Over the past two decades, a quiet revolution in developmental psychology and related fields has demonstrated that children have skills and concepts relevant to mathematics learning that are present early in life, and that most children enter school with a wealth of knowledge and cognitive skills that can provide a foundation for mathematics learning. At the same time, these foundational skills are not enough—children need rich mathematical interactions, both at home and at school in order to be well prepared for the challenges they will meet in elementary school and beyond. (Chapter 4 discusses supporting children’s mathematics at home, and Chapters 5 and 6 discuss children’s mathematical development and related instructional practices.) The knowledge and interest that children show about number and shape and other mathematics topics provide an important opportunity for parents and preschool teachers to help them develop their understanding of mathematics (e.g., Gelman, 1980; Saxe, Guberman, and Gearhart, 1987; Seo and Ginsburg, 2004).
In this chapter we review research on the mathematical development of infants and young children to characterize both the resources that most children bring to school and the limitations of preschoolers’ understanding of mathematics. Because this literature is vast, it is not possible to do it justice in a single chapter. However, we attempt to provide an overview of key issues and research findings relevant to early childhood education settings. These include
What is the nature of early universal starting points? These are generally thought to provide an important foundation for subsequent
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3
Cognitive Foundations for Early
Mathematics Learning
Over the past two decades, a quiet revolution in developmental psy-
chology and related fields has demonstrated that children have skills and
concepts relevant to mathematics learning that are present early in life,
and that most children enter school with a wealth of knowledge and cog-
nitive skills that can provide a foundation for mathematics learning. At
the same time, these foundational skills are not enough—children need
rich mathematical interactions, both at home and at school in order to be
well prepared for the challenges they will meet in elementary school and
beyond. (Chapter 4 discusses supporting children’s mathematics at home,
and Chapters 5 and 6 discuss children’s mathematical development and
related instructional practices.) The knowledge and interest that children
show about number and shape and other mathematics topics provide an
important opportunity for parents and preschool teachers to help them
develop their understanding of mathematics (e.g., Gelman, 1980; Saxe,
Guberman, and Gearhart, 1987; Seo and Ginsburg, 2004).
In this chapter we review research on the mathematical development
of infants and young children to characterize both the resources that most
children bring to school and the limitations of preschoolers’ understand-
ing of mathematics. Because this literature is vast, it is not possible to do
it justice in a single chapter. However, we attempt to provide an overview
of key issues and research findings relevant to early childhood education
settings. These include
• What is the nature of early universal starting points? These are gen-
erally thought to provide an important foundation for subsequent
5
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60 MATHEMATICS LEARNING IN EARLY CHILDHOOD
mathematical development (e.g., Barth et al., 2005; Butterworth,
2005; Dehaene, 1997; but see Holloway and Ansari, 2008, and Rips,
Bloomfield, and Asmuth, 2008, for contrasting views). We examine
two domains that are foundational to mathematics in early child-
hood: (1) number, including operations, and (2) spatial thinking,
geometry, and measurement.
• What are some of the important developmental changes in math-
ematical understandings in these domains that occur during the
preschool years?
• What is the relation of mathematical development to more general
aspects of development needed for learning mathematics, such as the
ability to regulate one’s behavior and attention?
EVIDENCE FOR EARLY UNDERSTANDING OF NUMBER
Preverbal Number Knowledge
Delineating the starting points of knowledge in important domains is a
major goal in developmental psychology. These starting points are of theo-
retical importance, as they constrain models of development. They are also
of practical importance, as a basic tenet of instruction is that teaching that
makes contact with the knowledge children have already acquired is likely
to be most effective (e.g., Clements et al., 1999). Thus, it is not surprising
that infant researchers have been actively mapping out the beginnings of
preverbal number knowledge—knowledge that appears to be shared by
humans from differing cultural backgrounds as well as with other species,
and thus part of their evolutionary endowment (e.g., Boysen and Berntson,
1989; Brannon and Terrace, 1998, 2000; Brannon et al., 2001; Cantlon and
Brannon, 2006; Dehaene, 1997; Dehaene, Dehaene-Lambertz, and Cohen,
1998, Meck and Church, 1983). A large body of research has examined
a set of numerical skills, including infants’ ability to discriminate between
different set sizes, their ability to recognize numerical relationships, and
their ability to understand addition and subtraction transformations. The
study of numerical knowledge in infants represents a major departure
from previously held views, which were heavily influenced by Piaget’s
(1941/1965) number conservation findings and stage theory. These older
findings showed that children do not conserve number in the face of spatial
transformations until school age, and they led many to believe that before
this age children lack the ability to form concepts of number (see Mix,
Huttenlocher, and Levine, 2002, for a review). Although Piaget recognized
that children acquire some mathematically relevant skills at earlier ages,
success on the conservation task was widely regarded as the sine qua non
of numerical understanding.
