Early Risk Factors for Later
Mathematics Difficulties
Paul L. Morgan, Ph.D., Population Research Institute,
The Pennsylvania State University
George Farkas, Ph.D., University of California, Irvine
Steve Maczuga, M.S., Population Research Institute, The
Pennsylvania State University
1
This work is supported by grant #R324A07270, National Center for Special Education, Institute of
Education Sciences
No official endorsement should be inferred
Illustrative data from NAEP (2005)
Those falling below “basic” level of mathematics proficiency
2
Grade level
Disabled
Non-disabled
4th grade
43%
17%
8th grade
68%
27%
12th grade
83%
36%
Why mathematics difficulties is
something to prevent
 Being poorly skilled in mathematics lowers an adult’s
employability & wages, over & above poor reading, low IQ, &
many other factors (Rivera-Batiz, 1992)
 Even adults with good literacy skills are more likely to be
unemployed (& less likely to be promoted when employed) if
they have poor mathematics skills (Parsons & Bynner, 1997)
 Being poorly skilled in mathematics may have negative “carry
over” effects on children’s socio-emotional development
 Children who are poorly skilled in mathematics can also begin to
report internalizing psychopathology (Morgan et al., in review)
3
Field’s limited knowledge about MD
 A preliminary search of PsychInfo yields the following:
 “Mathematics disabilities” and “longitudinal” and school
children” = 32 citations
 “Reading disabilities” and “longitudinal” and “school
children” = 340 citations
 Much of the extant work on MD has used small convenience
samples identified with varying criteria and followed over
fairly short time periods
 “Figuring this out” has implications for practice
 Wait & fail vs. Wait & see”
4
Hypothesized or identified risk factors for
MD
 Child’s socio-demographic characteristics
 Age, gender, SES (e.g., Jordan et al., 2007; Lachance & Mazzocco,
2006; Mazzocco & Thompson, 2005)
 More “educationally relevant factors”
 Retention, reading difficulties, inattention & other learning-related
behavior problems, disability (Cirino et al., 2007; Fuchs et al.,
2006; Miller & Mercer, 1997)
 Some contradictory results from these investigations
 Gender: Aunola et al. (2004); Jordan et al. (2006) vs. Lachance &
Mazzocco (2006)
 SES: Jordan et al. (2007) vs. Mazzocco & Thompson (2005)
 Retention: Hong & Raudenbush (2005) vs. Jordan et al. (2003)
5
Repeated learning failure as a
proposed “core characteristic”
 Identifying children as having MD using low score on a single
measure’s administration is likely to identify a heterogeneous group,
whose low skill level may result from distinct mechanisms
 Some children may perform poorly due to neurological-based disorders,
while other’s may perform poorly due to economic disadvantage or
fewer opportunities to learn mathematics informally in the home (e.g.,
Jordan et al., 2006; Starkey et al., 2004)
 Repeated failure to learn mathematics has been put forth as a criterion for
mathematics disability, but the predictive utility of this criterion, as well as
risk factors for repeated learning failure has yet to be established (Geary et
al., 2000; Mazzocco & Myers, 2003)
6
Organization to today’s seminar
 Investigating two, inter-related questions
 Research Question #1, Study 1: To what extent does the timing and
persistence of mathematics difficulties during kindergarten elevate
children’s risk of displaying mathematics difficulties throughout the
remainder of elementary school?
 Research Question #2, Study 2: If the persistence of mathematics
difficulties “matters,” then which children are especially at risk for
experiencing persistent mathematics difficulties?
7
Study 1’s purpose
 What factors predict of children’s initial mathematics knowledge
& skills growth over time?
 Study 1 characterized these risk factors as belonging to one of
three “blocks”
 The relative severity of the child’s MD (i.e., its timing &
persistency)
 The child’s socio-demographics (e.g., SES)
 The child’s more “educationally relevant” factors (e.g., his or her
relative reading skill, the frequency of the child’s learning-related
behaviors, the child’s disability status)
8
Study 1’s theoretical categorizations of
MD
 “Experiential disadvantage”
 Some children might display an early onset of MD (i.e., at K entry)
due to “or a lack of high-quality interactions with siblings, parents,
& other adults during preschool (e.g., Klibanoff et al., 2006)
 “Instructional disadvantage”
 Other children might display a later onset of MD (i.e., at the end of
K), which may be due to or a lack of response to typically
instruction offered by general classroom teachers (e.g., Jordan et
al., 2006)
 “Doubly disadvantaged”
 Still other children might display repeated MD, despite their
preschool & K experiences, and so may be more likely to have
mathematics disabilities
9
Study 1’s categorizations of the timing
& persistency of the child’s MD
 MD as a score in the bottom 10% of scores on the Mathematics




1
0
Test at a particular survey wave
D10: Children who displayed MD at the fall but not spring of K
D01: Children who displayed MD at the spring but not fall of K
D11: Children who displayed MD at both the fall & spring of K
D00: Children who displayed MD at neither the fall nor the
spring of K
Study 1’s major research question
Does the onset &
relative persistency of
MD in K (D10, D01,
D11)…
1
1
….after statistically
controlling for sociodemographic & more
educationally relevant
factors measured
during K…
…predict children’s
initial skill level
(spring of 1st grade) &
over time growth
(spring of 1st grade to
spring of 3rd to spring
of 5th grade) in
mathematics?
