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OCR for page 95
4
Developmental Variation,
Sociocultural Influences, and
Difficulties in Mathematics
There is evidence that most children bring foundational resources and
knowledge about mathematics to school. However, this is not the whole
story. Research findings reveal enormous discrepancies in young children’s
levels of mathematics competence, and these discrepancies appear to be
larger in the United States than they are in some other countries (e.g.,
China) (Starkey and Klein, 2008). This chapter describes the kinds of dif-
ferences that exist and reviews what is known about the nature and sources
of developmental variations among children.
Most children bring core number sense or number competencies to
school (National Research Council, 2001). Number sense refers to in-
terconnected knowledge of numbers and operations. Although preverbal
number sense begins in infancy and appears to be universal, preschool and
kindergarten number sense involves understanding of number words and
symbols, which is heavily influenced by experience and instruction. The
number sense children bring to kindergarten is highly predictive of their
later mathematics achievement. The term “number sense” means different
things in different fields of research, and almost no two researchers define
it in exactly the same way (Gersten, Jordan, and Flojo, 2005; Jordan et al.,
2006). The term “number sense” is used in this chapter because much of the
research summarized here uses it. When the discussion is more general, the
term “number competencies” is used along with number sense to remind
the reader that we are talking about knowledge and skills that can be taught
and learned. The word “competencies” is used as a balanced term mean-
ing both knowledge and skills. The competencies encompassed by the term
“number sense” as used here are described more fully in Chapter 5.
5
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6 MATHEMATICS LEARNING IN EARLY CHILDHOOD
Despite strong universal starting points, striking individual differences
in number sense emerge early in life and are present by the time children
enter preschool (e.g., Klibanoff et al., 2006). These differences are apparent
both on standardized tests (e.g., Arnold et al., 2002; Starkey, Klein, and
Wakeley, 2004) and on specific measures tapping early number competen-
cies, such as determining set size, comparing sets, and carrying out calcu-
lations (e.g., Entwisle and Alexander, 1990; Ginsburg and Russell, 1981;
Griffin, Case, and Siegler, 1994; Jordan, Huttenlocher, and Levine, 1992;
Levine et al., in preparation; Saxe, Guberman and Gearheart, 1987). The
level of number sense in kindergarten is highly of predictive future math-
ematics success in first through third grades (Fuchs et al., 2007; Jordan,
Glutting, and Ramineni, in press; Locuniak and Jordan, in press; Mazzocco
and Thompson, 2005) as well as into the later school years (Duncan et al.,
2007).
In this chapter, we explore individual differences in children’s math-
ematics competence. We begin by describing the differences associated
with key social groups specifically defined by socioeconomic status, gender,
race/ethnicity, and English language ability. We then discuss the contextual
factors and early experiences that appear to be linked to these differences,
giving particular attention to the role of the family and language. We then
discuss learning disabilities. We end with a brief discussion of potential
intervention.
GROUP DIFFERENCES IN MATHEMATICS PERFORMANCE
Researchers have explored several key social factors that are linked
to systematic, average differences in children’s mathematical performance.
Socioeconomic status (SES), which includes income level as well as level of
parental education, is strongly linked to differences in mathematics com-
petence. Evidence related to gender differences in mathematics competence
is less clear, although some differences have been found.
Socioeconomic Status
Mathematical skills of young children from low-income families lag
behind those of their middle-income peers. Preschoolers who attend Head
Start Programs perform significantly below children who attend preschools
serving middle-income children on standardized tests of mathematical read-
iness (Ehrlich and Levine, 2007). The gulf between low- and middle-income
children is wide and includes spatial/geometric and measurement as well
as number competencies (Clements, Sarama, and Gerber, 2005; Klein and
Starkey, 2004; Saxe et al., 1987).
Jordan and colleagues (Jordan et al., 2006, 2007) found that low-
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DEVELOPMENTAL VARIATION, SOCIOCULTURAL INFLUENCES
income children enter kindergarten far behind their middle-income peers
on tasks assessing counting skills, knowledge of number relations (e.g.,
recognizing which number is smaller), and number operations. Moreover,
longitudinal assessment over six data points revealed that low-income chil-
dren were four times more likely than their middle-income peers to show
flat growth in these areas throughout kindergarten and early first grade.
