictures (a total of 20 pictures) as quickly and
accurately as possible. The test consisted of pictures of three different
objects (cat, flower, tree), and the child’s task was to identify and mark
with a pencil every picture in which a particular image (cat) appeared. A
score for visual attention was computed by subtracting the total number of
wrong answers from the number of correct answers (maximum: 20) and
dividing this by the time it took for the child to finish the task. The time
limit was 3 min, after which the test was automatically terminated. The
Cronbach’s alpha for this test has been shown to vary between .73 and .81
(Korkman et al., 1997).
Metacognitive knowledge. The Metacognitive Knowledge Test (MKT)
of the Diagnostic Tests for Metacognitions and Mathematics test battery
(Salonen et al., 1994; see also Annevirta & Vauras, 2001) consisted of four
analogous tasks. In each of the four tasks, the child was shown either two
or three cards. Each card depicted a different boy or girl using a different
approach to learning a task. Next, the child was asked to point out the card
in which the boy or girl was depicted using the most effective learning
method (e.g., “Look, three girls have been given five pictures each. The
pictures tell a story about a polar bear. Now, each girl is doing different
things [three different pictures]. Which of the girls do you think learns the
story best? 1. Is it the girl who puts the pictures on a table, and makes a
beautiful circle out of them, and then admires the pictures? 2. Is it the girl
who puts the pictures on a table in a line and then looks at the pictures
carefully one by one? 3. Or is it the girl who puts the pictures on a table,
and then arranges them into a story, and finally looks at the pictures for a
little while?”). Finally, after making a choice, the child was asked to
explain the reasons for selecting that particular picture. The explanations
given by the child were written down by the experimenter.
The scoring of the MKT consisted of two procedures. First, two separate
scores were calculated for each of the four tasks. The first score (MKT1)
was calculated on the basis of the child’s choice of pictures. The alternative
pictures provided scores from 1 to 3 depending on the effectiveness of the
learning strategy described in them (1 the least effective way to learn,
2a moderate way to learn, 3the most effective way to learn).
The second score (MKT2) was based on the analysis of the explanation
given by the child. To calculate this score, we content-analyzed the
explanation given by the child into four categories: (a) trivial or naive
explanation, (b) implicit, slight reference to his or her own mental pro-
cesses, (c) fairly adequate reference to his or her own mental processes, and
(d) more explicit explanations referring to relevant mental processes. The
reliability of the content analysis, measured by the rate of agreement
between three independent raters (kappas), ranged between .60 and .92.
Second, the final score for the MKT was calculated as follows. Each
combination of the MKT1 and MKT2 scores (four combinations) was
given a weighted score, following the recommendations of the developers
of the test (Salonen et al., 1994). The weighted score was determined both
by the choice of the picture and by the quality of the explanation, that is,
how well the child was able to connect the choice of the picture to mental
processes. The scores were weighted so as to give more weight to the
explanations than to the choice of the right picture. For example, children
who made a wrong choice of picture but who provided a good explanation
were given a higher score than those who made the right choice but who
could not give an adequate explanation. This was consistent with the
theoretical idea that metacognitive knowledge in particular reflects under-
standing of cognitive processes (Flavell, 1976). Finally, these scores were
summed across the four tasks to form the final sum score for metacognitive
knowledge.
The Cronbach alpha reliability for the entire instrument has been shown
to be .80 among Finnish preschoolers (Annevirta & Vauras, 2001). The
test–retest reliability for the test was .72 in the present sample. A similar
test for metacognitions has previously been used by Swanson (1990).
Listening comprehension. Children’s listening comprehension abilities
were measured with a listening comprehension test (Korpilahti, 1998). In
this test, the child was shown three pictures and, at the same time, read a
sentence (e.g., “A girl is reading a recipe in order to know how to bake a
cake”). Then, the child was asked to choose the picture that best matched
the meaning of the sentence. There were a total of 10 items. One point was
703
DEVELOPMENTAL DYNAMICS OF MATH PERFORMANCE
given for each correct answer. The maximum score for the test was thus 10.
Children’s performance in the Listening Comprehension Test has been
shown to correlate .59 with their performance in the Auditory Reasoning
test from the Finnish version of the Illinois Test of Psycholinguistic
Abilities (Kuusinen & Blåfield, 1974).
Results
The statistical analyses were performed in six steps. First, a
simplex model for the observed math performance variables was
tested to examine the stability of math performance and changes in
its variance over time (Jo¨reskog, 1970). Second, latent growth
curve modeling was used to investigate the growth rate of math
performance and the association between the level of math per-
formance and its developmental trend across time (Duncan, Dun-
can, Strycker, Li, & Alpert, 1999; B. O. Muthe´n & Khoo, 1998).
Third, individual differences in the level and the rate of growth of
math performance were predicted by different cognitive anteced-
ents and sex. Fourth, growth mixture modeling (GMM; B. O.
Muthe´n, 2001a, 2001b; B. Muthe´n & Muthe´n, 2000) was applied
to examine the extent to which heterogeneity existed in develop-
mental trajectories, that is, whether there were naturally occurring
homogeneous groups of children that differed according to the
children’s level and growth rate of math performance. GMM both
estimates mean growth curves for each trajectory class and cap-
tures individual variation around these growth curves by estimat-
ing the growth factor variances for each class (B. Muthe´n &
Muthe´n, 2000). Fifth, class membership was predicted using cog-
nitive antecedents and sex as covariates. Finally, to examine
whether the antecedent variables might be different in the different
trajectory classes, we used cognitive antecedents and sex to predict
the level and growth in math performance within classes.
All the analyses were performed within a structural equation
modeling framework using the Mplus statistical package (Version
2.13; L. K. Muthe´n & Muthe´n, 1998 –2003). Goodness of fit was
evaluated using four indicators:
2
/df, Bentler’s (1990) compara-
tive fit index (CFI), the Tucker-Lewis index (TLI), and the stan-
dardized root-mean-square residual (SRMR). The sample correla-
tion and covariance matrices and the means for the observed
variables are shown in Table 1.
Simplex Model
First, to examine the stability of math performance and its
variance across the measurements, we created a simplex model for
the math performance construct across Time 1 to Time 6. In this
model, the latent math performance constructs each consisted of
one indicator, that is, the observed math performance variable. To
end up with an identifiable model, we estimated the measurement
error variances of the observed math performance variables at
Time 1 and Time 2 as equal and, similarly, the measurement error
variances of observed math performance at Time 5 and Time 6 as
equal. Because the residual term of the latent math performance
construct at Time 2 was found to be negative, this parameter was
further fixed at 0. The fit of the model was good,
2
(7, N194)
9.87, p.20, CFI 1.00, TLI 0.99, SRMR 0.01. The results
are shown in Figure 1.