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COGNITIVE FOUNDATIONS FOR EARLY MATHEMATICS LEARNING
Beginning in the 1960s and 1970s, researchers began to actively ex-
amine early numerical competencies, which led to a revised understanding
of children’s numerical competence. This research identified a great deal
of competence in preschool children, including counting and matching
strategies that children use on Piaget’s conservation of number task (see the
discussion in Chapter 5).
As we detail, infant and toddler studies have largely focused on the
natural numbers (also called counting numbers). However, they have also
examined representations of fractional amounts and proportional relations
as well as geometric relationships, shape categories, and measurement.
Moreover, although there is some disagreement in the field about the in-
terpretation of the findings of infant and toddler studies as a whole, these
findings are generally viewed as showing strong starting points for the
learning of verbal and symbolic mathematical skills.
Infants’ Sensitiity to Small Set Size
Infant studies typically use habituation paradigms to examine whether
infants can discriminate between small sets of objects, either static or mov-
ing (Antell and Keating, 1983; Starkey and Cooper, 1980; Strauss and
Curtis, 1981; Van Loosbroek and Smitsman, 1990; Wynn, Bloom, and
Chiang, 2002). In a typical habituation study, infants are repeatedly shown
sets containing the same number of objects (e.g., 2) until they become bored
and their looking time decreases to a specified criterion. The infant is then
shown a different set size of objects or the same set size, and looking times
are recorded. Longer looking times indicate that the infant recognizes
that the new display is different from an earlier display. Results show that
infants (ranging in age from 1 day old to several months old) can discrimi-
nate a set of two objects from a set of three objects, yet they are unable to
discriminate four objects from six objects, even though the same 3:2 ratio
is involved. These findings indicate that infants’ ability to discriminate small
set sizes is limited by number rather than by ratio. Huttenlocher, Jordan,
and Levine (1994) suggest that infants’ ability to discriminate small sets (2
versus 3) could be based on an approximate rather than on an exact sense
of number.
Several studies suggest that the early quantitative sensitivity displayed
by infants for small set sizes is actually based on their sensitivity to amount
(surface area or contour length) which covaries with numerosity, rather
than on number per se (Clearfield and Mix, 1999, 2001). That is, unless
these variables are carefully controlled, the more items there are, the greater
the amount of stuff there is. In studies that independently vary number and
amount, Clearfield and Mix (1999, 2001) found that infants ages 6 to 8
months detected a change in amount (contour length or area) but not a
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62 MATHEMATICS LEARNING IN EARLY CHILDHOOD
change in number. Thus, if they were habituated to a set of two items, they
did not dishabituate to a set of three items if that set was equivalent to the
original set in area or contour length.
However, recent findings indicate that infants are sensitive to both
continuous quantity and to number (Cordes and Brannon, 2008, in press;
Kwon et al., 2009). Furthermore, Cordes and Brannon (2008) report that,
although 6-month-old infants are sensitive to a two-fold change in number,
they are sensitive to a three-fold change only in cumulative area across
elements, suggesting that early sensitivity to set size may be more finely
tuned than early sensitivity to continuous quantity. Other studies that
provide support for early number sensitivity include a study showing that
6-month-old infants can discriminate between small sets of visually pre-
sented events (puppet jumps) (e.g., Wynn, 1996). This result is not subject
to the alternative explanation of discrimination based on amount rather
than number, like the findings involving sets of objects. However, it is
possible that even though the rate and duration of the events have been
controlled in these studies, infants’ discrimination is based on nonnumerical
cues, such as rhythm (e.g., Demany, McKenzie, and Vurpillot, 1977; Mix
et al., 2002). Indeed, in one study in which the rate of motion was not a
reliable cue to numerosity, 6-month-olds did not discriminate old and new
numerosities (Clearfield, 2004).