Underlining the importance of early number sense to school success, the
researchers found that level of performance on a battery assessing number
sense in kindergarten as well as rate of growth between kindergarten and
first grade accounted for 66 percent of the variance in mathematics learn-
ing at the end of first grade (Jordan et al., 2007). In other words, number
sense in kindergarten in strongly related to competence in mathematics at
the end of first grade and the rate of growth over the first grade year. In-
come status, gender, age, and reading ability did not account for additional
variance in first grade mathematics outcomes over and above initial per-
formance and growth in number sense. This suggests that SES differences
found at the end of first grade are due to initial differences in number sense
in kindergarten.
Several studies indicate that SES differences in preschoolers’ number skills
are more marked on tasks tapping number skills without objects (called ver-
bal tasks) than on tasks tapping number skills with objects (called nonverbal
tasks). When kindergarten and first grade children are presented with verbal
calculation problems with no objects, either as number combination prob-
lems (“How much is 3 and 2?”) or story problems (“Mike had 3 pennies.
Jen gave him 2 more pennies. How many pennies does Mike have now?”),
middle-income children perform much better than do low-income children
(Jordan et al., 2006; Jordan, Huttenlocher, and Levine, 1992; Jordan, Levine,
and Huttenlocher, 1994). Middle-income children also achieve at a faster
rate on calculation problems without objects in kindergarten (Jordan et al.,
2006, 2007). In contrast, SES differences are smaller if the same calculations
are presented in a nonverbal format with objects (e.g., the child is shown 3
disks that are then hidden with a cover. The tester then slides 2 disks under
the cover and the child indicates how many are now hidden).
Jeong and Levine (2005) have shown that knowing number words is
associated with very early performance on numerosity matching tasks that
do not require verbal responses (e.g., matching arrays of visual dots). Spe-
cifically, performance on these tasks is more exact for children who have
acquired the meaning of a few number words. For instance, 2- to 3-year-
olds were more exact in their ability to match small set sizes when they have
better knowledge of the cardinal meanings of number words. Although
low-income children performed worse than middle-income children on
such numerosity matching tasks, this difference was eliminated if answers
that were plus or minus 1 from the correct answer were counted as correct
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MATHEMATICS LEARNING IN EARLY CHILDHOOD
(Ehrlich, Levine, and Goldin-Meadow, 2006). Thus, low-SES preschoolers
appear to have approximate representations of set sizes and number words
at a time when their higher SES peers have gained exact representations.
Therefore, low-SES preschoolers need experiences to learn number words
and to use them to help on these matching tasks.
The sources of these differences are difficult to pinpoint. Research on
children’s early experiences point to the amount of support for mathematics
at home as well as other language and contextual factors. Some findings
show that young children from low-income families receive less support for
mathematics in their home environment than do their middle-income peers
(Blevins-Knabe and Musun-Miller, 1996; Holloway et al., 1995; Saxe et al.,
1987; Starkey et al., 1999). Compounding the situation, public preschool
programs serving low-income families tend to provide fewer learning op-
portunities and supports for mathematical development than ones serving
middle-income families (Clements and Sarama, 2008). These factors are
discussed in greater detail in the section on the influence of context and
experience.
Gender
Results and opinions vary regarding gender differences in early math-
ematics. Some studies have no revealed gender differences in mathematics
performance (e.g., Clements and Sarama, 2008; Lachance and Mazzocco,
2006; Levine, Jordan, and Huttenlocher, 1992; Sarama et al., 2008). Some
have found differences favoring boys: Jordan et al. (2006) found small but
statistically significant gender effects on calculation with objects and on
numerical estimation. In particular, boys had an edge over girls even when
income level, age, and reading ability were controlled for in the analyses,
and there were more boys than girls in the highest performing group. How-
ever, Coley’s (2002) analysis of the Early Childhood Longitudinal Study da-
tabase indicated small advantages in kindergarten in different areas for each
gender: Girls were somewhat better in recognizing numbers and shapes, and
boys were somewhat better in numerical operations.