The results show, first, a high stability in math performance
across the six measurements. Second, the results reveal that the
variance of the latent math performance constructs (see Figure 1)
Table 1
Sample Correlation (Below the Diagonal) and Covariance (Above the Diagonal) Matrixes Between Manifest Variables and Their Means (n 194)
Observed variable 1234567891011M
1. Math performance: Measurement 1 20.31 24.95 22.96 21.65 24.93 25.35 11.59 0.11 6.44 3.00 0.25 12.27
2. Math performance: Measurement 2 .81*** 46.92 37.42 34.29 40.77 43.91 16.91 0.17 9.85 3.72 0.32 22.71
3. Math performance: Measurement 3 .72*** .77*** 50.02 42.61 49.78 49.61 18.72 0.23 9.11 3.76 0.48 28.75
4. Math performance: Measurement 4 .62*** .65*** .78*** 59.23 61.98 58.77 19.34 0.18 7.47 3.64 0.09 40.66
5. Math performance: Measurement 5 .58*** .63*** .74*** .85*** 89.72 76.05 22.34 0.20 7.73 4.79 0.01 48.19
6. Math performance: Measurement 6 .58*** .66*** .72*** .79*** .83*** 94.70 21.59 0.22 8.90 4.08 0.00 55.97
7. Counting ability: Measurement 1 .72*** .68*** .74*** .70*** .66*** .62*** 12.86 0.07 4.09 1.70 0.15 6.82
8. Visual attention: Measurement 1 .24*** .24*** .31*** .22** .20** .22** .18* 0.01 0.07 0.05 0.02 0.30
9. Metacognitive knowledge:
Measurement 1
.45*** .45*** .40*** .30*** .25*** .29*** .36*** .21** 10.31 1.86 0.37 12.01
10. Listening comprehension:
Measurement 1
.38*** .31*** .30*** .27*** .29*** .24*** .27*** .24*** .33*** 3.08 0.16 6.69
11. Sex
a
.12 .09 .14* .02 .00 .00 .08 .33*** .23** .19** 0.25 1.53
a
1girl, 2 boy.
*p.05. ** p.01. *** p.001.
704 AUNOLA, LESKINEN, LERKKANEN, AND NURMI
increased throughout the measurements. Overall, these results sug-
gest that the development of math performance showed a cumu-
lative pattern across time.
The results concerning the mean structure of math performance
(see Figure 1) show that the changes in the level of math perfor-
mance across time were all positive and statistically significant,
with one exception: children’s math performance improved, on
average, throughout the study period except for the period from
Time 4 (end of Grade 1) to Time 5 (beginning of Grade 2).
Latent Growth Curve Model
To investigate the growth dynamics of math performance and
the strength of any association between the level of math perfor-
mance and its developmental trend, we created a latent growth
curve model for the math performance measurements across Time
1 to Time 6.
First, a model with two growth factor components, that is, (a)
the intercept growth factor (Level) and (b) the linear growth rate
(Slope), was estimated. The model was constructed by fixing the
loadings of the observed math performance variables across Time
1 to Time 6 to 1 on the intercept factor (Level) and to 0, 1, 2, 3,
4, and 5 on the Slope factor. The residual variances of the observed
math performance variables were first allowed to be freely esti-
mated. However, because the residual variance of observed math
performance at Time 1 was found to be negative, this was further
fixed to 0.
The fit of the model was
2
(17, N194) 166.06, p.001,
CFI 0.86, TLI 0.88, SRMR 0.10. Inspection of the
modification indices suggested that freeing the Slope factor load-
ings would improve the fit of the model. Consequently, another
model was constructed. In this model, the first loading on the
Slope factor was set at 0 and the last at 1, and the loadings between
these two time points were allowed to be estimated. By so doing,
the slope was defined as the change from Time 1 to Time 6, that
is, the mean of the slope was the mean difference between Time 1
and Time 6 (L. K. Muthe´n & Muthe´n, 2001). Moreover, on the
basis of the modification indices, the estimated residual terms
between the measurements at Time 2 and Time 3, between Time
3 and Time 4, and between Time 4 and Time 5 were allowed to
correlate. The fit of this model was good,
2
(10, N194)
21.98, p.02, CFI 0.99 TLI 0.98, SRMR 0.06. The
results of this final model are shown in Figure 2.
The mean of the Level of math performance at the beginning
(Measurement 1) was positive and statistically significant (M
12.29, SE 0.32, p.001), as was the mean of the Slope
(average rate of growth: M43.75, SE 0.59, p.001). The
results reveal that the variance of Level (20.22, p.001) and the
variance of the Slope (54.48, p.001) were both statistically
significant, indicating that there were significant individual differ-
ences in these two growth components. The results (see Figure 2)
show further that the covariance between latent Level and Slope
was positive and statistically significant (cov 5.48, SE 2.58,
p.05, r.17). This result suggests that the development of
math performance across the six measurements shows a cumula-
tive pattern: The higher the initial level of math performance, the
faster its rate of growth from preschool to Grade 2. Conversely, the
Figure 1. Simplex model of math performance. Var(
1
) to Var(
6
)variances of the latent math performance
constructs at different measurement points;
the estimated mean performance level at Time 1;
1
to
5
the
changes in the level of the latent math performance (standard errors in parentheses).
705
DEVELOPMENTAL DYNAMICS OF MATH PERFORMANCE
lower the initial level of math performance was, the less improve-
ment there was. Overall, both the simplex and the latent growth
curve models suggest that the development of math performance
showed a cumulative pattern.
Predicting Level and Growth in Math Performance
The next aim of the study was to investigate the extent to which
various cognitive variables, that is, counting ability, visual atten-
tion, listening comprehension, metacognitive knowledge, and sex,
would predict the Level and Slope components of mathematical
performance. Therefore, the predicting variables were added to the
previous model as covariates. The fit of the model was good,
2
(30, N194) 59.71, p.001, CFI 0.98, TLI 0.97,
SRMR 0.05. After omitting nonsignificant paths, the fit of the
model was
2
(34, N194) 62.06, p.002, CFI 0.98, TLI
0.97, SRMR 0.05. The results are shown in Figure 3.
The results show (see Figure 3), first, that the Level of math
performance was predicted by three of the predicting covariates:
The higher the levels of counting ability, metacognitive knowl-
edge, and listening comprehension children showed at the begin-
ning of the preschool year, the higher was their level of math
performance. Second, two of the cognitive antecedents predicted
the Slope (i.e., rate of growth) of math: The higher the level of
counting ability and visual attention that children had at the be-
ginning, the faster the rate of growth they showed in math. Third,
the child’s sex also contributed to the rate of growth in math
performance: Boys showed a faster rate of growth than girls.
Further examination of the sex differences showed that there
were no mean differences between boys and girls in observed math
performance variables at any of the measurement points. At each
of the measurement points, there was, however, more variance
among boys than among girls in these variables.
Growth Mixture Modeling
Next, we investigated whether different groups of children could
be identified on the basis of the developmental trajectories they
showed across the six measurements of their math performance.
To identify different developmental trajectories, we performed
mixture models with different numbers of latent classes. In these
analyses, the latent trajectory classes are formed on the basis of the
growth factor means (i.e., the means of Level and Slope) so that
each class defines a different trajectory over time (B. O. Muthe´n,
2001b). Moreover, the variances of growth factors and the covari-
ance between them are allowed to vary between classes.