Database & analytical sample
 ECLS-K database, a large, nationally representative sample of children
entering K in 1998-1999, and tracked through their elementary school
years
 Analytical sample = 7,892 children
 Sample included those children whose parent-identified race/ethnicity
was White, non-Hispanic or Black/African-American, non-Hispanic
 Other race/ethnic groups excluded because of substantial amounts of missing data
on the K administration of the Reading Test
 Sample also included those children who were retained and those who
changed schools between survey waves
 Socio-demographics of the ECLS-K full & analytical samples fairly
equivalent
1
2
Study 1’s measures
 Mathematics achievement & MD
 IRT-scaled, untimed & individually administered Mathematics Test
(Reliabilities of thetas = .89-.94)
 Socio-demographic factors
 Age, SES, Gender
 Educationally relevant factors
13
 K retention
 K reading difficulties (RD)
 IRT-scaled untimed & individually administered Reading Test (Reliability of K
theta = .91) Difficulties as score <10%
 Learning-related behavior problems
 Teacher rating, SSRS Approaches to Learning subscale (e.g., attention, task
persistence) Difficulties as score <10%
 Disability status
Study 1’s selected descriptive statistics
Fall K
Spring K
Spring
Grade
1st
Spring
Grade
3rd
Spring
Grade
5th
MD in fall K only
N
Mean (SD)
326
326
323
259
211
12.96(1.18)
26.18(4.12)
47.12(10.79)
76.45(16.43)
96.93(17.45)
MD in spring K only
N
Mean (SD)
325
325
323
245
180
17.18(2.10)
19.30(1.77)
39.96(10.38)
67.04(14.29)
88.55(20.34)
MD in both fall and spring K
N
Mean (SD)
465
465
461
352
239
11.80 (1.62)
17.09(2.72)
34.99(9.72)
59.86(10.97)
79.35(17.83)
No MD in either fall or spring K.
N
Mean (SD)
6776
6776
6718
5764
4578
25.48(8.40)
36.67(11.03)
62.60(15.54)
98.31(19.06)
118.71(18.42)
Note. MD = Mathematics Difficulties. Estimates are weighted by child-level sampling weights.
Mean Mathematics Test score, by
relative MD groupings
1
5
Relative persistency of these MD
categorizations
16
D Spring of 1st
grade
D Spring of 3rd
grade
D Spring of 5th
grade
D10
78/323
(24%)
68/259
(26%)
59/211
(28%)
D01
139/323
(43%)
113/245
(46%)
82/180
(46%)
D11
314/461
(68%)
245/352
(70%)
156/239
(65%)
D00
252/6718
(3%)
236/5764
(4%)
224/4578
(4%)
Study 1’s analytical method & models
 HLM using repeated observations over time as Level 1 &
individual child as Level 2 estimating a quadratic growth
curve:
 Level 1: Yti = π0i + π1it + π2it2 + eti
 Predictor models
 Model 1: the baseline or unconditional model
 Model 2: Adds the dummy variables using D10, D01, D11, with
D00 as reference group
 Model 3: Adds to Model 2 the child’s socio-demographic
characteristics
 Model 4: Adds to Model 3 the more educationally relevant
characteristics (e.g., child’s reading difficulties, disability status)
17
Study 1’s growth modeling results
Model 2
Model 1
Model 3
Model 4
Intercept
Slope of
time
Intercept
Slope
of Time
Intercept
Slope of
Time
Intercept
Slope of
Time
59.63*
20.82*
62.79*
21.10*
59.58*
20.45*
60.80*
20.55*
D10
-16.55*
-1.57*
-12.54*
-1.57*
-11.74*
-1.30*
D01
-24.25*
-1.90*
-18.90*
-1.45*
-17.51*
-1.29*
D11
-29.17*
-2.91*
-22.62*
-2.40*
-19.69*
-1.96*
Age in month
.71*
-.19*
.72*
-.20*
SES
5.69*
.55*
5.46*
.53*
White
6.29*
1.25*
6.24*
1.35*
Female
-3.43*
-.70*
-4.19*
-.75*
Repeat K
-4.33
-.59
-3.81
-.62
-.12
-.50
Fall K. Approaches
difficulty
-6.44*
-.55
IEP
-6.70*
-.06
Intercept
Fall K. Reading
difficulty
Study 1’s contributions
 Young children displaying persistent-type MD averaged 5th grade
Mathematics Test scores over 2 SD below those who had not
displayed MD in K (even those displaying variable-type MD
averaged 5th grade scores 1-1.5 SD below their typical peers)
 Indicate the need to systematically monitoring young children’s
mathematics knowledge
 Young children repeatedly displaying MD likely need intensive
early intervention to be mathematically skilled by end of 5th
grade, to avoid “wait to fail”
19
Study 1’s limitations
 Study 1 identified RMD as a relatively stable phenomenon, in
that children, even those only who were kindergarten-aged,
were unlikely to “catch up” with those who were displaying
RMD, and that the relative persistency of RMD “mattered”
 Yet Study 1 fails to identify risk factors that may increase
children’s likelihood of experiencing RMD.