Some research with older children indicates that girls in the primary
grades may tend to use less advanced strategies than do boys (Fennema
et al., 1998), and other work suggests no gender differences in the math-
ematics performance of older students (Hyde et al., 2008). Recent research
(e.g., Carr et al., 2007) suggests spatial skills may promote the use of more
advanced computational strategies, and boys seem to have an advantage
in the more general area of spatial cognition, even in preschool. There are
differences in the mean level of performance of boys and of girls on mental
rotation tasks by 4½ years of age, ranging from small but significant dif-
ferences (Levine et al., 1999) to large differences with girls performing at
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DEVELOPMENTAL VARIATION, SOCIOCULTURAL INFLUENCES
chance levels (Rosser et al., 1984). Preschool boys also perform better than
preschool girls on solving problems involving mazes (e.g., Fairweather and
Butterworth, 1977; Wechsler, 1967; Wilson, 1975) and are faster at copy-
ing a three-dimensional Lego (plastic blocks) model (Guiness and Morley,
1991). However, it appears that at least some of these differences are cre-
ated by lack of particular types of experiences (Ebbeck, 1984).
Spatial skill may reflect or at least interact with greater engagement of
boys than girls in spatial activities, such as building with Legos (Baenninger
and Newcombe, 1989). Young boys typically spend more time playing with
Legos and putting puzzles together than do girls, suggesting that engage-
ment in spatial activities promotes skill development (Levine et al., 2005).
The amount of puzzle play for both boys and girls was related to the men-
tal transformation performance (McGuinness and Morley, 1991). Parents’
spatial language may be more important for girls than for boys; use of such
language by parents related to mental transformation performance of girls
but not of boys (Cannon, Levine, and Huttenlocher, 2007). Boys tend to be
more interested in movement and action from the first year of life and girls
more focused on social interactions (e.g., Lutchmaya and Baron-Cohen,
2002). Boys also may gesture more on spatial tasks (e.g., Ehrlich, Levine,
and Goldin-Meadow, 2006), indicating that encouraging gesture, especially
for girls, may be helpful in spatial learning.
Given the finding that boys seem to have an advantage in spatial cog-
nition and that this seems to result partly from the number of experiences
they have that support such learning, it seems particularly important for
both numerical and spatial learning that girls be given opportunities for
spatial learning. Importantly, intervention studies with preschoolers using
a research-based mathematics curriculum did not find an interaction with
gender, indicating that girls can learn as much as boys in both numeri-
cal and spatial tasks (Clements and Sarama, 2008; Sarama et al., 2008).
Simple modifications to everyday preschool activities, such as block build-
ing (Kersh, Casey, and Young, in press) and the use of stories about spatial
topics (Casey et al., 2008), have been shown to be effective in developing
girls’ spatial cognition. Teachers should ensure that girls play with blocks
and provide them with challenges that ensure that they extend their block-
building skills, such as building windows, bridges, and arches.
Race and Ethnicity
Over the past several decades, research has found differences in chil-
dren’s mathematics learning outcomes as a function of their race/ethnicity
(e.g., Ginsburg and Russell, 1981). This section discusses differences in
mathematics learning outcomes, but readers should keep in mind that using
a fixed trait based on a single dimension can lead to a cultural deficit model
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100 MATHEMATICS LEARNING IN EARLY CHILDHOOD
(Lubienski, 2007). Racial/ethnic groups are heterogeneous, and children in
particular racial/ethnic groups have mathematical knowledge and skills that
range from low to high mastery levels.
Generally, African American, Hispanic, and American Indian/Alaska
Native children achieve at lower levels than their white peers in mathe-
matics (National Center for Education Statistics, 2007). Few data exist
on early childhood mathematics teaching and learning in relation to race/
ethnicity, but one can extrapolate from K-12 studies. Findings suggest that
this achievement disparity is related to differences in mathematics learning
before school entry and fewer meaningful pedagogical experiences once
children of color enter school (Magnuson and Waldfogel, 2008). For ex-
ample, the National Assessment of Educational Progress (NAEP) survey
data show that fourth grade black and Hispanic students and those with
low SES report that mathematics mainly consists of memorizing facts, a
belief that is negatively correlated with achievement even after controlling
for race/ethnicity and SES (Lubienski, 2006, 2007). Furthermore, teachers’
reports indicate that black and Hispanic children were more likely to be
routinely assessed with multiple-choice tests than white students (Lubienski,
2006). These practices do not represent the best pedagogy for high-quality
mathematics education (National Council of Teachers of Mathematics,
2000).
Teachers who build on children’s everyday mathematical experiences
promote genuine mathematics learning (Civil, 1998; Ladson-Billings, 1995).
