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Early Predictors of High School Mathematics Achievement

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  • Columbia University Teachers College

Abstract and Figures

Identifying the types of mathematics content knowledge that are most predictive of students' long-term learning is essential for improving both theories of mathematical development and mathematics education. To identify these types of knowledge, we examined long-term predictors of high school students' knowledge of algebra and overall mathematics achievement. Analyses of large, nationally representative, longitudinal data sets from the United States and the United Kingdom revealed that elementary school students' knowledge of fractions and of division uniquely predicts those students' knowledge of algebra and overall mathematics achievement in high school, 5 or 6 years later, even after statistically controlling for other types of mathematical knowledge, general intellectual ability, working memory, and family income and education. Implications of these findings for understanding and improving mathematics learning are discussed.
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Psychological Science
23(7) 691 –697
© The Author(s) 2012
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DOI: 10.1177/0956797612440101
http://pss.sagepub.com
Knowledge of mathematics is crucial to educational and finan-
cial success in contemporary society and is becoming ever
more so. High school students’ mathematics achievement pre-
dicts college matriculation and graduation, early-career earn-
ings, and earnings growth (Murnane, Willett, & Levy, 1995;
National Mathematics Advisory Panel, 2008). The strength
of these relations appears to have increased in recent decades,
probably because of a growing percentage of well-paying jobs
requiring mathematical proficiency (Murnane et al., 1995).
However, many students lack even the basic mathematics
competence needed to succeed in typical jobs in a modern
economy. Children from low-income and minority back-
grounds are particularly at risk for poor mathematics achieve-
ment (Hanushek & Rivkin, 2006).
Marked individual and social-class differences in mathemat-
ical knowledge are present even in preschool and kindergarten
(Case & Okamoto, 1996; Starkey, Klein, & Wakeley, 2004).
These differences are stable at least from kindergarten through
fifth grade; children who start ahead in mathematics generally
stay ahead, and children who start behind generally stay behind
(Duncan et al., 2007; Stevenson & Newman, 1986). There are
substantial correlations between early and later knowledge in
other academic subjects as well, but differences in children’s
mathematics knowledge are even more stable than differences
in their reading and other capabilities (Case, Griffin, & Kelly,
1999; Duncan et al., 2007).
These findings suggest a new type of research that can con-
tribute both to theoretical understanding of mathematical devel-
opment and to improving mathematics education. If researchers
can identify specific areas of mathematics that consistently pre-
dict later mathematics proficiency, after controlling for other
types of mathematical knowledge, general intellectual ability,
and family background variables, they can then determine why
those types of knowledge are uniquely predictive, and society
can increase efforts to improve instruction and learning in those
areas. The educational payoff is likely to be strongest for areas
that are strongly predictive of later achievement and in which
many children’s understanding is poor.
Corresponding Author:
Robert S. Siegler, Carnegie Mellon University–Psychology, 5000 Forbes Ave.,
Pittsburgh, PA 15213
E-mail: rs7k@andrew.cmu.edu
Early Predictors of High School
Mathematics Achievement
Robert S. Siegler
1
, Greg J. Duncan
2
, Pamela E. Davis-Kean
3,4
,
Kathryn Duckworth
5
, Amy Claessens
6
, Mimi Engel
7
,
Maria Ines Susperreguy
3,4
, and Meichu Chen
4
1
Department of Psychology, Carnegie Mellon University;
2
Department of Education, University of California,
Irvine;
3
Department of Psychology, University of Michigan;
4
Institute for Social Research, University of Michigan;
5
Quantitative Social Science, Institute of Education, University of London;
6
Department of Public Policy,
University of Chicago; and
7
Department of Public Policy and Education, Vanderbilt University
Abstract
Identifying the types of mathematics content knowledge that are most predictive of students’ long-term learning is essential
for improving both theories of mathematical development and mathematics education. To identify these types of knowledge,
we examined long-term predictors of high school students’ knowledge of algebra and overall mathematics achievement.
Analyses of large, nationally representative, longitudinal data sets from the United States and the United Kingdom revealed
that elementary school students’ knowledge of fractions and of division uniquely predicts those students’ knowledge of
algebra and overall mathematics achievement in high school, 5 or 6 years later, even after statistically controlling for other
types of mathematical knowledge, general intellectual ability, working memory, and family income and education. Implications
of these findings for understanding and improving mathematics learning are discussed.