As recommended by B. Muthe´n and Muthe´n (2000) and B. O.
Muthe´n (2001a, 2001b, 2003), three different criteria were used to
decide about the number of latent classes: (a) the fit of the model
as evaluated by the Bayesian information criterion (BIC; Schwartz,
1978) statistics (the lower the BIC value, the better the model) and
the Vuong-Lo-Mendell-Rubin (VLMR) test of fit (compares solu-
tions with different numbers of classes; a low [.05] pvalue
indicates that the k1 class model has to be rejected in favor of
a model with at least kclasses), (b) the classification quality that
can be determined by examining the posterior probabilities and
entropy values (entropy values range from zero to one, with values
close to one indicating a clear classification), and (c) the usefulness
and interpretativeness of the latent classes in practice (e.g., the
number of individuals in each class, the number of estimated
parameters).
The results show that the BIC indices strongly supported a
two-class solution. The fit of this solution (BIC 6,928.88) was
better than that of either three-class (BIC 6,954.27) or four-class
(BIC 6,993.42) solutions. The VLMR test suggested that a
two-class solution was better than a one-class solution (3,430.52,
p.01). According to the VLMR, a three-class solution would
also fit the data well (3,415.94, p.01). However, because the
third group contained only 13 children and because the BIC value
supported a two-class solution, the two-class solution was selected
for further analysis. The results of this final model, that is, the
estimated values for the latent growth components, their variances,
and estimated class probabilities for each latent class, are shown in
Table 2.
Figure 2. Latent growth curve model for math performance. *Fixed.
706 AUNOLA, LESKINEN, LERKKANEN, AND NURMI
The first class consisted of 73 children who were typified by a
high level of math performance and a high and positive rate of
growth. This group was labeled high performers. The second class
consisted of 121 children characterized by a low level of math
performance and a positive but lower (than Group 1) level of
growth rate. This group was labeled low performers. The variances
of the level and the growth rate were statistically significant in
both groups (see Table 2). Overall, the results accord well with
those of the simplex and latent growth curve models, suggesting a
cumulative pattern of math performance. That is to say, there were
two groups of children in the sample: those who performed well in
math at the beginning and who showed a fast rate of growth and
those who started at a lower level and who showed a slower rate
of growth.
Predicting Class Membership
Next, cognitive predictors, that is, counting ability, visual atten-
tion, metacognitive knowledge, listening comprehension, and sex,
were added to predict children’s group membership (two-class
solution). The results show that two of the cognitive antecedents,
plus sex, predicted children’s class membership: The higher the
Figure 3. Predictors of latent Level and Slope of math performance (standardized estimates). *p.05. **p
.01. ***p.001.
Table 2
Estimated Means and Variances for the Level and Slope of Math Performance in the Different
Classes (Two-Class Solution) and Estimated Class Probabilities and Class Sizes (Standard
Errors in Parentheses)
Class
Math performance
Probability
Level Slope
MVariance MVariance
Class 1: High performers 16.00 (0.30) 2.10 (0.67) 48.81 (0.94) 31.72 (7.59) .38 (n73)
Class 2: Low performers 10.40 (0.41) 18.88 (2.00) 41.14 (0.73) 43.60 (7.60) .62 (n121)
Whole Data 12.29 (0.32) 20.22 (2.05) 43.75 (0.59) 54.48 (6.38)
707
DEVELOPMENTAL DYNAMICS OF MATH PERFORMANCE
level of counting ability (estimate 1.56, SE 0.47, p.01) and
visual attention (estimate 16.28, SE 7.12, p.05) that
children showed at the beginning, the more likely they were to be
in the high performers group. Moreover, being a boy increased the
likelihood of membership in the high performers group (esti-
mate 2.66, SE 1.10, p.05).
Predicting Growth Within Classes
Finally, we were interested in examining the extent to which the
development of mathematical competence might have different
antecedents depending on the trajectory class (i.e., group member-
ship). Consequently, we tested a model in which antecedent vari-
ables of level and slope were allowed to vary between the two
groups. The results are shown in Table 3.
The results show, first, that among the high performers, the level
of math performance was predicted by visual attention and meta-
cognitive knowledge, whereas the slope was predicted by counting
ability and sex. Second, among low performers, the level of math
performance was predicted by counting ability, and the slope was
predicted by counting ability and visual attention.
Because the VLMR test suggested that a three-class solution
would also fit the data, this is also briefly described. The results
concerning the first two groups are highly similar to those found in
the two-class solution. The third group (n13) differed from
these two groups in showing a lower initial level of math perfor-
mance (M3.64) and slower rate of growth (M37.91),
suggesting that this third group was the most disadvantaged group
in terms of math performance, that is, potentially a learning dis-
abled group. The fact that the levels of the predictor variables were
systematically lower in this group (mean for counting skills was
0.43, for listening comprehension was 5.12, for visual attention
was 0.21, and for metacognitive knowledge was 8.93) than in the
other two groups further supports this notion.
Discussion
Although a considerable amount of research has been carried
out on children’s mathematical performance, most of it has fo-
cused on math-related learning difficulties, clinical samples, or
certain subcomponents of math performance (Geary, 1993). The
present study made an effort to complement the understanding of
the development of children’s math performance by examining the
developmental dynamics of overall math performance among a
nonselected sample of children during their transition from pre-
school to Grade 2. Similarly, the cognitive antecedents of this
development and the heterogeneity observable in math develop-
mental trajectories were examined. The results show that individ-
ual differences in mathematical performance grew larger as the
children moved from preschool to primary school. Moreover, the
growth of math competence was faster among those children who
entered preschool already equipped with a higher level of mathe-
matical skills and was slower among those whose initial level of
skills was lower. Both the initial level of math performance and its
growth were predicted by children’s counting ability.
Development of Math Performance
The first aim of the present study was to investigate the devel-
opment of math performance among a normal, nonselected popu-
lation of children during their transition from preschool to Grade 2.
The results show, first, that the development of mathematical
performance during preschool and the first 2 years of primary
school followed a cumulative pattern: Individual differences in
math performance were stable from the very beginning of the
preschool year to the end of Grade 2; they also grew larger across
time, and initially high-performing children showed a greater
increase in performance compared with those who started from a
lower level. These results are in accordance with previous studies
carried out among older school pupils and adolescents, which have
shown that individual differences in mathematical achievement
increase across time (B. O. Muthe´n & Khoo, 1998; Williamson et
al., 1991). The present study adds to the earlier literature by
showing that the accumulation of math performance begins at
preschool before formal instruction in math has started.
The GMM analyses provide further evidence for these findings.
There appeared to be two groups of children: those who started out
with a high level of math skills and who showed an accelerating
developmental trajectory and those who started out with a low
level of performance and who showed relatively less rapid devel-
opment. These results add to previous research in which learning
disabled children have been compared with normally achieving
children (Jordan et al., 2002, 2003) by showing that a widening of
individual differences in math performance also occurs among a
nonselected population of children.
