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Early Predictors of High School Mathematics Achievement

Authors: Robert S. Siegler1, Greg J. Duncan2, Pamela E. Davis-Kean3, Kathryn Duckworth4,

Amy Claessens5, Mimi Engel6, Maria Ines Susperreguy3, and Meichu Chen3

Summary: Elementary school students knowledge of fractions and division uniquely predicts

their high school mathematics achievement, even after controlling for a wide range of relevant

variables, suggesting that efforts to improve mathematics education should focus on improving

students learning in those areas.

Word Count = 2617 1Carnegie Mellon University, 2University of California, Irvine, 3University of Michigan, 4Institute of Education, London, 5University of Chicago, 6Vanderbilt University

Siegler, et al., page 2

Abstract

Improving theories of mathematical development, as well as mathematics education, requires

identifying the types of content knowledge that are most predictive of students long-term

learning. To identify these types of knowledge, we examined both concurrent and long-term

predictors of high school students knowledge of algebra and overall mathematics achievement.

Analyses of large, nationally-representative, longitudinal data sets from the U.S. and the U.K.

revealed that elementary school students knowledge of fractions and division uniquely predict

those students knowledge of algebra and overall mathematics achievement in high school, five

or six 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.

Siegler, et al., page 3

Early Predictors of High School Mathematics Achievement

Knowledge of mathematics is crucial to educational and financial success in

contemporary society and is becoming ever more so. High school students mathematics

achievement predicts college matriculation and graduation, early-career earnings, and earnings

growth (Murname, Willett, & Levy, 1995; National Mathematics Advisory Panel (NMAP),

2008). The strength of these relations appears to have increased in recent decades, probably due

to a growing percentage of well-paying jobs requiring considerable mathematical proficiency

(Murname, 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

backgrounds are particularly at risk for poor mathematics achievement (Hanushek & Rivkin,

2006).

Marked individual and social class differences in mathematical 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). Substantial correlations between early and

later knowledge are also present in other academic subjects, but differences among children in

mathematics knowledge are even more stable than in reading and other areas (Case, Griffin, &

Kelly, 1999; Duncan, et al., 2007).

These findings suggest a new type of research that can contribute both to theoretical

understanding of mathematical cognition and development and to improving mathematics

education. If we can identify specific areas of mathematics that are most consistently predictive

of later mathematics proficiency, after controlling for other types of mathematical knowledge,

Siegler, et al., page 4

general intellectual ability, and family background variables, we can then determine why those

types of knowledge are uniquely predictive and can increase efforts to improve instruction and

learning in those areas. The educational payoff is likely strongest in the case of areas that both

strongly predict later achievement and in which many childrens understanding is poor.

In the present study, we examine 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 mathematical development. The 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 functions 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 ordered and assigned specific locations on number lines and that can be

combined arithmetically. This idea resembles Case and Okamotos (1996) proposal that the

central conceptual structure for whole numbers, a mental number line, is eventually extended to

other types of numbers, including rational numbers and negatives. The theory also proposes that

a complementary, and equally crucial, part of numerical development is learning that many

properties that are true of whole numbers having unique successors, being countable, including

finite number of entities within any given interval, never decreasing with addition and

multiplication, etc. are not true of numbers in general.

One implication of this theory of numerical development is that acquisition of fractions

knowledge is crucial to numerical development, rather than being of secondary importance. For

Siegler, et al., page 5

most children, fractions provide the first opportunity to learn that a variety of salient and

invariant properties of whole numbers are not true of all numbers (e.g., that multiplication does

not necessarily produce answers greater than the multiplicands that produce them). This

understanding does not come easily; although children receive repeated instruction on fractions

starting in third or fourth grade (National Council of Teachers of Mathematics (NCTM), 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 the large majority of theories in the area, which focus on whole

numbers and relegate knowledge of fractions and other types of numbers to secondary status or

ignore them altogether. 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 knowledge of

mathematics.