Keywords
mathematics achievement, cognitive development, childhood development, fractions, division
Received 10/26/11; Revision accepted 1/31/12
Research Report
692 Siegler et al.
In the present study, we examined sources of continuity in
mathematical knowledge from fifth grade through high school.
We were particularly interested in testing the hypothesis that
early knowledge of fractions is uniquely predictive of later
knowledge of algebra and overall mathematics achievement.
One source of this hypothesis was Siegler, Thompson, and
Schneiders (2011) integrated theory of numerical develop-
ment. This theory proposes that numerical development is a
process of progressively broadening the class of numbers that
are understood to possess magnitudes and of learning the func-
tions that connect those numbers to their magnitudes. In other
words, numerical development involves coming to understand
that all real numbers have magnitudes that can be assigned
specific locations on number lines. This idea resembles Case
and Okamoto’s (1996) proposal that during mathematics
learning, the central conceptual structure for whole numbers, a
mental number line, is eventually extended to rational num-
bers. The integrated theory of numerical development also
proposes that a complementary, and equally crucial, part of
numerical development is learning that many properties of
whole numbers (e.g., having unique successors, being count-
able, including a finite number of entities within any given
interval, never decreasing with addition and multiplication)
are not true of numbers in general.
One implication of this theory is that acquisition of fractions
knowledge is crucial to numerical development. For most chil-
dren, fractions provide the first opportunity to learn that several
salient and invariant properties of whole numbers are not true of
all numbers (e.g., that multiplication does not necessarily pro-
duce answers greater than the multiplicands). This understand-
ing does not come easily; although children receive repeated
instruction on fractions starting in third or fourth grade (National
Council of Teachers of Mathematics, 2006), even high school
and community-college students often confuse properties of
fractions and whole numbers (Schneider & Siegler, 2010;
Vosniadou, Vamvakoussi, & Skopeliti, 2008).
This view of fractions as occupying a central position
within mathematical development differs substantially from
other theories in the area, which focus on whole numbers and
relegate fractions to secondary status. To the extent that such
theories address development of understanding of fractions at
all, it is usually to document ways in which learning about
them is hindered by whole-number knowledge (e.g., Gelman
& Williams, 1998; Wynn, 1995). Nothing in these theories
suggests that early knowledge of fractions would uniquely
predict later mathematics proficiency.
Consider some reasons, however, why elementary school
students’ knowledge of fractions might be crucial for later
mathematics—for example, algebra. If students do not under-
stand fractions, they cannot estimate answers even to simple
algebraic equations. For example, students who do not under-
stand fractions will not know that in the equation 1/3X = 2/3Y,
X must be twice as large as Y, or that for the equation 3/4X = 6,
the value of X must be somewhat, but not greatly, larger than
6. Students who do not understand fraction magnitudes also
would not be able to reject flawed equations by reasoning
that the answers they yield are impossible. Consistent with
this analysis, studies have shown that accurate estimation of
fraction magnitudes is closely related to correct use of frac-
tions arithmetic procedures (Hecht & Vagi, 2010; Siegler
et al., 2011). Thus, we hypothesized that 10-year-olds’ knowl-
edge of fractions would predict their algebra knowledge
and overall mathematics achievement at age 16, even after we
statistically controlled for other mathematical knowledge,
information-processing skills, general intellectual ability, and
family income and education.
Method
To identify predictors of high school mathematics proficiency,
we examined two nationally representative, longitudinal data
sets: the British Cohort Study (BCS; Butler & Bynner, 1980,
1986; Bynner, Ferri, & Shepherd, 1997) and the Panel Study
of Income Dynamics-Child Development Supplement (PSID-
CDS; Hofferth, Davis-Kean, Davis, & Finkelstein, 1998).
Detailed descriptions of the samples and measures used in
these studies and of the statistical analyses that we applied are
included in the Supplemental Material available online; here,
we provide a brief overview.