Cognitive Antecedents
The second aim of the present study was to examine the extent
to which different cognitive antecedents would predict the level
and growth of children’s math performance. The results show,
first, that the level of counting ability at the beginning of preschool
predicted both the children’s math performance level and growth
rate: The higher the level of counting ability children showed at the
beginning of the preschool year, the higher was their level of math
performance and the faster the rate of growth they showed in their
subsequent skill development. Previous results from cross-
sectional and experimental studies have shown that counting abil-
Table 3
Estimated Standardized Regression Coefficients of the
Covariates for Level and Slope Within Classes
Covariate
High performers
(n57)
Low performers
(n136)
Level Slope Level Slope
Counting ability .58** .86*** .38**
Visual attention .29** .20*
Metacognitive knowledge .69*** — — —
Listening comprehension ————
Sex — .46*
R
2
.60 .50 .78 .19
Note. Because, in growth mixture modeling, information concerning
covariates contributes to class formation, the estimated class sizes changed
slightly in the model in which covariates were included (Table 3) compared
with a model without covariates (Table 2). Dashes indicate that there were
no predictions obtained from the particular covariate.
*p.05. ** p.01. *** p.001.
708 AUNOLA, LESKINEN, LERKKANEN, AND NURMI
ity is associated with performance in mathematics and that chil-
dren with learning disabilities in math show deficiency in their
counting abilities (Geary, 1993; Geary et al., 1992; Ohlsson &
Rees, 1991). The results of the present study contribute to an
understanding of the development of math performance by show-
ing that counting ability not only is associated with the simulta-
neous level of math performance but also predicts subsequent
growth in it.
There are at least three possible explanations for these results.
First, they may be due to the fact that mathematical ability devel-
ops in a hierarchical manner: To be able to handle more complex
mathematical tasks, lower level skills and procedures, such as
counting, have had to be learned and automatized (Entwisle &
Alexander, 1990; Resnick, 1989). For example, it has been sug-
gested that children’s basic conceptual understanding of how to
count objects and their knowledge of the order of numbers play an
important role in math performance because they promote the
automatic use of math-related information, allowing attentional
resources to be devoted to more complex mathematical problem
solving (Gersten & Chard, 1999; Resnick, 1989). The second
explanation for the results of the present study comes from
Siegler’s (1986; Lemaire & Siegler, 1995) strategy choice model.
According to this model, young children first use counting-based
procedures to solve arithmetical problems. Good counting skills
lead to increasingly accurate answers and thus foster not only a
movement to automatized fact retrieval but also accuracy in prob-
lem solving. By contrast, because repeated counting errors
strengthen the associations between incorrect solutions and the
specific problem, poor counting skills may lead to difficulties in
inhibiting the retrieval of irrelevant associations. The third expla-
nation is that because counting involves number words and the use
of the basic phonetic system, low counting ability may reflect a
broader spectrum of learning disabilities, which have been shown
to lead to slow progress in math performance (Jordan et al., 2002,
2003).
Second, the results of the present study show that children’s
level of metacognitive knowledge was related to their level of
math performance: The higher the level of metacognitive knowl-
edge at the beginning of the preschool year, the higher the initial
level of math performance. One explanation for this result is that
metacognitive knowledge reflects pupils’ abilities to adjust appro-
priately to varying problem-solving tasks and demands and there-
fore lays the basis for their math performance. Swanson (1990), for
example, showed that pupils with high metacognitive skills out-
performed those with lower metacognitive skills in problem solv-
ing. The results show further, however, that metacognitive knowl-
edge did not predict the growth of math performance. These results
suggest that although effective learning strategies, such as meta-
cognitions, may support the use of already acquired knowledge
and skills in a flexible way (Geary, 1990; Lucangeli et al., 1997),
they may be less important in the acquisition of new skills.
Similarly, Geary (1990) suggested that although the coordination
of fundamental processes in math might initially require metacog-
nitive processes, as the system becomes automatized, mindful
self-regulation (i.e., metacognitions) is no longer necessary.
The results show further that listening comprehension also con-
tributed to the level of math performance: The higher the level of
listening comprehension at the beginning of the preschool year, the
higher was the initial level of math performance. Previously, the
ability to handle verbal information has been shown to be associ-
ated with performance in story problems (Jordan et al., 2002). The
results of the present study add to the previous literature on the
topic by showing that overall math performance, too, is associated
with listening comprehension. In the present study, listening com-
prehension did not, however, contribute to the growth of math
performance. This result suggests that, like metacognitions, listen-
ing comprehension may be effective in supporting the use of
already acquired mathematical knowledge and skills but that it
does not contribute to the acquisition of new math skills.
The results show further that attentional resources, operational-
ized here as visual attention, were related to the rate of growth in
math performance: The higher the level of visual attention at the
beginning of preschool, the faster the subsequent rate of growth in
math performance. Previous results have shown that problems with
executive control and attentional allocation are related to learning
disabilities in mathematics (Ackerman et al., 2001; Geary et al.,
1999; McLean & Hitch, 1999). The present study contributes to
these findings by showing that visual attention also predicts an
increase in math performance. There are two possible explanations
for these results. First, Geary (1990) suggested that this association
is due to the fact that attentional resources and executive functions
have an impact on the ways in which children initiate and direct
their cognitive processing in different mathematical tasks and on
retrieval representations of to-be-remembered information. For
example, a high level of attentional ability, as a form of executive
control, may facilitate finding the right schemas for the tasks at
hand and, thus, proper problem-solving strategies. A low level of
attention, on the other hand, may lead to less effective problem-
solving strategies because of mental activation of the wrong sche-
mas. Another possibility is that the association between visual
attention and the development of math performance is due to a
third, shared factor, such as working memory capacity (McLean &
Hitch, 1999) or general cognitive ability (Conway, Cowan,
Bunting, Therriault, & Minkoff, 2002). For example, working
memory deficiencies have consistently been shown to be associ-
ated with low math performance and related disabilities (Geary,
1990, 1993). The present finding, that visual attention predicted
the rate of growth in math rather than its initial level, may have
been due to the fact that performing well in increasingly complex
mathematical tasks requires more attention and working memory
capacity overall than doing well in simpler tasks.
The results also show that the cognitive antecedents of math
performance varied depending on the stage of skill development
shown by the children. Among initially high-performing children,
the level of math performance was predicted by visual attention
and metacognitive knowledge, whereas the trend was predicted by
counting ability. Among initially low-performing children, the
level of math performance was predicted by counting ability, and
the trend was predicted by visual attention, in addition to counting
ability. These results suggest that the role of attentional resources
such as visual attention may be particularly important during the
early phases of development, when basic skills are to be learned
and automatized, but less so in the later phases, when the processes
of problem solving have become automatized. Counting ability, in
turn, seems to be a powerful predictor of math development
independent of the child’s stage of development; in other words,
both low- and high-achieving children benefit from initially good
counting abilities. Because previous studies on the antecedents and
709
DEVELOPMENTAL DYNAMICS OF MATH PERFORMANCE
correlates of math performance have mainly been cross-sectional
or based on a comparison of children with learning disabilities and
normally achieving children, they have not made it possible to
distinguish between the effects of independent factors on the level
and the effects of these factors on the trend. The results of the
present study suggest that the level of math performance and its
trend are in part predicted by different factors. Moreover, these
predictors are different depending on the phase of development of
math performance.