A set size limitation also is seen in the behavior of 10- to 14-month-olds
on search tasks (Feigenson and Carey, 2003, 2005; Feigenson, Carey, and
Hauser, 2002). For example, in one study 12-month-olds saw crackers
placed inside two containers. The toddlers chose the larger hidden quantity
for 1 versus 2 and 2 versus 3 crackers, but they failed to do so on 3 versus
4, 2 versus 4, and 3 versus 6 crackers (Feigenson, Carey, and Hauser, 2002).
The authors suggest that this failure is due to the set size limitation of the
object file system.1 When cracker size was varied, the toddlers based their
search on the total cracker amount rather than on number. Similarly, 12- to
14-month-olds searched longer in a box in which two balls had been hidden
after they saw the experimenter remove one ball, than they did in a box in
which one ball had been hidden and the experimenter removed one ball (in
actuality there were no more balls in either box, as the experimenter sur-
reptitiously removed the remaining ball). They also succeeded on 3 versus
2 balls but failed on 4 versus 2 balls. That is, they did not search longer in
a box in which four balls were hidden and they saw two removed than in a
box in which they had seen two hidden and two were removed. The failure
1 The object file system refers to the representation of an object in a set that consists of small
numbers, the objects are in a 1-to-1 correspondence with each mental symbol, and there is no
summary representation of set size (e.g., three items are represented as “this,” “this,” “this”
rather than “a set of three things”) (Carey, 2004).
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COGNITIVE FOUNDATIONS FOR EARLY MATHEMATICS LEARNING
on 2 versus 4, which has the same ratio as the 2 versus 1 problem, suggests
that they were using the object file system rather than the analog magnitude
system, which is second system that represents large set sizes (4 or more)
approximately. Furthermore, in this study, the toddlers based their search
on the number of objects they saw hidden rather than on the total object
volume. Thus, at least by 12 months of age, it appears that children can
represent the number of objects in sets up to three (Feigenson and Carey,
2003). A subsequent study shows that this set size limit can be extended to
four if spatiotemporal cues allow the toddlers to represent the sets as two
sets of two (Feigenson and Halberda, 2004).
Infants’ Sensitiity to Large Set Size
Recent studies have shown that infants can approximate the number
of items in large sets of visual objects (e.g., Brannon, 2002; Brannon,
Abbott, and Lutz, 2004; Xu, 2003; Xu and Spelke, 2000; Xu, Spelke, and
Goddard, 2005), events (puppet jumps) (Wood and Spelke, 2005), and
auditory sets (Lipton and Spelke, 2003) that are well beyond the range of
immediate apprehension of numerosity (subitizing range). Consistent with
the accumulator model, which refers to a nonverbal counting mechanism
that provides approximate numerical representations in the form of analog
magnitudes, infants’ discrimination of large sets is limited by the ratio of
the two sets being compared rather than by set size. Thus, at 6 months of
age, when infants are habituated to an array of dots, they dishabituate to a
new set as long as the ratio between two sets is at least 2:1. By 10 months
of age, infants are able to discriminate visual and auditory sets that differ by
a 2:3 ratio but not by a 4:5 ratio (Lipton and Spelke, 2003, 2004; Xu and
Arriaga, 2007). Importantly, these studies controlled for many continuous
variables, suggesting that the discriminations were based on number rather
than amount (e.g., Brannon, Abbott, and Lutz, 2004; Cordes and Brannon,
2008; Xu, 2003; Xu and Spelke, 2000).
Do Infants Hae a Concept of Number?
Infants may be able to discriminate between sets of different sizes but
have no notion that all sets that have the same numerosity form a category
or equivalence class (the mathematical term for such a category). This no-
tion is referred to as the cardinality concept (e.g., the knowledge that three
flowers, three jumps, three sounds, and three thoughts are equivalent in
number). Number covers such matters as the list of counting numbers (e.g.,
1, 2, 3, . . .) and its use in describing how many things are in collections. It
also covers the ordinal position (e.g., first, second, third, . . .), the idea of
cardinal value (e.g., how many are there?), and the various operations on
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6 MATHEMATICS LEARNING IN EARLY CHILDHOOD
number (e.g., addition and subtraction). The notion of 1-to-1 correspon-
dences connects the counting numbers to the cardinal value of sets. Another
important aspect of number is the way one writes and says them using the
base 10 system (see Chapters 2 and 5 for further discussion). Knowledge
of number is foundational to children’s mathematical development and
gradually develops over time, so not all aspects of the number are present
during the earliest years.