The BCS sample included 3,677 children born in the United
Kingdom in a single week of 1970. The tests of interest in the
present study were administered in 1980, when the children
were 10-year-olds, and in 1986, when the children were
16-year-olds. Mathematics proficiency at age 10 was assessed
by performance on the Friendly Maths Test, which examined
knowledge of whole-number arithmetic and fractions. Mathe-
matics proficiency at age 16 was assessed by the APU (Applied
Psychology Unit) Arithmetic Test, which examined knowl-
edge of whole-number arithmetic, fractions, algebra, and
probability. General intelligence was assessed at age 10 by
performance on the British Ability Scale, which included mea-
sures of verbal and nonverbal intellectual ability, vocabulary,
and spelling. Parents provided information about their educa-
tion and income and their children’s gender, age, and number
of siblings.
The PSID-CDS included a nationally representative sample
of 599 U.S. children who were tested in 1997 as 10- to 12-
year-olds and in 2002 as 15- to 17-year-olds. At both ages,
they completed parts of the Woodcock-Johnson Psycho-
Educational Battery-Revised (WJ-R), a widely used achieve-
ment test. The 10- to 12-year-olds performed the Calculation
Subtest, which included 28 whole-number arithmetic items (8
addition, 8 subtraction, 7 multiplication, and 5 division items)
and 9 fractions items. The 15- to 17-year-olds completed the
test’s Applied Problems Subtest, which included 60 items on
whole-number arithmetic, fractions, algebra, geometry, mea-
surement, and probability. Applied Problems items 29, 42, 43,
45, and 46 were used to construct the measure of fractions
knowledge, and items 34, 49, 52, and 59 were used to con-
struct the measure of algebra knowledge. Also obtained at
Early Predictors of High School Math Achievement 693
ages 10 to 12 were measures of working memory (as indexed
by backward digit span), demographic characteristics (gender,
age, and number of siblings), and family background (parental
education in years and log mean income averaged over 3
years). Two measures of literacy from the WJ-R, passage com-
prehension and letter-word identification (a vocabulary test),
were obtained both at age 10 to 12 and at age 15 to 17.
Results
The results yielded by bivariate and multiple regression analy-
ses are presented for the British sample in Table 1 and for the
U.S. sample in Table 2. In both tables, results are presented for
algebra scores (Models 1 and 2) and total math scores (Models
3 and 4).
Our main hypothesis was that knowledge of fractions at age
10 would predict algebra knowledge and overall mathematics
achievement in high school, above and beyond the effects of
general intellectual ability, other mathematical knowledge,
and family background. The data supported this hypothesis.
In the United Kingdom (U.K.) data, after effects of all other
variables were statistically controlled, fractions knowledge at
age 10 was the strongest of the five mathematics predictors of
age-16 algebra knowledge and mathematics achievement
(Table 1, Models 2 and 4). A 1-SD increase in early fractions
knowledge was uniquely associated with a 0.15-SD increase in
subsequent algebra knowledge and a 0.16-SD increase in total
math achievement (p < .001 for both coefficients). In the U.S.
data, after effects of other variables were statistically con-
trolled, the relations between fractions knowledge at ages 10
to 12 and high school algebra and overall mathematics
achievement at ages 15 to 17 were of approximately the same
strength as the corresponding relations in the U.K. data (Mod-
els 2 and 4 in Tables 1 and 2). As documented in the Supple-
mental Material (see Tables S5 and S6), in both data sets, the
predictive power of increments to fractions knowledge was
equally strong for children lower and higher in fractions
knowledge.