Sex
The results of the present study show no sex differences in the
level of children’s math performance. This result accords well with
the earlier literature showing that during the elementary school
years, there are no sex differences in math performance (Friedman,
1989; Hyde et al., 1990; Skaalvik & Rankin, 1994). The results of
the present study show, however, that the rate of growth was faster
among boys than among girls (see also Leahey & Guo, 2001; B. O.
Muthe´n & Khoo, 1998). The results show further that the faster
growth rate of boys was particularly evident among high-
performing children (Benbow, 1988; Friedman, 1989; Geary,
1996; Mills et al., 1993). Overall, the results of the present study
suggest that sex differences in math development start to emerge
during the first 2 years of primary school, particularly among
high-achieving children (see also Geary, 1996). One possible
explanation for these sex differences is that at some point during
the first years of school, boys start to show a higher level of
math-related motivation and self-concept of ability compared with
girls (Eccles et al., 1983; Eccles, Adler, & Meece, 1984; Eccles,
Wigfield, Harold, & Blumenfeld, 1993), reflecting, for example,
the socializing effects of parents and teachers (Eccles, Janis, &
Harold, 1990). This may be particularly true among high-
achieving children because demanding mathematical tasks require
more motivation and related effort than does the learning of basic
skills. Another explanation is that because there is more variance
in math performance among boys than among girls, it follows that
among high-achieving pupils, boys are more likely to outperform
girls (Geary, 1996; Royer, Tronsky, Marchant, & Jackson, 1999;
Willingham & Cole, 1997). No sex differences were found, how-
ever, among the low-performing children.
Pedagogical Implications
The results of the present study have at least four pedagogical
implications. First, the result showing that math performance de-
velops in a cumulative manner suggests that there might be a need
to focus more attention on learning math and related skills at
preschool age or even earlier to promote children’s future math
performance. For example, the major focus in the first 2 years of
primary school in Finland has been on developing literacy skills
rather than mathematics. Mathematics has usually received much
less attention. The results of the present study suggest that children
may benefit in the longer run from extra investment in teaching
math from the very beginning of preschool and primary school.
Second, the fact that children with initially low skills remained
increasingly behind those with high skills suggests that children
with different levels of skills might also benefit from a different
curriculum and methods of instruction. For example, Geary (1995)
suggested that the automatization of basic mathematical facts
needs drill and practice in addition to constructivist-based instruc-
tion, rather than constructivist-based instruction alone. In other
words, mathematical procedures should be practiced until the child
can automatically execute the procedure with different types of
problems. This may have particular importance for children per-
forming poorly in math. Only after this kind of instruction has been
given can attentional and working memory resources be devoted to
more complex features of the specific math problem. By contrast,
those who already know basic mathematics may benefit from a
focus on the acquisition of new conceptual competencies and
arithmetical knowledge rather than being limited to the practice of
basic procedures.
Third, the results of the present study show that the initial level
of counting ability, in particular, was a strong predictor of both the
level of math performance and its subsequent development. This
result suggests that supporting the development of counting abil-
ities—for example, by focusing on numeracy in preschool—might
assist children to perform better at math later on. It is possible, for
instance, that practicing counting skills and acquiring fluency in
them during the preschool year may be an effective strategy for
increasing children’s later accuracy in fact retrieval. Siegler (1986)
has suggested previously that children with good counting skills
are able to end up with the correct solutions to various mathemat-
ical problems, which also diminishes the possibility that a partic-
ular problem becomes associated with incorrect answers.
Fourth, the findings of the present study that metacognitive
knowledge and listening comprehension were associated with the
level of math performance and that visual attention skills predicted
its development, particularly among low achievers, suggest that
there is a need to focus, aside from math itself, also on more
general skills, such as metacognitions and attention, when seeking
to improve the learning of math. For example, Swanson (1990) has
suggested that high metacognitive skills can even compensate for
overall low ability because they lead to more efficient information
processing and the monitoring of right and wrong answers. This
might equally be true for listening comprehension and visual
attention skills. Supporting skills of this kind when teaching math
may increase the level of performance, thereby also promoting
subsequent progress in mathematics.
Limitations
There are at least four limitations that should be taken into
account in any attempts to generalize the findings of the present
study. First, the set of cognitive antecedents of math performance
included in the study was limited and did not include, for example,
memory span (Geary, 1993) and processing time (Bull & Johnston,
1997; Pellegrino & Goldman, 1987). Second, the present study
focused on the role of cognitive variables in children’s math
performance. Other factors not taken into account in this study,
such as motivational variables (Aunola, Leskinen, & Nurmi, 2004;
Vermeer, Boekaerts, & Seegers, 2000) and family factors (Aunola
& Nurmi, 2004; Aunola, Nurmi, Lerkkanen, & Rasku-Puttonen,
2003; Entwisle, & Alexander, 1990; Huntsinger, Jose, Larson,
Balsink-Krieg, & Shaligram, 2000), may also play a central role.
Third, the focus of the present study was on overall math perfor-
mance. However, because the developmental dynamics, the ante-
cedents, and the sex differences (Geary, 1996) may be different
710 AUNOLA, LESKINEN, LERKKANEN, AND NURMI
depending on the subcomponents of math (see, e.g., Jordan et al.,
2003; Leahey & Guo, 2001; Mills et al., 1993), further studies are
needed to understand the developmental differences in subcompo-
nents of math performance. Finally, the present study was carried
out in one particular culture, Finland. Because the structure of
number words (regular vs. irregular), as well as math instruction at
school, varies across cultures, it is possible that different results
might have been obtained in different cultural environments
(Geary, 1995). Moreover, the fact that Finnish children enter
school at a relatively late age may also have influenced the pattern
of findings of the present study.
Conclusion
Overall, the results of the present study suggest that individual
differences in math performance grow larger across time. This
accumulation seems to be already under way during the preschool
year, before formal instruction has begun: Children who entered
preschool with a high level of math skills showed rapid develop-
ment later on, whereas those who started at a lower skill level
showed relatively slower development. Although metacognitive
knowledge and listening comprehension were found to be related
to the level of math performance, early counting ability proved to
be the most powerful predictor of the development of math skills.
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Received November 20, 2003
Revision received March 2, 2004
Accepted April 18, 2004
New Editor Appointed for Journal of Occupational Health Psychology
The American Psychological Association announces the appointment of Lois E. Tetrick, PhD, as
editor of Journal of Occupational Health Psychology for a 5-year term (2006 –2010).