For example, Ladson-Billings (1995) found that urban and suburban stu-
dents’ community experiences shaped the way they approached a math-
ematics problem-solving task and that students’ differing approaches to
learning could be used by teachers to inform their instruction. Instructional
practices that extend children’s out-of-school experiences are more likely to
produce meaningful mathematics learning.
English Language Learners
Surprisingly little research has examined the mathematics performance
of English language learners. Findings for other subject areas show that
children who have limited proficiency in English perform more poorly than
their native English-speaking peers in other academic subjects (McKeon,
2005). A major issue for educating English language learners (ELL) is the
language of instruction (Barnett et al., 2007; Genesee et al., 2006). In re-
search conducted by Barnett and colleagues (2007) with 3- and 4-year-olds,
they tested whether children in a two-way immersion (English and Spanish)
or those in English-only programs made gains in English language measures
of mathematics, vocabulary development, and literacy. They found that
children in both types of programs made gains on all academic measures
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DEVELOPMENTAL VARIATION, SOCIOCULTURAL INFLUENCES
and the two-way immersion classrooms saw improvements in Spanish lan-
guage development for both ELL and English-speaking children without
losses to English language learning (Barnett et al., 2007). It is important to
note that classrooms in both types of program employed a licensed teacher
and an assistant with a child development associate credential. A review of
the K-12 literature on the language of instruction provides evidence that
conflicts with the findings of Barnett and colleagues; specifically, Lindholm-
Leary and Borasato (2006) suggest that bilingual education may be related
to more positive educational outcomes for older ELL students. Given these
disparate findings, additional research in high-quality early childhood set-
tings on this topic is warranted.
One of the few studies focused specifically on mathematics competence
with this population of students suggests there may not be performance dif-
ferences in mathematics. Secada (1991) found that first grade Hispanic stu-
dents were not at a disadvantage to their native English-speaking peers in
solving addition and subtraction word problems. However, with the grow-
ing number of ELL in the student population, it vital that more attention
be paid to the relationship between language status and early mathematics
learning so that early childhood education can effectively accommodate and
support these children.
INFLUENCE OF CONTEXT AND EXPERIENCE
As noted in the previous section, research has identified consistent,
average differences in mathematics competence and performance depend-
ing on membership in a particular social group. Why group membership
is linked to such differences is a complicated question. Research suggests
that early experiences play an important role in shaping the observed dif-
ferences. In this section we explore the contributions of context and early
experience. We begin with a general discussion of the role of families in
shaping early experience, including parents’ knowledge and beliefs about
mathematics, and the support they provide for mathematics through en-
gagement in mathematics activities. We then look more specifically at how
differences in experiences at home are linked to the observed SES differ-
ences in performance. Finally, we consider the role of language in math-
ematics learning.
Role of Families
Families are one of the critical social settings in which children develop
and learn (Bronfenbrenner, 2000; Iruka and Barbarin, 2008). Families influ-
ence children’s development in many ways, including parenting practices,
provision of resources, interactions with school, and involvement in the
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102 MATHEMATICS LEARNING IN EARLY CHILDHOOD
community (Weiss, Caspe, and Lopez, 2006; Woods and Kurtz-Costes,
2007). Parents have different attitudes, values, and beliefs in raising young
children, which result in difference emphasis on educational activities in
the home. Families support mathematics learning through their activities
at home, conversations, attitudes, materials they provide to their children,
expectations they have about their performance, the behaviors they model,
and the games they play. Parents also build connections with their children’s
educational settings—all of which can shape children’s early mathematics
development.
Parents’ Knowledge and Beliefs About Early Childhood Mathematics
Although there are only a few empirical studies about parental beliefs
and behaviors related to early mathematics, those that exist suggest that
parents place more importance on literacy development (Barbarin et al.,
2008). Barbarin and colleagues examined the beliefs of parents whose
children were enrolled in public prekindergarten regarding the skills chil-
dren need to be prepared for school. Mathematical skills and such tasks as
counting were rated less important than other social and cognitive tasks.
Specifically, language/early literacy was mentioned 50 percent of the time,
whereas numeracy was mentioned only 3.5 percent of the time (Barbarin
et al., 2008). Similarly, Cannon and Ginsburg (2008) found that mothers
thought it was more important that their children learn daily living skills
and develop language skills in preschool than that their children learn
mathematical skills. Most mothers in the study reported they themselves
spent more time teaching their children language skills than mathematics
skills at home.