If fractions knowledge continues to be a direct contributor to
mathematics achievement in high school, as opposed to having
influenced earlier learning but no longer being directly influen-
tial, we would expect strong concurrent relations between high
school students’ knowledge of fractions and their overall math-
ematical knowledge. High school students’ knowledge of frac-
tions did correlate very strongly with their overall mathematics
achievement, in both the United Kingdom, r(3675) = .81, p <
.001, and the United States, r(597) = .87, p < .001. Their frac-
tions knowledge also was closely related to their knowledge of
algebra in both the United Kingdom, r(3675) = .68, p < .001,
and the United States, r(597) = .65, p < .001. Although algebra
is a major part of high school mathematics and fractions consti-
tute a smaller part, the correlation between high school students’
Table 1. Early Predictors of High School Mathematics Achievement: British Cohort Study Data (N = 3,677)
Algebra score Total math score
Predictor
Model 1 (bivariate
regression)
Model 2 (multiple
regression)
Model 3 (bivariate
regression)
Model 4 (multiple
regression)
Age-10 math skills
Fractions 0.42*** (0.02) 0.15*** (0.02) 0.46*** (0.02) 0.16*** (0.02)
Addition 0.20*** (0.02) 0.00 (0.02) 0.26*** (0.02) 0.05** (0.02)
Subtraction 0.22*** (0.02) 0.04* (0.02) 0.24*** (0.02) 0.03 (0.02)
Multiplication 0.32*** (0.02) 0.06*** (0.02) 0.37*** (0.02) 0.08*** (0.02)
Division 0.37*** (0.02) 0.13*** (0.02) 0.40*** (0.02) 0.12*** (0.02)
Age-10 abilities
Verbal IQ 0.39*** (0.02) 0.11*** (0.02) 0.42*** (0.02) 0.10*** (0.02)
Nonverbal IQ 0.41*** (0.02) 0.17*** (0.02) 0.46*** (0.02) 0.19*** (0.02)
Demographic characteristics
Female gender –0.02 (0.02) 0.00 (0.02) –0.01 (0.02) 0.00 (0.01)
Age 0.01 (0.02) –0.03* (0.02) 0.01 (0.02) –0.03* (0.01)
Log mean household income 0.38*** (0.04) 0.08* (0.03) 0.40*** (0.04) 0.09* (0.04)
Parents’ education 0.27*** (0.02) 0.10*** (0.02) 0.29*** (0.02) 0.10*** (0.02)
Number of siblings –0.05** (0.02) –0.01 (0.01) –0.09 (0.02) –0.05*** (0.01)
Mean R
2
.29 .35
Note: This table presents results from regression models predicting algebra and total math scores at age 16 from math skills, cognitive
ability, and child and family characteristics at age 10. All predictors and dependent variables were standardized; therefore, although the
coefficients reported are unstandardized, they can be interpreted much like standardized coefficients. Parameter estimates and stan-
dard errors (in parentheses) are based on 20 multiply imputed data sets. The British Cohort Study data on which these analyses were
based are publicly available from the Centre for Longitudinal Studies, Institute of Education, University of London Web site: http://
www.cls.ioe.ac.uk/bcs70.
*p < .05. **p < .01. ***p < .001.
694 Siegler et al.
knowledge of fractions and their overall mathematics achieve-
ment was stronger than the correlation between their algebra
knowledge and their overall mathematics achievement in both
the U.K. data, r(3675) = .81 versus .73, χ
2
(1, N = 3,677) = 66.49,
p < .001, and the U.S. data, r(597) = .87 versus .80, χ
2
(1, N =
599) = 15.03, p < .001.
Early knowledge of whole-number division also was consis-
tently related to later mathematics proficiency. Among the five
mathematics variables derived from the elementary school tests,
early division had the second-strongest correlation with later
mathematics outcomes in the U.K. data (Table 1) and the stron-
gest correlation with later mathematics outcomes in the U.S.
data (Table 2). Concurrent correlations between high school stu-
dents’ knowledge of division and their overall mathematics
achievement were also substantial both in the United Kingdom,
r(3675) = .59, and in the United States, r(597) = .69, ps < .001.
To the best of our knowledge, relations between elementary
school children’s division knowledge and their mathematics
proficiency in high school have not been documented
previously.
Regressions like those in Tables 1 and 2 place no con-
straints on the estimated coefficients. Therefore, we reesti-
mated our regression models, first imposing an equality
constraint on the coefficients for fractions and division, and
then imposing an equality constraint on the coefficients for
addition, subtraction, and multiplication (see Table S4 in the
Supplemental Material). Finally, we tested whether the pooled
coefficients for these two sets of skills differed from each
other. The predictive relation was stronger for fractions and
division than for the other mathematical skills in both the U.K.
and the U.S. data—U.K.: F(1, 3664) = 36.92, p < .001, for
algebra and F(1, 3664) = 28.79, p < .001, for overall mathe-
matics; U.S.: F(1, 558) = 7.12, p < .01, for algebra and F(1,
558) = 9.72, p < .01, for overall mathematics (see Table S4).