As of January 1, 2005, manuscripts should be submitted electronically via the journal’s Manuscript
Submission Portal (www.apa.org/journals/ocp.html). Authors who are unable to do so should
correspond with the editor’s office about alternatives:
Lois E. Tetrick, PhD
Incoming Editor, JOHP
George Mason University
Department of Psychology, MSN, 3F5
4400 University Drive, Fairfax, VA 22030
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The current editor, Julian Barling, PhD, will receive and consider manuscripts through December
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DEVELOPMENTAL DYNAMICS OF MATH PERFORMANCE
... As such, it is well possible that mastery of higher order basic numerical skill (such as knowledge of cardinal numbers) may influence the development of lower order basic numerical skills (such as number identification; cf. Aunola, Leskinen, Lerkkanen, & Nurmi, 2004). ...
... This indicates that children at this age improve significantly in dealing with quantities within half a year. This is in line with the claim by Aunola et al. (2004), who argued that sharpening the understanding of quantities and their processing is an important developmental step at this age. ...
... As introduced above, we were specifically interested in the interrelations of the different basic numerical skills counting/ countingbased solutions, cardinality understanding, and awareness of structures as well as their developmental influences on each other. It was argued previously that mastery of higher order basic numerical skills may well influence the development of the lower order basic numerical skills (e.g., Aunola et al. 2004). This was substantiated by the findings of the present study. ...
Article
Basic numerical skills underlie children’s numerical development. In this follow-up study, we investigated the associations between the development of counting, cardinality understanding, and awareness of structures in quantities in children at the age of 4;7 and 5;3 years. We assumed that the ability to consider structures in quantities should allow children to capture the cardinality of non-symbolic quantities more efficiently and thus moderate the development of counting and cardinality understanding. We assessed 57 (32 girls) children in a pretest at about four and a half years of age and again in a posttest six months later. Results showed that structured quantities were captured more efficiently than unstructured quantities which might have been counted. Moreover, results substantiated the mediating role of considering structures in quantities on the development of children’s counting and cardinality understanding. As such, this follow-up study indicated beneficial contributions of awareness of structures on the development of counting and cardinality understanding. This may suggest that introducing structured quantities early on in elementary math education might be sensible.
... Some evidence suggests the developmental trajectories in language skills are different for boys and girls in the way that girls mature at an earlier age than boys and that boys increase more than girls around the age they enter formal schooling (Bornstein, Hahn, and Maurice Haynes 2004;Toivainen et al. 2017). Likewise, during the first two years of primary school, boys have shown a faster growth rate than girls in math skills, particularly among high-achieving children (Aunola et al. 2004). A recent Norwegian study also found that girls scored significantly higher than boys in overall literacy skills (including vocabulary and decoding skills) at the start of school (McTigue et al. 2021). ...
... Based on prior findings, we expected girls to have higher vocabulary and math scores in ECEC than boys Bornstein, Hahn, and Maurice Haynes 2004;Brandlistuen et al. 2021;Zambrana, Ystrom, and Pons 2012). We expected girls`advantage in these skills to decrease from ECEC to first grade (Aunola et al. 2004;Bornstein, Hahn, and Maurice Haynes 2004;Toivainen et al. 2017). Finally, we expected that boys might be more affected by maternal education than girls (Autor et al. 2016(Autor et al. , 2019Fan, Fang, and Markussen 2015;Zambrana, Ystrom, and Pons 2012). ...
... Results showed no significant differences between girls`and boys`changes in vocabulary and math skills from ECEC to first grade. Thus, our hypothesis, which was based on previous research indicating girls`advantage in these skills would decrease from ECEC to first grade (Aunola et al. 2004;Bornstein, Hahn, and Maurice Haynes 2004;Toivainen et al. 2017;McTigue et al. 2021), was not supported. As already mentioned, children's vocabulary and math skills were highly stable from ECEC to first grade, and there was practically no variance left to be explained by other factors, such as gender. ...
Article
Full-text available
Parental education and child gender are related to learning and development during childhood and adolescence. The present study investigated the role of mother’s education level and child gender for children’s vocabulary and math skills in Norway. Children’s vocabulary and math skills were assessed in Early Childhood Education and Care (ECEC) centers (Mage= 5.8; N = 243, 49.4% girls) and first grade (Mage= 6.8 years). Results showed that maternal education predicted children’s vocabulary and math skills in a play-based ECEC setting. There was a small gender difference (favoring girls) in math skills but not in vocabulary in ECEC. However, maternal education and gender did not significantly predict the change in vocabulary or math skills from ECEC to first grade, and gender did not moderate the relationship between maternal education and academic skills in young Norwegian children. Implications of these results are discussed.
... In den letzten zwei Jahrzehnten rückte zunehmend die Bedeutung spezifischer mathematischer Vorläuferfertig keiten unter dem Konzept des Number Sense mit unter schiedlichen Operationalisierungen in den Mittelpunkt (Aunio et al., 2006;Berch, 2005;Gersten, Jordan & Flojo, 2005): Angefangen von der Definition des Number Sense als angeborene Fähigkeit zur Unterscheidung von Men gen, von der ausgehend sich eine komplexere Zahlverar beitung im Sinne eines inneren mentalen Zahlenstrahls entwickelt (Dehaene, 1992), über die Zählkompetenz (EinszuEinsKorrespondenz, Kardinalitätsprinzip, Zäh len, Vorgänger und Nachfolger benennen), das Zahlen wissen (Vergleichen von Mengen und numerischen Größen), die Transforma tion von Zahlen (Addition und Subtraktion, Rechnen), das Schätzen von Mengen und das Erkennen von Zahlenmustern (Jordan, Kaplan, Oláh & Lokuniak, 2006;Lambert, 2015) bis hin zur Fähigkeit zur Verknüpfung von Beziehungen und Rechenprinzipien (Berch, 2005;Lambert, 2015). Mehrfach konnte die prädiktive Bedeutung früher Zählkompetenzen im Vor schulalter belegt werden (Aunola, Leskinen, Lerkkanen & Nurmi, 2004;Koponen, Aunola, Ahonen & Nurmi, 2007;Passolunghi, Vercelloni und Schadee, 2007;Zhang et al., 2020). Andernorts erbrachten breitere Variablen Sets (Zählfertigkeiten, Zahlwissen, Vorgänger, Nachfol ger, Vergleiche, nonverbales Rechnen, Addition und Subtraktion, Zahlzerlegung) die beste Prognoseleistung (Jordan, Kaplan, Locuniak und Ramieni, 2007;Jordan, Kaplan, Ramineni & Lokuniak, 2009). ...
... Grundsätzlich geht mit höherer Sensitivität immer eine abnehmende Spezifität einher (Tröster, 2009 (Aunola et al., 2004;Gomm, 2014;Jordan et al., 2007;Krajewski, 2008). Dass eine reduzierte Prädiktoren Anzahl zu ähnlich guten Prognoseergebnissen führt wie ein breiteres VariablenSet wurde bereits mehrfach belegt und hier ein weiteres Mal deutlich (Jordan et al., 2010, Walter, 2016b, Walter, 2020. ...