Engagement in Mathematics Actiities
Children’s mathematical competence is supported and shaped by the
math-related activities they engage in as part of their daily lives (Benigno
and Ellis, 2008). Parenting practices in which parents engage children in
conversations about number concepts, play with puzzles and shapes, en-
courage counting, and use number symbols to represent quantity in their
interactions in the physical world can facilitate mathematics learning (see
Box 4-1 for examples of how parents can engage children in mathematics
activities). Acquiring mathematics knowledge involves more than learning
numbers. It also includes learning shapes and patterns. It is facilitated by
conversations about what children are doing when they compute, solve
puzzles, and develop patterns and discussions of why they took a particular
approach to a problem.
In fact, one study demonstrates how parents and their children can
engage in mathematics-related activities. In a groundbreaking study of
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DEVELOPMENTAL VARIATION, SOCIOCULTURAL INFLUENCES
BOX 4-1
Supporting Children’s Mathematics at Home
Parents play an important role in supporting mathematics learning through the
mathematics-related activities in which they engage their children. Incorporating
mathematics-focused activities during play is one strategy for enhancing math-
ematics. Another is to capitalize on situations in which mathematics is a natural
part of everyday tasks, such as grocery shopping or cooking. During daily activi-
ties, parents can:
• bserve their children carefully, seeing what they do and encouraging and
O
extending their fledgling use of number symbols and processing.
• ay the number word list. For example, they can count small food items or
S
the number of cups at the table.
• sk children to tell them about their problem solving. For example, they can
A
ask “What did you mean by that?” or “Why did you do it that way?”
• ngage in activities that involve playing with blocks, building things, and
E
board games.
Given the prevalence of the Internet, television, and videogames in the lives
of children, even young children (for a review, see Fisch, 2008), these means of
communication provide interesting opportunities for impacting early mathematics
skills. Fisch (2008) provides a review of existing media that include a math-
ematical component. These include television shows, such as Sesame Street;
m
athematics-based software games, such as Building Blocks and Millie’s Math
House; websites that include mathematics content, such as that of Sesame Street
and Disney; and electronic, interactive toys.
The Internet can be a tool to help families devise mathematics-related activi-
ties for their young children. Such websites as FAMILY MATH, from the Lawrence
Hall of Science at the University of California, Berkeley, can provide this kind of
help. Although there are no effectiveness data available for this website, FAMILY
MATH offers fun activities that maintain mathematical integrity and uses inex-
pensive materials that families may already have at home (see http://sv.berkeley.
edu/showcase/pages/fm_act.html).
early childhood mathematics in family contexts, Saxe and colleagues (1987)
found that many of the children in the 78 families they studied, both low
and middle income, were spontaneously engaging in number-related activi-
ties (counting toys, using numbers in play, etc.), but the nature of their nu-
merical knowledge and environment differed. Mothers in the study reported
that both they and their children had a high level of interest in number play,
but middle-income children performed better than low-income children on
both the cardinality and arithmetic tasks.
There are numerous opportunities on a daily basis for children and
families to explore mathematical terms and concepts. These include meal-
times, shopping, playtime, sports, television, and reading (Benigno and Ellis,
2008). In fact, Blevins-Knabe and Musun-Miller (1996) provide evidence
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10 MATHEMATICS LEARNING IN EARLY CHILDHOOD
to support the effects of parental modeling, reporting a relation between
parental participation in number activities and children’s involvement in
similar activities. Moreover, they found that parental reports of children’s
number activities at home predicted their scores on a standardized test of
early mathematical ability.
Several studies suggest that exposure to the language and symbol sys-
tem of mathematics powerfully extends the universal starting points of
children’s quantitative knowledge and contributes to observed differences in
mathematics competence. This is true in terms of exposure to the language
of mathematics in preschool (Klibanoff et al., 2006) as well as at home be-
tween ages 14 and 30 months (Levine et al., in preparation). These studies
show that the range of number words used in these settings is enormous.
For example, in the home study, a longitudinal project in which families
were visited every 4 months for five 90-minute sessions during which they
were asked to go about their normal activities, the use of number words
ranged from a low of 3 to a high of 175 instances. Similarly, in the class-
room studies, the amount of number input provided by teachers during
a 1-hour period that included circle time ranged from 1 to 104 coded
instances.