The greater predictive power of knowledge of fractions and
knowledge of division was not due to their generally predict-
ing intellectual outcomes more accurately. When Models 2
and 4 in Tables 1 and 2 were applied to predicting high school
students’ literacy (spelling and vocabulary from the British
Ability Scale in the BCS; passage comprehension and letter-
word identification from the WJ-R in the PSID-CDS), only
two of the eight predictive relations between fractions and
division knowledge, on the one hand, and literacy, on the
other, were significant: Fractions knowledge predicted vocab-
ulary in the U.K. data, and division knowledge predicted
letter-word identification in the U.S. data (see Tables 3 and 4).
Moreover, in all cases but one, the pooled predictive effect of
fractions and division knowledge on literacy was no greater
than the pooled predictive effect of addition, subtraction, and
multiplication knowledge on literacy (see Table S4 in the Sup-
plemental Material). The one exception was that fractions and
division knowledge more accurately predicted vocabulary in
Table 2. Early Predictors of High School Mathematics Achievement: Panel Study of Income Dynamics Data (N = 599)
Algebra score Total math score
Predictor
Model 1 (bivariate
regression)
Model 2 (multiple
regression)
Model 3 (bivariate
regression)
Model 4 (multiple
regression)
Early math skills
Fractions 0.41*** (0.06) 0.17* (0.08) 0.49*** (0.05) 0.18** (0.06)
Addition 0.26*** (0.06) 0.09 (0.06) 0.30*** (0.06) 0.05 (0.05)
Subtraction 0.26*** (0.05) 0.04 (0.05) 0.39*** (0.05) 0.12* (0.05)
Multiplication 0.31*** (0.05) 0.00 (0.06) 0.43*** (0.05) 0.02 (0.05)
Division 0.40*** (0.05) 0.19*** (0.06) 0.53*** (0.05) 0.26*** (0.06)
Early abilities
Backward digit span 0.29*** (0.06) 0.10 (0.06) 0.33*** (0.05) 0.08 (0.05)
Passage comprehension 0.38*** (0.05) 0.11 (0.06) 0.51*** (0.05) 0.20*** (0.05)
Demographic characteristics
Female gender –0.06 (0.05) –0.09 (0.05) –0.08 (0.05) –0.13*** (0.04)
Age 0.04 (0.05) –0.18*** (0.05) 0.06 (0.05) –0.22*** (0.04)
Log mean family income
(1994–1996)
0.31*** (0.06) 0.05 (0.06) 0.38*** (0.06) 0.12* (0.06)
Parents’ education 0.39*** (0.05) 0.19*** (0.05) 0.41*** (0.06) 0.11 (0.06)
Number of siblings –0.17** (0.06) –0.03 (0.05) –0.18** (0.06) –0.03 (0.04)
Mean R
2
.35 .52
Note: This table presents results from regression models predicting algebra and total math scores at age 15 to 17 from math skills,
cognitive abilities, and child and family characteristics at age 10 to 12. All predictors and dependent variables were standardized;
therefore, although the coefficients reported are unstandardized, they can be interpreted much like standardized coefficients.
Parameter estimates and standard errors (in parentheses) are based on 20 multiply imputed data sets. The data on which these
analyses were based came from the Panel Study of Income Dynamics public-use data set, available at http://psidonline.isr.umich.edu.
*p < .05. **p < .01. ***p < .001.