Article
Full-text available
Zusammenfassung. Hintergrund: Einen wichtigen Baustein im Rahmen der Prävention von Rechenschwierigkeiten stellen Screening-Verfahren dar, die ein individuelles Entwicklungsrisiko zuverlässig und frühzeitig aufzeigen. Die meisten Instrumente zur Prognose solcher Schwächen im Grundschulalter sind überwiegend als vergleichsweise zeitaufwändige Einzelverfahren konzipiert. Das Ziel der vorliegenden Studie ist die Entwicklung und Evaluation eines gruppenbasierten Screening-Verfahrens für den Einsatz am Schulanfang. Methode: Im vorliegenden Beitrag werden die Entwicklung und Evaluation eines Filter-Screenings an einer Stichprobe von insgesamt 174 Erstklässlern beschrieben. Ein breites Variablen-Set aus domänenspezifischen und domänenunspezifischen Prädiktoren der mathematischen Leistung wurde in die Analyse einbezogen. Resultate: Auf der Basis der logistischen Regressionsanalyse konnte ein durch eine Kreuzvalidierung abgesichertes Vier-Variablen- Prognosemodell (Mengenschätzen, Vorgänger benennen, Zahlen lesen, Matrizen-Test) identifiziert werden, das sehr gute AUC-Werte (bis zu >> .90) aufweist. Diskussion: Die Ergebnisse liefern wertvolle Erkenntnisse hinsichtlich der Implementation eines validen und Schuleingangsscreenings als Gruppenverfahren.
... Reading, spelling, and math (i.e., addition and subtraction) skills are essential academic skills from the start of Grade 1 onward. Early reading and spelling skills become relatively stable at the beginning of primary school and continue to improve thereafter (Entwisle et al., 2005;Duncan et al., 2007); the same is true for math skills (Aunola et al., 2004). Conceptually, it has been suggested that in addition to general instructional support, emotional support is also important to meet students' needs for belonging and competence Deci, 2000, 2020), which, in turn, enhances academic skills (see Hamre and Pianta, 2001). ...
Article
Full-text available
Our study aimed to investigate the patterns of children’s relationships with their parents and teachers, the development of these relationships during Grade 1, and respective links to children’s learning (in task persistence and performance). Parents of 350 children answered questionnaires about the quality of their relationships with their children; 25 teachers answered questions about children’s task persistence at school and the quality of their relationships with their students; 350 children completed literacy and math performance tests; and six testers evaluated children’s task persistence when completing those tests. All measures were administered twice: at the start and end of Grade 1. Latent profile analyses found two meaningful child profiles that were similar at the beginning and end of Grade 1: average relationship (89% at T1, 85% at T2) and conflictual relationship (11% at T1, 15% at T2) with parents and teachers. These profiles were highly stable throughout Grade 1, except for 15 children who moved from an average relationship to a conflictual relationship profile. This declining trajectory can be characterized by poor relationships with teachers and low task persistence at the end of Grade 1, although they did not perform any worse than other children. Finally, children exhibiting conflictual relationships with their parents and teachers at the beginning of Grade 1 performed worse on spelling and subtraction tasks and demonstrated lower task-persistent behavior at the end of Grade 1 than those with average (good) relationships with parents and teachers.
... First, both kindergarten and first grade students made significant academic growth in reading and math, but kindergarten students made significantly more growth than first grade students in the current study. During these early years, students are progressing from alphabet knowledge to contextual reading ability (Morris et al., 2003) and from counting to basic computational and problem-solving skills (Aunola et al., 2004). Thus, significant opportunities for growth exist in both subject areas, but perhaps kindergarten students demonstrate more growth due to their concurrent development of attentional regulatory skills (Janvier and Testu, 2007). ...
Article
The purpose of this study was to examine the relationship between learning and creativity in early elementary students using both static and growth achievement scores in reading and mathematics. Participants were kindergarten and first grade students from the Midwestern United States. Initial correlations demonstrated significant positive relationships between students’ performance on the Torrance Test of Creative Thinking –Figural (TTCT-F) and static academic achievement scores in both reading and mathematics, but that same relationship did not exist with academic growth scores. Specifically, when academic growth was examined further using Generalized Additive Models (GAMs), a complex picture emerged, such that grade level (i.e., kindergarten v. first grade) and subscale type (e.g., Fluency v. Originality) influenced the significance and nature of the relationship (i.e., linear v. nonlinear). In general, as students increased in creativity performance, they demonstrated less academic growth. Future work should explore the underlying mechanisms explaining these relationships to better help students leverage their creative abilities for positive academic gains in the classroom setting.
... Previous studies (e.g. Aunola et al. 2004;Jordan et al. 2007;Jordan, Glutting, and Ramineni 2010;Reys and Yang 1998) have indicated that counting and number sense skills are a strong predictor of future maths achievement in kindergarten and elementary school. ...
Article
The aim of this study was to examine the number sense performance of 8th-grade Turkish students. In this study, the students’ performances on number sense were also examined in terms of gender, school location, parents’ education level, maths achievement, perceived student satisfaction with the maths teacher, perceived importance of maths and perceived parental support for maths. Participants of this study consisted of 306 eighth-grade students (13- to 14-year-olds) from the western part of Anatolia in Turkey. The data were collected via the ‘Number Sense Test (NST)’ consisting of 38 multiple-choice questions and six number sense components. Data were analysed by using the Statistical Package for Social Sciences (SPSS) 17.0. The results from this study revealed that most of the students performed poorly on the number sense test and its components. No significant difference was found between female and male performances in number sense, but their performances significantly differed according to school location and parents’ education level. The results of this study also indicated that a significant positive correlation was between the number sense performances of students and their maths achievement, perceived student satisfaction with maths teachers, perceived importance of maths and perceived parental support for maths.
Article
Adaptive serious mathematical games in kindergarten were used to investigate whether kindergarteners could grasp mathematics topics. A pretest–posttest-follow up design with two conditions. (Condition 1 educational kindergarten games on the computer, focusing on counting and comparison, Condition 2 educational kindergarten games on the computer, focusing on memory, counting and comparison) and one active control group (playing educational kindergarten games without mathematical content) was set up dealing with 45 preschoolers with a mean age of 68.78 months ( SD = 4.46). Children were matched in kindergarten on their early mathematical and language skills as well as on their intelligence before the interventions took place. The study revealed that playing mathematical games in kindergarten had the potential to enhance the early mathematical skills. Children with initial weak mathematical skills in kindergarten caught up with their average performing peers, pointing to the importance of serious numerical games as “opportunities” in kindergarten. Both boys and girls benefitted, with a sustained effect in grade 1, revealing promising potential effects of offering opportunities to focus on mathematics even in very young children.