While research suggests that families do incorporate mathematics into
their everyday lives, they may also need reminders of the importance of
mathematics. An observational study of 39 preschoolers and their fami-
lies (Tudge and Doucet, 2004) found that the children engaged in a very
low rate of explicit mathematics lessons over the course of a day and also
demonstrated low levels of mathematics-related play. Of the mathematics
lessons that were observed, the most common were lessons involving num-
bering, and the most common types of mathematical play involved toys
that featured numbers (puzzles, computer programs, etc.). Furthermore,
parents may overestimate their children’s mathematical skills. Fluck and
colleagues (2005) found that parents believed their children had a much
better grasp of the concept of cardinality (beyond mere counting) than the
children actually displayed.
Differences in Children’s Experiences and Learning
Opportunities as a Function of Socioeconomic Status
Evidence suggests that SES differences in children’s mathematics com-
petence are linked to parallel differences in experiences provided in the
home. For parents in some low-SES families, involvement in fostering the
acquisition of mathematics skills in their children may be hampered by
multiple factors. Poverty and uncertainty related to inadequate resources
and residential instability can easily become all-consuming, leaving room
for little else. Parents in low-SES families, though concerned about their
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DEVELOPMENTAL VARIATION, SOCIOCULTURAL INFLUENCES
children’s education, may feel less ready to assist them due to limitations
in their own education, the strains of inadequate financial resources, unmet
mental health needs, and specific discomfort with their own mathematical
skills and a lack of awareness of the importance of early mathematics devel-
opment (for research on the effects of poverty on parenting see, e.g., Knitzer
and Lefkowitz, 2006; McLoyd, 1990; see Clements and Sarama, 2007, for
a specific discussion of low-income families and mathematics).
Research shows that low-income parents provide fewer mathematics
activities than middle-class parents (Starkey et al., 1999). This includes free
activities, such as those that are integrated into everyday experiences and
made-up games, suggesting that, to some extent, lack of financial resources
does not explain the difference. Starkey and Klein (2008) suggest that the
difference may instead stem from educational background and exposure
to mathematics courses. The difference may be resource-based as well.
Ramani and Siegler (2008), in a study of board game activities, found that,
although 80 percent of middle-class preschool-age children reported play-
ing one or more board games outside preschool, only 47 percent of Head
Start children did so. However, such board games could easily be made and
used at home.
It is also vital to remember that, in many cases, children and families
from low-SES backgrounds are involved with many more agencies and
programs than their more well-off peers. “Exploring the contribution of
these additional settings is important because interpreting SES effects as
emanating exclusively from the family or the child means that policy and
program interventions may focus too narrowly as they attempt to improve
the educational outcomes of low-SES children” (Aikens and Barbarin,
2008, p. 236). Policy makers, researchers, and practitioners should not
neglect the importance of the interactions and experiences of the multiple
contexts and the nature of development in everyday life. Thus, at the level
of a mother and child interacting in a larger social context unique to cul-
tural environments, the entire dynamic may influence a child’s learning and
specifically reinforce or hinder the development of mathematical thinking
and understanding.
The SES gap prior to preschool entry suggests that the home environ-
ment plays a major role, yet it is important to note that formal preschool
programs do not appear to be ameliorating it. In fact, the gap widens during
the preschool years. “In the United States, neither the home nor preschool
learning environments of low-SES children provide sufficient enrichment
to close or even maintain early SES-related differences in mathematical
knowledge” (Starkey and Klein, 2008, p. 266). The issue of how to better
support low-income children in mathematics and address the gap is taken
up in detail in Chapter 7.
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110 MATHEMATICS LEARNING IN EARLY CHILDHOOD
task used is asking children to represent the cardinal value associated with
a given number name using sets of blocks representing units and tens. Chil-
dren whose native language is Chinese, Korean, or Japanese are consistently
more likely to represent numbers as sets of tens and ones as either a first or
second choice than are children whose native language is English, French,
or Swedish.