Several studies (e.g., Starkey, Spelke, and Gelman, 1990; Strauss and
Curtis, 1984) examined whether infants understand that small sets that
share their numerosity but contain different kinds of entities form a cat-
egory (e.g., two dogs, two chicks, two jumps, two drumbeats). Starkey and
colleagues (1990) examined this question by habituating infants to sets of
two or three aerial photographs of different household objects. At test,
infants were shown novel photographs that alternated between sets of two
and sets of three. Infants dishabituated to the novel set size, suggesting that
they considered different sets of two (or three) as similar. Whereas these
studies might be regarded as suggesting that infants form numerical equiva-
lence classes over visual sets containing disparate objects, these studies may
have tapped infants’ sensitivity to continuous amount rather than number,
as described above (Clearfield and Mix, 1999, 2001). That is, unless careful
controls are put in place, sets with two elements will on average be smaller
in amount than sets of three elements (e.g., Clearfield and Mix, 1999, 2001;
Mix et al., 2002).
Findings showing that infants consider two objects and two sounds to
form a category would not be subject to this criticism and thus could be
considered as strong evidence for abstract number categories. In an impor-
tant study, Starkey, Spelke, and Gelman (1983) tested whether infants have
such categories. While the results seemed to indicate that 7-month-olds
regarded sets of two (or three) objects and drumbeats as similar, several
attempts to replicate these important findings have called them into ques-
tion (Mix, Levine, and Huttenlocher, 1997; Moore et al., 1987). Thus,
whether infants have an abstract concept of number that allows them to
group diverse sets that share set size remains an open question. The find-
ings, reviewed below, showing that 3-year-olds have difficulty matching
visual and auditory sets on the basis of number, and that this skill is related
to knowledge of conventional number words, suggest that the ability to
form equivalence classes over sets that contain different kinds of elements
may depend on the acquisition of conventional number skills. Kobayashi,
Hiraki, Mugitani, and Hasegawa (2004) suggest that the methods used may
be too abstract to tap this intermodal knowledge and that when the sounds
made are connected to objects, for example, the sound of an object landing
on a surface, evidence of abstract number categories may be revealed at
younger ages, perhaps even in infants.
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COGNITIVE FOUNDATIONS FOR EARLY MATHEMATICS LEARNING
Infant Sensitiity to Changes in Set Size
Several studies report that infants track the results of numerically
relevant transformations—adding or taking away objects from a set. That
is, when an object is added to a set, they expect to see more objects than
were previously in the set and when an object is taken away, they expect
to see fewer objects than were previously in the set. Wynn (1992a) found
that after a set was transformed by the addition or subtraction of an object,
5-month-old infants looked longer at the “impossible” result (e.g., 1 + 1 =
1) than at the “correct” result. However, as for numerical discrimination,
subsequent studies suggest that their performance may reflect sensitivity to
continuous (cumulative size of objects) amount rather than to numerosity
(Feigenson, Carey, and Spelke, 2002). For the problem 1 + 1, infants looked
longer at 2, the expected number of objects, when the cumulative size of the
two objects was changed than at three, the impossible number of objects,
when the cumulative size of the objects was correct—that is, when the
cumulative area of the three objects was equivalent to the area that would
have resulted from the 1 + 1 addition.
Cohen and Marks (2002) suggested an alternative explanation for
Wynn’s results. In particular, they suggest that the findings could be at-
tributable to a familiarity preference rather than to an ability to carry out
numerical transformations. For the problem 1 + 1 = 2, they point out that
infants more often see one object, as there was a single object in the first
display of every trial and thus, based on familiarity, look more at 1 (the
incorrect answer). A similar argument was made for looking more at 2 for
the problem 2 − 1.
Although their findings support this hypothesis, a more recent study by
Kobayashi et al. (2004) provides evidence that infants look longer at 1 + 1
= 3 and 1 + 2 = 3 than at 1 + 1 + 2 and 1 + 2 = 3 when the first addend is
a visual object and the second addend consists of a tone(s). This paradigm
cannot be explained by the familiarity preference because, for each prob-
lem, infants see only one element on the stage.