Early Predictors of High School Math Achievement 695
Table 3. Results From Regression Models Predicting Literacy at Age 16 From Math Skills and Child and Family
Characteristics at Age 10: British Cohort Study Data (N = 3,677)
Spelling Vocabulary
Predictor
Model 1 (bivariate
regression)
Model 2 (multiple
regression)
Model 3 (bivariate
regression)
Model 4 (multiple
regression)
Age-10 math skills
Fractions 0.16*** (0.02) 0.02 (0.02) 0.38*** (0.02) 0.09*** (0.02)
Addition 0.10*** (0.02) 0.00 (0.02) 0.21*** (0.02) 0.03 (0.02)
Subtraction 0.12*** (0.02) 0.04 (0.02) 0.17*** (0.02) 0.00 (0.02)
Multiplication 0.17*** (0.02) 0.05* (0.02) 0.27*** (0.02) 0.03 (0.02)
Division 0.16*** (0.02) 0.03 (0.02) 0.28*** (0.02) 0.04 (0.02)
Age-10 abilities
Verbal IQ 0.19*** (0.02) 0.09*** (0.02) 0.49*** (0.02) 0.30*** (0.02)
Nonverbal IQ 0.20*** (0.02) 0.08*** (0.02) 0.38*** (0.02) 0.10*** (0.02)
Demographic characteristics
Female gender 0.14*** (0.02) 0.14*** (0.02) 0.00 (0.02) 0.03 (0.02)
Age 0.01 (0.02) 0.00 (0.02) 0.02 (0.02) –0.02 (0.02)
Log mean household income 0.19*** (0.03) 0.05 (0.04) 0.39*** (0.03) 0.05 (0.03)
Parents’ education 0.13*** (0.02) 0.04* (0.02) 0.32*** (0.02) 0.14*** (0.05)
Number of siblings –0.09*** (0.02) –0.07*** (0.02) –0.12*** (0.02) –0.05*** (0.02)
Mean R
2
.09 .30
Note: All predictors and dependent variables were standardized; therefore, although the coefficients reported are unstandardized,
they can be interpreted much like standardized coefficients. Parameter estimates and standard errors (in parentheses) are based
on 20 multiply imputed data sets. The British Cohort Study data on which these analyses were based are publicly available from the
Centre for Longitudinal Studies, Institute of Education, University of London Web site: http://www.cls.ioe.ac.uk/bcs70.
*p < .05. ***p < .001.
the U.K. data, F(1, 3664) = 5.53, p < .05. (See Tables 3 and 4
for a summary of predictors of literacy in the two data sets.)
Discussion
These findings demonstrate that elementary school students’
knowledge of fractions and whole-number division predicts
their mathematics achievement in high school, above and
beyond the contributions of their knowledge of whole-number
addition, subtraction, and multiplication; verbal and nonverbal
IQ; working memory; family education; and family income.
Knowledge of fractions and whole-number division also had a
stronger relation to math achievement than did knowledge of
whole-number addition, subtraction, and multiplication; verbal
IQ; working memory; and parental income. These results were
consistent across data sets from the United Kingdom and the
United States. The fact that the relations of the predictor vari-
ables to algebra knowledge and overall mathematics achieve-
ment were similar in strength in the two samples, despite
differences in the samples, the tests, and the times at which the
data were obtained, is reason for confidence in the generality of
the findings.
The correlation between knowledge of fractions in elemen-
tary school and achievement in algebra and mathematics over-
all in high school was expected, but the relation between early
division knowledge and later mathematical knowledge was
not. Fractions and division are inherently related (N/M means
N divided by M), but the finding that early knowledge of
fractions and early knowledge of division accounted for inde-
pendent variance in later algebra knowledge and overall math-
ematics achievement indicated that neither relation explained
the other.
There are several likely reasons why knowledge of division
uniquely predicted later mathematics achievement. Mastery of
whole-number division, like mastery of fractions, is required
to solve many algebra problems (e.g., to apply the quadratic
equation). Also, as is the case with fractions, high percentages
of students fail to master division; thus, when high school stu-
dents in the PSID-CDS were presented with a seemingly easy
problem in which a boy wants to fly on a plane that travels
400 miles per hour in order to visit his grandmother who lives
1,400 air miles away, only 56% of the students correctly
indicated how long the flight would take. More speculatively,
poor knowledge of both division and fractions might lead stu-
dents to give up trying to make sense of mathematics, and
instead to rely on rote memorization in subsequent mathemat-
ics learning.
An alternative interpretation is that the unique predictive
value of knowledge of fractions and knowledge of division
stems from those operations being more difficult than addi-
tion, subtraction, and multiplication, and thus measuring more
advanced thinking. Some of our results are inconsistent with
this interpretation, however. First, knowledge of fractions and
knowledge of division were not uniquely predictive of most
subsequent literacy skills (see Tables 3 and 4), as should have
been the case if their predictive value was due solely to their
696 Siegler et al.
greater difficulty. Second, spline tests (see Tables S5 and S6 in
the Supplemental Material) showed that the predictive strength
of early knowledge of fractions and division did not differ
between students with greater and lesser mathematics achieve-
ment in high school. Thus, the unique predictive value of early
fractions and division knowledge seems to be due to many
students not mastering fractions and division and to those
operations being essential for more advanced mathematics,
rather than simply to fractions and division being relatively
difficult to master.