 That is, who are the children who are likely to be displaying RMD
 Is there a “common core” of risk factors for RMD that might be
incorporated into the field’s screening, monitoring, and prevention
efforts?
20
Study’s 2’s purpose and suppositions
 Study 2’s purpose
 Is there a “common core” of factors that increase a child’s risk of
experiencing repeated learning difficulties in mathematics?
 Study’s suppositions
 Identifying risk factors “early” is better than identifying these
factors “late”
 Doing so helps guide earlier screening, monitoring, and intervention
efforts
 Children who repeatedly fail to attain mathematical proficiency
should be of elevated concern
 These children are consistently non-responsive to the instructional
practices and routines being provided
21
Study 2’s brief overview
 We used two population-based, longitudinal datasets (i.e.,
the ECLS-K, the ECLS-B) to identify early risk factors for
later, repeated mathematics difficulties (RMD)
 We estimated the predicted effects for a wide range of risk
factors
 We were particularly interested in potentially malleable and
“educationally relevant” factors
 We statistically controlled for the “autoregressor” and strong
confounds in the analyses to more conservatively estimate
predicted effects
22
Study’s 2 two datasets
 Two NCES-maintained datasets
 Early Childhood Longitudinal Study-Kindergarten Cohort (ECLS-K)
 Kindergarten-8th grade longitudinal, nationally representative sample
 Early Childhood Longitudinal Study-Birth Cohort (ECLS-K)
 Birth-Kindergarten longitudinal, nationally representative sample
 Both datasets include individually-administered, adaptive
measures of:
 academic achievement
 direct observation ratings of learning-related behaviors
 multi-source surveys of the children’s socio-demographic,
gestational, and birth characteristics
23
Analytical samples, time
periods,measures,
operationalizations
ECLS-K
ECLS-B
Analytical samples
N=5,838
N=5,650*
Time periods
Spring of Kindergarten, 3rd, 5th,
& 8th grade
24, 48, & 60 months
Measures
Socio-demographics, birth
characteristics, reading and
mathematics achievement, &
behavior
Socio-demographics, gestational
& birth characteristics, cognitive
functioning, vocabulary, reading
and mathematics achievement,
& behavior
Repeated Mathematics
Difficulties (RMD)
Score below 25% cut off at
spring of 3rd, 5th, & 8th grade
administrations of ECLS-K
Mathematics Test
Score below 25% at both
Preschool & Kindergarten
administrations of modified
ECLS-K Mathematics Test
RMD % of analytical samples
16.44% (n=960)
15.68% (n=900*)
24
*Sub-sample rounded to nearest 50
Analytical methods
ECLS-K
Descriptive statistics
Child- and family-level socio-demographics, child-level learner
characteristics
Logistic regression (Odds
Ratios as the effect size metric)
Step 1: Dichotomized as “0” &
“1,” with “1” as being in the
group of children with scores in
the lowest 25% of the score
distribution of the spring of 3rd,
5th, & 8th grade administrations
of the the Mathematics Test, and
“0” as not not being in this
group
Step 2: Predicted the child’s
group membership, using a
range of socio-demographic,
birth, & learner characteristics,
and controlling for the
autoregressor, at spring of
kindergarten
25
ECLS-B
Step 1: Dichotomized as “0” &
“1,” with “1” as being in the
group of children with scores in
the lowest 25% of the score
distribution of the 48 & 60
month administrations of the
the Mathematics Test, and “0” as
not not being in this group
Step 2: Predicted the child’s
group membership, using a
range of socio-demographic,
gestational & birth, & learner
characteristic, and controlling
for a strong confound (i.e.,
cognitive delay), at 24 months
Study’s longitudinal designs
Predictors measured by
Criterions measured by
ECLS-K