Order Relations
A few studies have examined infants’ sensitivity to numerical order
relations (more than, less than). One habituation study showed that 10- and
12-month-olds discriminated equivalent sets (e.g., a set of two followed
by another set of two) from nonequivalent sets (e.g., a set of two fol-
lowed by a set of three) (Cooper, 1984). In another study, Cooper (1984)
habituated 10-, 12-, 14-, and 16-month-old infants to sequences that were
nonequivalent. In the “less than” condition, the first display in the pair
was always less than the second (e.g., infants were shown two objects
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66 MATHEMATICS LEARNING IN EARLY CHILDHOOD
followed by three objects). The reverse order was shown for the “greater
than” condition. At test, the 14- and 16-month-olds showed more inter-
est in the opposite relation than the one that was shown, suggesting that
they represented the less than and greater than relations, whereas 10- and
12-month-olds did not. However, Brannon (2002) presents evidence that
infants are sensitive to numerical order relations by 11 months of age.
Summary
The results of infant studies using small set sizes show that, very early
in life, infants have a limited ability to discriminate sets of different sizes
from each other (e.g., 2 versus 3 but not 4 versus 6). The set size limitation
has been interpreted as reflecting one of two core systems for number—the
object file system. They also expect the appropriate result from small num-
ber addition and subtraction transformations (e.g., 1 + 1 = 2 and 2 − 1 = 1),
at least when amount covaries with number. Somewhat later, by 10 months
of age, infants discriminate equivalent from nonequivalent sets, and by 14
months of age they discriminate greater than from less than relationships.
Because many of these studies did not control for continuous variables
that covary with number (i.e., contour length and surface area), the basis
of infant discriminations is debated. However, recent studies indicate that
infants are sensitive to both number of objects in small sets and to continu-
ous variables, and they may be more sensitive to number than to cumulative
surface area. Infant studies also have examined sensitivity to approximate
number by using larger sets of items (e.g., 8 versus 16). These studies have
found that infants can discriminate between sets with a 2:1 ratio by age 6
months and between those with 2:3 ratios by age 9 months as long as all
set sizes involved are greater than or equal to 4, that is, 6-month-olds fail
to discriminate 2 versus 4.
We also note that infants’ early knowledge of number is largely implicit
and has important limitations that are discussed below. There were no
number words involved in any of the studies described above. This means
that learning the number words and relating them to sets of objects is a
major new kind of learning done by toddlers and preschoolers at home and
in care and education centers. This learning powerfully extends numerical
knowledge, and children who acquire this knowledge at earlier ages are
provided with a distinct advantage.
Mental Number Representations in Preschool Children
Just as much of the infant research has a focus on theorizing about and
researching the nature of infant representations of number, so, too, does
some research on toddlers and preschool children. The goal is to understand
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COGNITIVE FOUNDATIONS FOR EARLY MATHEMATICS LEARNING
how and when young children represent small and larger numbers. To do
this, special tasks are used that involve hidden objects, so that children must
use mental representations to solve the task. Sometimes objects are shown
initially and are then hidden, and sometimes objects are never shown and
numbers are given in words. These tasks are quite different from situations
in which young children ordinarily learn about numbers in the home or in
care and educational centers, and they can do tasks in home and naturalis-
tic settings considerably earlier than they can solve these laboratory tasks
(e.g., Mix, 2002). In home and in care and educational settings, numbers
are presented with objects (things, fingers), and children and adults may see,
or count, or match, or move the objects. The objects do not disappear, and
they are not hidden. Children’s learning under these ordinary conditions is
described in Chapter 5. Here we continue to focus on theoretical issues of
representations of numbers.
Small Set Sizes
Like infants, 2- to 3-year-olds show more advanced knowledge of
number than would be predicted by previous views. As noted previously,
conservation of number was considered to be a hallmark of number de-
velopment (Piaget, 1941/1965). However, Gelman’s (1972) “magic experi-
ment” showed that much younger children could conserve number if the
spatial transformation was less salient and much smaller set sizes were used.
In this study, 3- to 6-year-olds were told that either a set of two mice or a set
of three mice was the “winner.” The two sets were then covered and moved
around. After children learned to choose the winner, the experimenter al-
tered the winner set, either by changing the spatial arrangement of the mice
or by adding or subtracting a mouse. Even the 3-year-olds were correct in
recognizing that the rearrangement maintained the status of the winner,
whereas the addition and subtraction transformations did not.
Huttenlocher, Jordan, and Levine (1994) examined the emergence of
exact number representation in toddlers. They posited that mental models
representing critical mathematical features—the number of items in the set
and the nature of the transformation—were needed to exactly represent
the results of a calculation. Similarly, Klein and Bisanz (2000) suggest that
young children’s success in solving nonverbal calculations depends on their
ability to hold and manipulate quantitative representations in working
memory as well as on their understanding of number transformations.