Ho and Fuson (1998) compared the performance of Chinese-speaking
preschool children in Hong Kong with English-speaking children in Britain
and the United States. They found that half of the Chinese-speaking 5-year-
olds (but none of the English-speaking children) who could count to at least
50 were able to take advantage of the base-ten structure of number names
to quickly determine the answer to addition problems of the form “10 + n =
?,” compared with other problems. Fuson and Kwon (1992) argued that the
Chinese number-naming structure facilitates the use of a tens-complement
strategy for early addition. In this approach, when adding numbers whose
sum is greater than 10 (e.g., 8 + 7), the smaller addend is partitioned into
the tens-complement of the first addend (2) and the remainder (5); the
answer is 10 plus that remainder (10 + 5). In Chinese-structured number-
naming systems, the answer corresponds to the result of the calculation
(“shi wu” − “10 5”); in English, there is an additional step as the answer
is converted into a different number name (“fifteen”). Fuson and Kwon
reported that most Korean first graders they tested used this method before
it was explicitly taught in school. Explicit instruction may be required for
English-speaking children, but there is evidence that it can be quite success-
ful, even with children from at-risk populations. Fuson and her colleagues
(Fuson, Smith, and Lo Cicero, 1997) report success with explicitly teaching
low-SES urban first graders about the base-ten structure of numbers, with
the result that their end-of-year arithmetic performance approximated that
reported for East Asian children.
LEARNING DISABILITIES IN MATHEMATICS
Mathematics learning disabilities appear in 6 to 10 percent of the el-
ementary school population (Barberisi et al., 2005). Many more children
struggle in one or more mathematics content area at some point during
their school careers (Geary, 2004). Although less research has been devoted
to mathematical than to reading disabilities (Geary and Hoard, 2001;
Ginsburg, 1997), considerable progress has been made over the past two
decades with respect to understanding the nature of the mathematics dif-
ficulties and disabilities that children experience in school (Gersten, Jordan,
and Flojo, 2005).
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DEVELOPMENTAL VARIATION, SOCIOCULTURAL INFLUENCES
Characteristics of Learning Difficulties
Poor computational fluency is a signature characteristic of mathematics
learning disabilities in elementary school (e.g., Geary, 2004; Hasselbring,
Goin, and Bransford, 1988; Jordan and Montani, 1997; Jordan, Hanich,
and Kaplan, 2003a, 2003b; Ostad, 1998; Russell and Ginsburg, 1984).
Computational fluency refers to accurate, efficient, and flexible computa-
tion with basic operations. Weak knowledge of facts reduces cognitive and
attentional resources that are necessary for learning advanced mathematics
(Goldman and Pellegrino, 1987). Computational fluency deficits can be
reliably identified in the first few years of school and, if not addressed, are
very persistent throughout elementary and middle school (Jordan, Hanich,
and Kaplan, 2003b).
Children around the world move through a learning path of levels of
solution methods for addition and subtraction problems. These levels be-
come progressively more abstract, abbreviated, embedded, and complex.
As they move through the levels, many children use a mix of strategies that
vary according to number size and aspects of the problem situation (Geary
and Burlinghman-Dubree, 1989; Siegler and Jenkins, 1989; Siegler and
Robinson, 1982; Siegler and Shipley, 1995).
In contrast, young children with a mathematics learning disability rely
on the most primitive Level 1 methods for extended periods in elementary
school, do not use efficient counting procedures (e.g., counting on from
the larger addend), and make frequent counting errors while learning to
add and subtract (Geary, 1990). They also lag behind other children in the
accuracy and linearity of their number line estimates (Geary et al., 2007).
Researchers have differentiated children with a specific mathematics learn-
ing disability from those with a comorbid learning disability in both math-
ematics and reading. Jordan and colleagues (Hanich et al., 2001; Jordan,
Hanich, and Kaplan, 2003a; Jordan, Kaplan, and Hanich, 2002) as well as
other researchers (e.g., Geary, Hamson, and Hoard, 2000; Landerl, Bevan,
and Butterworth, 2004) suggest that the nature of the mathematical deficits
is similar for both groups, although children with the comorbid condi-
tion show lower performance overall. What differentiates children with a
mathematics-only disability from those with combined mathematics and
reading learning disabilities is that the former group performs better on
word problems in mathematics, which depend on language comprehension
as well as calculation facility. The potential for catching up in mathematics
is much better for children with a mathematics-only disability, who can
exploit their relative strength in general language to compensate for their
deficiencies with numbers.
