Content uploaded by Robert Siegler

Author content

All content in this area was uploaded by Robert Siegler

Content may be subject to copyright.

Psychological Science

23(7) 691 –697

© The Author(s) 2012

Reprints and permission:

sagepub.com/journalsPermissions.nav

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

Schneider’s (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

References

Butler, N., & Bynner, J. M. (1980). British Cohort Study: Ten-Year

Follow-up, 1980. UK Data Archive [distributor]. Retrieved from

http://www.esds.ac.uk/findingData/bcs70.asp

Butler, N., & Bynner, J. M. (1986). British Cohort Study: Sixteen-

Year Follow-up, 1986. UK Data Archive [distributor]. Retrieved

from http://www.esds.ac.uk/findingData/bcs70.asp

Bynner, J., Ferri, E., & Shepherd, P. (Eds.). (1997). Twenty-something

in the 1990s: Getting on, getting by, getting nowhere. Aldershot,

England: Ashgate.

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.

Early Predictors of High School Math Achievement 697

Case, R., Griffin, S., & Kelly, W. M. (1999). Socioeconomic gra-

dients in mathematical ability and their responsiveness to inter-

vention during early childhood. In D. P. Keating & C. Hertzman

(Eds.), Developmental health and the wealth of nations: Social,

biological, and educational dynamics (pp. 125–152). New York,

NY: Guilford Press.

Case, R., & Okamoto, Y. (1996). Exploring the microstructure of

children’s central conceptual structures in the domain of number.

Monographs of the Society for Research in Child Development,

61(1–2, Serial No. 246), 27–58.

Duncan, G. J., Dowsett, C. J., Claessens, A., Magnuson, K., Huston,

A. C., Klebanov, P., . . . Japel, C. (2007). School readiness and

later achievement. Developmental Psychology, 43, 1428–1446.

Gelman, R., & Williams, E. (1998). Enabling constraints for cogni-

tive development and learning: Domain specificity and epigen-

esis. In W. Damon, D. Kuhn, & R. S. Siegler (Eds.), Handbook of

child psychology: Vol. 2. Cognition, perception & language (5th

ed., pp. 575–630). New York, NY: Wiley.

Hanushek, E. A., & Rivkin, S. G. (2006). School quality and the

black-white achievement gap (NBER Working Paper No. 12651).

Washington, DC: National Bureau of Economic Research.

Hecht, S. A., & Vagi, K. J. (2010). Sources of group and individual

differences in emerging fraction skills. Journal of Educational

Psychology, 102, 843–858.

Hofferth, S., Davis-Kean, P. E., Davis, J., & Finkelstein, J. (1998).

The Child Development Supplement to the Panel Study of Income

Dynamics: 1997 user guide. Ann Arbor: University of Michigan,

Survey Research Center, Institute for Social Research. Retrieved

from https://psidonline.isr.umich.edu/CDS/cdsi_userGD.pdf

Ma, L. (1999). Knowing and teaching elementary mathematics:

Teachers’ understanding of fundamental mathematics in China

and the United States. Mahwah, NJ: Erlbaum.

Moseley, B. J., Okamoto, Y., & Ishida, J. (2007). Comparing U.S. and

Japanese elementary school teachers’ facility for linking rational

number representations. International Journal of Science and

Mathematics Education, 5, 165–185.

Murnane, R. J., Willett, J. B., & Levy, F. (1995). The growing impor-

tance of cognitive skills in wage determination. Review of Eco-

nomics and Statistics, 78, 251–266.

National Council of Teachers of Mathematics. (2006). Curriculum

focal points for prekindergarten through grade 8 mathematics: A

quest for coherence. Washington, DC: Author.

National Mathematics Advisory Panel. (2008). Foundations for suc-

cess: The final report of the National Mathematics Advisory

Panel. Washington, DC: U.S. Department of Education.

Schneider, M., & Siegler, R. S. (2010). Representations of the magni-

tudes of fractions. Journal of Experimental Psychology: Human

Perception and Performance, 36, 1227–1238.

Siegler, R. S., Thompson, C. A., & Schneider, M. (2011). An inte-

grated theory of whole number and fractions development. Cog-

nitive Development, 62, 273–296.

Starkey, P., Klein, A., & Wakeley, A. (2004). Enhancing young

children’s mathematical knowledge through a pre-kindergarten

mathematics intervention. Early Childhood Research Quarterly,

19, 99–120.

Stevenson, H. W., & Newman, R. S. (1986). Long-term prediction

of achievement and attitudes in mathematics and reading. Child

Development, 57, 646–659.

Vosniadou, S., Vamvakoussi, X., & Skopeliti, I. (2008). The frame-

work theory approach to conceptual change. In S. Vosniadou

(Ed.), International handbook of research on conceptual change

(pp. 3–34). Mahwah, NJ: Erlbaum.

Wynn, K. (1995). Infants possess a system of numerical knowledge.

Current Directions in Psychological Science, 4, 172–177.