Huttenlocher, Jordan, and Levine (1994) gave children ages 2 to 4 a
numerosity matching task and a calculation task with objects (called non-
verbal; see Box 3-1). On the matching task, children were shown a set of
disks that was subsequently hidden under a box. They were then asked to
lay out the same number of disks. On the calculation task, children were
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6 MATHEMATICS LEARNING IN EARLY CHILDHOOD
BOX 3-1
Clarifying Experimental Misnomers
Researchers have used tasks in which two conditions vary in two important
ways, such as in Huttenlocher, Newcombe, and Sandberg (1994). In one condi-
tion, children are first shown objects, and then the objects are hidden. Number
words are not used in this condition. In the other condition, children never see
objects but must imagine or generate them (e.g., by raising a certain number of
fingers). Here the numbers involved are conveyed by using number words, either
as a story problem or just as words (e.g., “2 and 1 make what?”). In their reports,
researchers call the first condition nonverbal and the second condition verbal. But
these labels are a bit misleading, because they sound as if nonverbal and verbal
are describing the children’s solution methods. In this report we use language
that mentions both aspects that were varied: with objects (called nonverbal) and
without objects (called verbal).
shown a set of disks that was subsequently covered. Following this, items
were either added or taken away from the original set. The child’s task was
to indicate the total number of disks that were hidden by laying out the
same number of disks (“Make yours like mine”).
On both the matching and transformation tasks, performance increased
gradually with age. Children were first successful with problems involving
low numerosities, such as 1 and 2, gradually extending their success to
problems involving higher numerosities. Importantly, when children re-
sponded incorrectly, their responses were not random, but rather were ap-
proximately correct. Approximately correct responses were seen in children
as young as age 2, the youngest age group included in the study. On the
basis of these findings, Huttenlocher, Jordan, and Levine (1994) argue that
representations of small set sizes begin as approximate representations and
become more exact as children develop the ability to create a mental model.
Exactness develops further and extends to larger set sizes when children
map their nonverbal number representations onto number words.
Toddlers’ performance on numerosity matching tasks indicates that,
as they get older, they get better at representing quantity abstractly. This
achievement appears to be related to the acquisition of number words (Mix,
2008). Mix showed that preschoolers’ ability to discriminate numerosi-
ties is highly dependent on the similarity of the objects in the sets. Thus,
3-year-olds could match the numerosities of sets consisting of pictures of
black dots to highly similar black disks. Between ages 3 and 5, children
were able to match the numerosities of increasingly dissimilar sets (e.g.,
black dots to pasta shells and black dots to sequential black disks at age
3½; black dots to heterogeneous sets of objects at age 4).
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COGNITIVE FOUNDATIONS FOR EARLY MATHEMATICS LEARNING
The abstractness of preschoolers’ numerical representations was also
assessed in a study (Mix, Huttenlocher, and Levine, 1996) examining their
ability to make numerical matches between auditory and visual sets, an
ability that Starkey, Spelke, and Gelman (1990) had attributed to infants.
The researchers presented 3- and 4-year-olds with a set of two or three
claps and were asked to point to the visual array that corresponded to the
number of claps. The 3-year-olds performed at chance on this task, but by
age 4, the children performed significantly above chance. In contrast, both
age groups performed above chance on a control task that involved match-
ing sets of disks to pictures of dots. Another study assessed the effect of the
heterogeneity of sets on the ability of 3- to 5-year-olds to make numerical
matches and order judgments. The results replicate Mix’s (1999b) finding
that the heterogeneity of sets decreases children’s ability to make equiva-
lence matches. However, heterogeneity versus homogeneity of sets did not
affect their ability to make order judgments (i.e., to judge which of two sets
is smaller) (Cantlon et al., 2007).
Mix (2002) has also examined the emergence of numerical knowledge
through a diary study of her son, Spencer. In this study, she found indica-
tions of earlier knowledge than the experiments described above might
indicate. Spencer was able to go into another room and get exactly two
dog biscuits for his two dogs at 21 months of age, long before children
succeed on the homogeneous or heterogeneous set matching tasks described
above. Indeed, Spencer himself had failed to perform above chance on
these laboratory tasks. Thus, it appears that early knowledge of numerical
equivalence may arise piecemeal, and first in highly contextualized situa-
tions. For Spencer, his earliest numerical equivalence matches occurred in
social situations (e.g., biscuits for dogs, sticks for guests). Whether this is
a general pattern or whether there are wide individual differences in such
behaviors is an open question (also see Mix, Sandhofer, and Baroody, 2005,
for a review).