Article
Mathematics skills relate to lifelong career, health and financial outcomes. Individuals’ cognitive abilities predict mathematics performance and there is growing recognition that environmental influences, including differences in culture and variability in mathematics engagement, also affect mathematics performance. In this Review, we summarize evidence indicating that differences between languages, exposure to maths-focused language, socioeconomic status, attitudes and beliefs about mathematics, and engagement with mathematics activities influence young children’s mathematics performance. These influences play out at the community and individual levels. However, research on the role of these environmental influences for foundational number skills, including understanding of number words, is limited. Future research is needed to understand individual differences in the development of early emerging mathematics skills such as number word skills, examining to what extent different types of environmental input are necessary and how children’s cognitive abilities shape the impact of environmental input. Children’s individual abilities and environment influence their mathematics skills. In this Review, Silver and Libertus examine how language, socioeconomic status and other environmental factors influence mathematics skills across childhood, with a focus on number word acquisition.
Article
There is an increased demand for useful measures that capture students' math learning during intervention. Similarly, there is an awareness of the importance of researchers observing guidelines for study quality in publishing intervention results, including information related to outcome measures. We investigated the characteristics of outcome measures to assist researchers and practitioners in selecting appropriate outcome measures for early numeracy interventions. We also explored the level of quality regarding how studies reported outcome measure information. To do this, we analyzed 94 outcome measures of math achievement across 25 kindergarten early numeracy intervention studies. Overall, studies met 84% of the quality indicators related to outcome measures. Fewer studies met the recommendation to include multiple measures (i.e., proximal and distal measures; 60%) or provide validity information about measures (39%). Ultimately, the results of this study provide researchers with valuable information for developing and selecting outcome measures to determine the effectiveness of early numeracy interventions. Overall, the 94 math outcome measures met 84% of the quality indicators. Fewer studies met the recommendation to include multiple measures (i.e., proximal and distal measures; 60%) or provide validity information about measures (39%). Future studies should aim to include validity information as this indicator was the least reported in intervention studies among the seven outcome measure quality indicators. Overall, the 94 math outcome measures met 84% of the quality indicators. Fewer studies met the recommendation to include multiple measures (i.e., proximal and distal measures; 60%) or provide validity information about measures (39%). Future studies should aim to include validity information as this indicator was the least reported in intervention studies among the seven outcome measure quality indicators.
Article
Student–teacher relationships are crucial for adolescents’ adjustment in the school context. The aim of the present study was to examine the role of teacher closeness in academic emotions and achievement among adolescents with and without learning difficulties during the first year in lower secondary school. Students’ learning difficulties (LDs) were identified based on tested reading and math skills. In addition, students evaluated their teacher relationships and rated academic emotions in literacy and math domains. The results indicated that higher teacher closeness was related to increasing positive emotions and increasing literacy achievement during seventh grade, whereas lower levels of teacher closeness were associated with increasing learning-related anger and boredom. The results were mostly similar for students with and without LDs, which indicates that students in general benefit from close teacher relationships during the first year in lower secondary school.
Data
Chapter Data, Program Inputs and Outputs for all LGM Examples in the textbook "An Introduction to Latent Variable Growth Curve Modeling: Concepts, Issues, and Applications, Second Edition". Model specifications are included providing program syntax for Amos, EQS, LISREL, and Mplus software programs. The files are arranged by chapter and include syntax, data, and output files for all examples a particular software program is capable of estimating. The first three chapters (specification of the LGM, LGM and repeated measures ANOVA, and multivariate representations of growth and development) cover the development of the LGM. These are followed by three chapters involving multiple group issues and extensions (analyzing growth in multiple populations, accelerated designs, and multilevel longitudinal approaches), and followed by the chapter on growth mixture modeling, which addresses multiple-group issues from a latent class perspective. The remainder of the book covers 'special topics' (chapters on interrupted time series approaches to LGM analyses, growth modeling with ordered categorical outcomes, Missing data models, a latent variable framework for LGM power analyses and Monte Carlo estimation, and latent growth interaction models). The zipfile is quite large (1MB) since it contains all files for the various software programs.
Conference Paper
Forty European American (EA; 20 girls, 20 boys) and 40 second-generation Chinese American (CA; 20 girls, 20 boys) preschool and kindergarten children (mean age at Time 1 = 5.7 years) and their mothers, fathers, and teachers participated in 3 data collections (1993, 1995, and 1997) to investigate sociocultural and family factors that contribute to children's academic achievement. CA children outscored EA children in mathematics at all 3 times. Initially, EA children outscored CA children in receptive English vocabulary, but CA children caught up to EA children at Time 3. CA children were better readers than EA children at Time 3. According to parental self-reports, CA parents structured their children's time to a greater degree, used more formal teaching methods, and assigned their children more homework. Parents' work-oriented methods and child-specific beliefs at Time 1 influenced children's mathematics performance at Time 3.
Article
This study focused on gender differences in growth in mathematics achievement in relation to various social-psychological factors such as attitude toward mathematics, self-esteem, parents' academic encouragement, mathematics teachers' expectations, peer influence, and so on. The study was based primarily on the data collected by the Longitudinal Study of American Youth (Miller, Hoffer, Suchner, Brown, & Nelson, 1992) and focused on students from Grade 7 to Grade 10. The key methodology used in the study was 3-level longitudinal and multilevel modeling. Results indicated that gender differences in growth in mathematics varied by one's initial status in mathematics. For those who started low, girls started higher than boys, but their average growth rate was slightly lower than boys. Although the average gender gap in growth rate was not statistically significant, the gap varied across schools. In some schools girls' average growth rate was higher, whereas in other schools boys' average growth rate was higher. For those who started high, there were no gender differences in initial status and growth rate. However, the effect of math attitude and math teacher encouragement on mathematics differed for boys and girls who started high. The effect of math attitude on mathematics was stronger for boys than for girls. The effect of math teacher encouragement on mathematics varied across schools for boys, but no math teacher encouragement effect was found for girls. Results also show that home resources, individual behavior problems, and attitude toward mathematics were related to growth in mathematics. In addition, aggregated school resources had a significant effect on growth in mathematics. The effect of math teacher encouragement on mathematics varied across schools for boys and girls who started low as for boys who started high. Implications of these results are discussed.
Article
This study involved administering for two academic years Reading Comprehension and Mathematics tests of a standardized battery to 253 Grade 10 students from one school. Factor matrices using microcomponents of the two tests for boys and girls were highly similar. While reading comprehension was a significant predictor of the problem-solving component for boys, it was not for girls. Also, significant differences in favor of boys were found on all three process components, on all but two content components, and on five microcomponents. Patterns of observed differences were similar for the two years
Article
Differences in boys' and girls' mathematical problem-solving behavior were studied in relation to 2 types of mathematics tasks: computations and applications. Participants were 79 boys and 79 girls of the 6th grade from 12 regular schools. In 2 separate individual sessions, cognitive and motivational variables were examined before, during, and after task execution. Differences in mathematical problem-solving behavior were dependent on the contents of the mathematics tasks and on gender. Interactions between type of task and gender were also noted. With respect to applied problem solving, girls rated themselves lower on confidence than boys and attributed bad results more often to lack of capacity and to the difficulty of the task. No gender differences were observed in relation to computations. Unexpectedly, girls had higher persistence than did boys, but only during applied problem solving. (PsycINFO Database Record (c) 2012 APA, all rights reserved)