Spring of Kindergarten
Spring of 3rd, 5th, and 8th grade
ECLS-B
24 months
Preschool (48 months) and
Kindergarten (60 months)
Datasets
26
Study 2’s ECLS-K measures
 ECLS-K Mathematics Test
 Individually-administered, untimed IRT measure measure of a range of age-
and grade-appropriate mathematics skills (e.g., identify numbers and shapes,
sequence, multiply, use fractions)
 Reliabilities of the IRT scaled scores ranged from .89 to .94
 “Low” score as having a score in the lowest 25% of the score distribution of
the spring of kindergarten Mathematics Test distribution
 ECLS-K Reading Test
 Individually-administered, untimed IRT measure measure children’s basic
skills (e.g., print familiarity, letter recognition, decoding), vocabulary
(receptive vocabulary), and comprehension (e.g., making interpretations)
 Reliabilities of the IRT scaled scores ranged from .91 to .96
 “Low” score as having a score in the lowest 25% of the score distribution of
the spring of kindergarten Reading Test administration
27
ECLS-K measures (cont.)
 Modified version of the Social Skills Rating Scale
 Kindergarten teacher rated the frequency of that the child engaged in
the particular behavior
 Strong split half reliabilities in kindergarten (e.g., .89, learningrelated behaviors)
 Three sub-scales, using “worst” 25% cut-off criterion
 Learning-related behavior problems (e.g., displays attentiveness, persists at
tasks)
 Externalizing problem behaviors (e.g., argues, disturbs the class)
 Internalizing problem behaviors (e.g., seems anxious, lonely)
 Survey data of children’s socio-demographics, birth characteristics
(e.g., low birthweight, mother’s education level)
28
Descriptive statistics for RMD and nonRMD groups, ECLS-K continuous data
RMD
Non-RMD
Mean (SD)
Mean (SD)
SD Unit Differences
Mathematics Test Score
25.63 (5.69)
40.22 (11.53)
-1.3
Reading Test Score
36.87 (7.21)
49.47 (14.19)
-.89
Approaches to Learning
2.66 (0.67)
3.24 (0.59)
-.98
Externalizing Problem Behavior
1.85 (0.68)
1.63 (0.56)
.39
Internalizing Problem Behavior
1.65 (0.51)
1.53 (0.46)
.26
Kindergarten Predictors
29
Logistic regression of 3rd, 5th, and 8th grade
RMD (ORs) using kindergarten predictors
Kindergarten Predictors
Low Kindergarten Math
30
Model 1
19.79 ***
Model 2
Model 3
Model 4
16.90 ***
16.94 ***
9.76 ***
Child is Male
0.52 ***
0.52 ***
0.38 ***
Child Age at Assessment
1.06 **
1.06 **
1.06 **
Mother’s Education, Less than High
School Grad.
Mother’s Education, High School Grad.
5.00 ***
5.00 ***
4.89 ***
1.94 ***
1.93 ***
1.94 ***
Mother’s Age at Birth > 35 years
0.82
0.82
0.84
Black
2.85 ***
2.86 ***
2.75 ***
Hispanic
0.76
0.76
0.82
Other
0.92
0.93
0.90
Birth Weight <= 1500 grams
1.23
0.99
Moderately Low Birth Weight
0.89
1.03
Low Kindergarten Reading
2.00 ***
Low Approaches to Learning
2.03 **
High Externalizing Behavior
1.61
High Internalizing Behavior
1.28
Study 2’s ECLS-K results
 Potentially malleable and educationally relevant risk factors
by the end of kindergarten for 3rd-8th grade RMD include
earlier history of MD, earlier history of RD, and earlier
history of learning-related behavior problems
 These risk factors are not mediated by the child’s or family’s
socio-demographics, or the child’s birth characteristics,
despite their sometimes strong predicted effects
 The onset of MD by kindergarten is an especially strong risk
factors for MD through the elementary and middle school
years
31
Study 2’s ECLS-B measures
 Modified Bayley
 Individually-administered measure of children’s age-appropriate cognitive
functioning as manifested in memory, habituation, preverbal communication,
problem-solving and concept attainment. The interviewers ask children to
complete specific tasks (e.g., “turn pages in a book,” “look for contents of a
box,” “put three cubes in a cup”).