Levine, Jordan, and Huttenlocher (1992) compared the ability of pre-
school children to carry out calculations involving numerosities of up to six
with objects (called nonverbal) and without objects (called verbal) (the for-
mer calculations were similar to those described above in the Huttenlocher,
Jordan, and Levine, 1994 study). The calculations without objects (called
verbal) were given in the form of story problems (“Ellen has 2 marbles and
her father gives her 1 more. How many marbles does she have altogether?”)
and in the form of number combinations (e.g., “How much is 2 and 1?”).
Children ages 4 to 5½ performed significantly higher on the calculation task
when they could see objects and transformations than on the calculation
tasks when they could not see objects or transformations. This was true
for both addition and subtraction calculations. This difference in perfor-
mance between nonverbal and verbal calculations was particularly marked
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MATHEMATICS LEARNING IN EARLY CHILDHOOD
Heads-to-Toes task asks children to do the opposite of what the instructor
tells them. So, for example, if the instructor asks the children to touch their
head, they are to touch their toes. This task measures behavioral regula-
tion (a component of self-regulation), in that it requires children to employ
inhibitory control, attention, and working memory. The researchers found
that behavioral regulation scores significantly predicted emergent math
scores. The researchers conclude that “strengthening attention, working
memory, and inhibitory control skills prior to kindergarten may be an ef-
fective way to ensure that children also have a foundation of early academic
skills” (p. 956). Espy and colleagues (2004) specifically studied the roles
of working memory and inhibitory control with almost 100 preschoolers.
They found that both components of executive function contributed to the
children’s mathematical proficiency, with inhibitory control being the most
prominent. Passolunghi and colleagues (2007) studied 170 6-year-olds in
Italy. They examined the roles of working memory, phonological ability,
numerical competence, and IQ in predicting math achievement. They found
that working memory skills significantly predicted math learning at the
beginning of elementary school (primary school in Italy).
SUMMARY
This chapter underscores that young children have more mathematics
knowledge, in terms of number and spatial thinking, than was previously
believed. Very early in life, infants can distinguish between larger set sizes,
for example 8 versus 16 items, but their ability to do so is only approxi-
mate and is limited by the ratio of the number of items in the sets. The set
size limitation is thought to reflect one of the two core systems for number
(Feigenson, Dehaene, and Spelke, 2004; Spelke and Kinzler, 2007). Further-
more, young infants’ early knowledge of quantity is implicit, in that they
do not use number words, which means that learning number words and
relating them to objects is one of the major developmental tasks to occur
during early childhood.
Toddlers and preschool children move from the implicit understand-
ing of number seen during infancy to formal number knowledge. Spoken
number words, written number symbols, and cultural solution methods are
important tools that support this developmental progression.
Young children also learn about space, including shapes, locations,
distances, and spatial relations, which also go through major development
during the early childhood years. Children’s acquisition of spatial language
plays an important role in the development of spatial categories and skills.
In addition to learning about number and shape, early childhood also
includes development of measurement, which is a fundamental aspect of
mathematics that connects geometry and number. Young children’s under-
standing of measurement begins with length, which is perceptually based,
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5
COGNITIVE FOUNDATIONS FOR EARLY MATHEMATICS LEARNING
and an important feature of their learning during this period is that they
have difficulty understanding units of measure. Young children can become
successful at this when given appropriate instruction.
It is also important to note that across early childhood, mathematical
development that is situated in an environment that promotes regulation of
cognitive activities and behavior can improve mathematical development.
More specifically, when young children have an opportunity to practice
staying on task, to keep information in mind while manipulating or chang-
ing it mentally, and to practice shifting between differing tasks, mathematics
learning is improved and in turn improves these regulatory processes.
Although we discuss universal starting points for mathematics devel-
opment in this chapter, there are, of course, differences in children’s math-
ematical development. The next chapter explores variation in children’s
mathematical development and learning outcomes and the sources of this
variation. We also discuss the role of the family and informal mathematics
learning experiences in supporting children’s mathematical development.
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