Some research shows that mathematics learning disabilities can be
traced to early weaknesses in number, number relationships, and number
OCR for page 112
112 MATHEMATICS LEARNING IN EARLY CHILDHOOD
operations as opposed to more general cognitive deficits (e.g., Gersten et al.,
2005; Malofeeva et al., 2004). Weak number competency is reflected in
poorly developed counting procedures, slow fact retrieval, and inaccurate
computation, all characteristics of the disability (Geary et al., 2000; Jordan,
Hanich, and Kaplan, 2003a). Skill with number combinations is tied to
fundamental number knowledge (Baroody and Rosu, 2006; Locuniak and
Jordan, in press). Accurate and efficient counting procedures can lead to
strong connections between a problem and its solution (Siegler and Shrager,
1984). Developmental dyscalculia, a severe form of mathematics disability
that has a known neurological basis, is explained more by domain-specific
impairments in number knowledge than by domain-general deficits related
to memory, spatial processing, or language (Butterworth and Reigosa,
2007). Although debate continues about the underpinnings of mathemat-
ics learning disabilities and diagnostic criteria (e.g., Geary et al., 2007),
weakness in number sense appears to be a common theme in the literature.
This finding has instructional implications for young children’s mathematics
education. Specifically, early interventions that focus on number sense have
the potential to improve children’s mathematics learning outcomes.
Helping High-Risk Children
Early number competencies serve as a foundation for learning formal
mathematics (Griffin et al., 1994; Miller, 1992). Deficits in these can pre-
vent children from benefiting from formal mathematics instruction when
they enter school, regardless of whether they are associated with environ-
mental disadvantages or with genuine learning differences or disabilities
(Baroody and Rosu, 2006; Griffin, 2007). In a recent study, Jordan and
colleagues (in press) found that poor mathematics achievement is mediated
by low number sense regardless of children’s social class. That is, deficits
in number sense are a better predictor of poor mathematics achievement
than SES when all else is equal. Implications of this work suggest that chil-
dren from low-income backgrounds and those with mathematics difficul-
ties would benefit from a mathematics intervention during the early years
(Jordan et al., in press).
Number competencies appear to have neurological origins, with their
core components (e.g., subitization and approximate number representa-
tions) developing without much formal instruction (Berch, 2005; Dehaene,
1997; Feigenson, Dehaene, and Spelke, 2004). These early foundations
provide support for learning more complex number skills involving number
words, number comparisons, and counting. Children with mathematics dif-
ficulties seem to have problems with the symbolic system of number, rather
than the universal analog magnitude system. Knowledge of the symbolic
number system is heavily influenced by experience and instruction (Geary,
OCR for page 113
11
DEVELOPMENTAL VARIATION, SOCIOCULTURAL INFLUENCES
1995; Levine et al., 1992). Engaging young children in number activities
(e.g., a mother or preschool teacher asking a child to give her 4 cookies) and
simple games (e.g., board games that emphasize 1-to-1 correspondences,
counting, and moving along number paths) are important for strengthen-
ing foundations and building conventional number knowledge (Gersten
et al., 2005, Klibanoff et al., 2006; Levine et al., in preparation). Case and
Griffin (1990) report that number sense learning is closely associated with
children’s home experiences with number concepts (e.g., reading number
books with children). Moreover, efforts to teach number-related skills to
high-risk kindergartners show promise for improving mathematics achieve-
ment (Griffin et al., 1994). In a recent study, Ramani and Siegler (2008)
showed that playing a number board game that involved counting on
squares on a number path improved the performance of 5-year-olds from
low-income backgrounds on counting, numeral identification, numerical
magnitude estimation, and number line estimation, and that the gains held
after a follow-up several weeks later. Importantly, children playing this
game said the number words written on the squares as they counted on one
or two more, rather than saying “one” or “two” as they counted on. Play-
ing games to help children master basic number, counting, and arithmetic
concepts and skills has long been advocated by mathematics educators (e.g.,
Baroody, 1987; Ernest, 1986; Wynroth, 1986)—a proposition that is sup-
ported by research (for reviews, see, e.g., Baroody, 1999; Bright, Harvey,
and Wheeler, 1985).
The effects of weaknesses in early mathematics, if not addressed, are
likely to be felt throughout the school years and beyond. There is good
reason to believe that early intensive instruction, both at home and at
school, will give children the background they need to achieve at grade
level in elementary school mathematics and help “shape the course of their
mathematical journey” (Griffin, 2007, p. 392).
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