 IRT reliability coefficient for the BSF-R mental scale at 24 months was .88
(NCES, 2007)
 “Low” as having a score in the lowest 25% of the score distribution
 Modified McArthur Communication Development Inventory (CDI)
 Child’s parents asked if the child is saying each of 50 vocabulary words (e.g.,
“meow,” “shoe,” “mommy,” “chase”)
 CDI recently reported to classify children into language status groups with
97% accuracy (Skarakis-Doyle et al., 2009)
 “Low” as having a total score in the lowest 25% of the score distribution
32
ECLS-B measures (cont.)
 Learning-related behavior problems
 Modified version of the Bayley’s Behavior Rating System
 Field staff administering the Bayley also rated the children’s
behavior on a frequency scale (e.g., 1=“constantly off task,”
5=“constantly attends”)
 Cronbach alpha of .92 for the behavioral items (Raikes et al.,
2007)
 “High” as having a score in the highest 25% of the distribution of
total scores for “inattentive,” “not persistent,” “no interest”
 Birth certificate data and parental survey on a range of socio-
demographic, gestational, and birth characteristics (e.g.,
preterm, low birthweight, congenital anomalies)
33
Descriptive statistics for RMD and nonRMD groups, ECLS-B continuous data
RMD
Non-RMD
Mean (SD)
Mean (SD)
SD Unit Differences
Modified Bayley
Score
121.39 (9.01)
128.79 (10.35)
-.71
Modified CDI Word
Score
23.67 (10.85)
30.35 (11.62)
-.57
24 months
34
Logistic regression of 48-60 month RMD
using 24 month predictors
24 Month Predictors
Low Bayley at 24
Months
Child’s Age at 60
month Assessment
Male
35
Model 1
3.64 ***
Model 2
Model 3
Model 4
3.02 ***
2.95 ***
2.23 ***
0.79 ***
0.79 ***
0.71 ***
1.18
1.22
1.12
African-American
1.35 *
1.32 *
1.34 *
Hispanic
1.18
1.21
1.24
Other
1.19
1.16
1.13
Mother’s
Education, no
diploma
Mother’s
Education, High
School Graduate
Mother’s Age over 35
at Child’s Birth
Mother Not Married
at Child’s Birth
4.66 ***
4.47 ***
4.40 ***
2.28 ***
2.22 ***
2.24 ***
0.89
0.86
0.84
1.24
1.22
1.22
Logistic regression of 48-60 month RMD
using 24 month predictors (cont.)
36
24 Month
Predictors
Model 3 (cont.)
Model 4 (cont.)
Very Pre-Term
1.15
1.09
Moderately Pre-Term
1.34
1.27
Very Low Birth
Weight
Moderately Low
Birth Weight
Labor Complications
1.77
1.65
1.54 *
1.58 **
0.75 *
0.74 *
Medical Risk Factors
1.03
1.01
Behavioral Risk
Factors
Obstetric Procedures
1.14
1.17
0.93
0.94
Congenital Anomalies
0.80
0.79
Low Word Score at
24 Months
High L-R
Behaviors at 24
Months
1.58 **
1.41 **
Study 2’s ECLS-B results
 Potentially malleable and educationally relevant risk factors
by 24 months for 48-60 month RMD include earlier history
of cognitive delay, language delay, and learning-related
behavior problems
 These risk factors are not mediated by the child’s or family’s
socio-demographics, or the child’s gestational or birth
characteristics, despite their sometimes strong predicted
effects
37
Collectively, what do Study 1’s and
Study 2’s analyses tell us?
 RMD is a relatively stable condition, in that children who are experiencing MD
“early” are likely to still be experiencing MD “late.”
 A “common core” of factors that increase a child’s risk of RMD may exist, that
includes:




Prior history of MD or an early onset of cognitive delay
Reading or language difficulties
Learning-related behavior problems
Being raised by a mother with a low level of education
 Prior history of learning difficulties and learning-related behavior problems may
be particularly educationally relevant, and potentially malleable
 The effects of these risk factors are robust, and can be detected early, by
children’s kindergarten or even toddler years
 Early screening, monitoring, and intervention efforts may need to be “multifaceted” so as to account for the multiple developmental pathways that may
result in children experiencing RMD
38
Thank you!
 For additional questions, please contact:
Paul L. Morgan
Department of Educational Psychology, School Psychology, and
Special Education
The Pennsylvania State University
University Park, PA 16802
(814) 863-2285
paulmorgan@psu.edu
39