Over 30 years of nationwide standardized testing, mathe-
matics scores of U.S. high school students have barely budged
(National Mathematics Advisory Panel, 2008). The present
findings imply that mastery of fractions and division is needed
if substantial improvements in understanding of algebra and
other aspects of high school mathematics are to be achieved.
One likely reason for students’ limited mastery of fractions
and division is that many U.S. teachers lack a firm conceptual
understanding of fractions and division. In several studies, the
majority of elementary and middle school teachers in the
United States were unable to generate even a single explana-
tion for why the invert-and-multiply algorithm (i.e., a/b ÷
c/d = ad × bc) is a legitimate way to solve division problems
with fractions. In contrast, most teachers in Japan and China
generated two or three explanations in response to the same
question (Ma, 1999; Moseley, Okamoto, & Ishida, 2007).
These and the present results suggest that improved teaching
of fractions and division could yield substantial improvements
in students’ learning, not only of fractions and division but of
more advanced mathematics as well.
Declaration of Conflicting Interests
The authors declared that they had no conflicts of interest with
respect to their authorship or the publication of this article.
Funding
National Science Foundation Grant 0818478 and Department of Edu-
cation, Instructional and Educational Sciences Grants R324C100004
and R305A080013 supported this research.
Supplemental Material
Additional supporting information may be found at http://pss.sagepub
.com/content/by/supplemental-data
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Table 4 . Results From Regression Models Predicting Literacy at Age 15 to 17 From Math Skills and Child and Family
Characteristics at Age 10 to 12: Panel Study of Income Dynamics Data (N = 599)
Letter-word identification Passage comprehension
Predictor
Model 1 (bivariate
regression)
Model 2 (multiple
regression)
Model 3 (bivariate
regression)
Model 4 (multiple
regression)
Early math skills
Fractions 0.40*** (0.05) 0.03 (0.05) 0.41*** (0.05) 0.04 (0.06)
Addition 0.22*** (0.05) –0.05 (0.04) 0.19*** (0.06) –0.06 (0.05)
Subtraction 0.38*** (0.06) 0.07 (0.06) 0.35*** (0.05) 0.05 (0.05)
Multiplication 0.47*** (0.06) 0.14* (0.06) 0.43*** (0.05) 0.11* (0.05)
Division 0.49*** (0.06) 0.14* (0.07) 0.44*** (0.05) 0.08 (0.05)
Early abilities
Backward digit span 0.30*** (0.05) 0.03 (0.04) 0.36*** (0.05) 0.10* (0.04)
Passage comprehension 0.65*** (0.06) 0.46*** (0.06) 0.65*** (0.04) 0.43*** (0.05)
Demographic characteristics
Female gender 0.15** (0.05) 0.05 (0.04) 0.13* (0.05) 0.03 (0.04)
Age 0.17*** (0.05) –0.08 (0.05) 0.13** (0.05) –0.08 (0.04)
Log mean family income
(1994–1996)
0.33*** (0.05) 0.08 (0.05) 0.39*** (0.06) 0.05 (0.05)
Parents’ education 0.32*** (0.07) 0.05 (0.06) 0.47*** (0.05) 0.23*** (0.05)
Number of siblings –0.06 (0.06) 0.06 (0.04) –0.15*** (0.05) 0.03 (0.04)
Mean R
2
.51 .53
Note: All predictors and dependent variables were standardized; therefore, although the coefficients reported are unstandardized,
they can be interpreted much like standardized coefficients. Parameter estimates and standard errors (in parentheses) are based on
20 multiply imputed data sets. Models were weighted by 2002 child-level weights and adjusted for the clustering of children within
the same family. The data on which these analyses were based came from the Panel Study of Income Dynamics public-use data set,
available at http://psidonline.isr.umich.edu.
*p < .05. **p < .01. ***p < .001.
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