Consider some reasons, however, why elementary school students knowledge of

fractions might be crucial for later mathematics learning, for example of algebra. To the extent

that instruction focuses on conceptual understanding of fractions, this focus tends to be in

elementary school (Kilpatrick, Swafford, & Findell, 2001). Lacking understanding of fractions,

students cannot estimate answers even to simple algebraic equations. For example, students who

do not understand 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 be more likely to

Siegler, et al., page 6

use flawed equations, because they could not reject them by reasoning that the answers they

yielded were impossible. Consistent with this analysis, accurate estimation of fraction

magnitudes is closely related to correct use of fractions arithmetic procedures (Hecht & Vagi,

2010; Siegler, et al., 2011). Thus, we hypothesized that 10-year-olds knowledge of fractions

would predict 16-year-olds algebra knowledge and overall mathematics achievement, even after

statistically controlling 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) (Bynner, Ferri,

& Shepherd, 1997) and the Panel Study of Income Dynamics-Child Development Study (PSID-

CDS) (Hofferth, Davis-Kean, Davis, & Finkelstein, 1998). Detailed descriptions of the samples

and measures used in these studies and the statistical analyses that we applied to the data are

included in the online supplementary materials; here, we provide a brief overview.

The BCS sample included 3,677 British children born in a single week of 1970. The

children completed the Friendly Maths Test and British Ability Scale as 10-year-olds and the

APU Arithmetic Test as 10- and 16-year-olds (in 1980 and 1986). At both ages, the Friendly

Maths Test included whole number arithmetic and fractions; at age 16, it also included algebra

and probability. The British Ability Scale included measures of childrens verbal and non-verbal

intellectual ability, vocabulary, and spelling. Parents provided information about their education

and income and their childrens 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

Siegler, et al., page 7

completed parts of the Woodcock-Johnson-Revised (WJ-R), a widely used achievement test.

The 10- to 12-year-olds performed the Calculation Subtest, which included whole number

arithmetic and fractions; the 15- to 17-year-olds completed the Applied Problems Subtest, which

included whole number arithmetic, fractions, algebra, geometry, measurement, and probability.

Also obtained at ages 10 to 12 were measures of childrens non-mathematical intellectual

capabilities (backward digit span and passage comprehension); demographic characteristics

(gender, age, and number of siblings); and family background (parental education and income).

Results

Our main hypothesis was that early knowledge of fractions would predict high school

mathematics knowledge, above and beyond the effects of general intellectual ability, other types

of mathematical knowledge, and family background. The data were consistent with this

hypothesis.

In the U.K. data (Table 1, Columns 2 and 4), 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. A one standard deviation

increase in early fractions knowledge was uniquely associated with a .15 standard deviation

increase in subsequent algebra knowledge and a .16 standard deviation increase in total math

achievement (p

Siegler, et al., page 8

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

influential, we would expect strong concurrent relations between high school students

knowledge of fractions and their overall mathematical knowledge. High school students

knowledge of fractions did correlate very strongly with their overall mathematics achievement,

in both the U.K., r (3675) = .81, p < .001, and in the U.S., r (597) = .87, p < .001. High school

students fractions knowledge also was closely related to their knowledge of algebra in both the

U.K., r (3675) = .68, p < .001, and the U.S., r (597) = .65, p < .001. Although algebra is a major

part of the high school achievement test and fractions a smaller part, the correlation between high

school students knowledge of fractions and their overall mathematics achievement 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, 2 = 66.49, p < .001, and in the U.S. set, r (597)

= .87 versus .80, 2 = 15.03, p < .001.

Early knowledge of whole number division also was consistently related to later

mathematics proficiency. Among the five mathematics variables in the elementary school tests,

the correlation between early division knowledge and later algebra knowledge and overall

mathematics achievement was the second strongest in the U.K. data (Table 1) and the strongest

in the U.S. data (Table 2). Concurrent correlations between high school students knowledge of

division and their overall mathematics achievement were also substantial: in the U.K., r (3675) =

.59; in the U.S., r (597) = .69, ps < .001. To the best of our knowledge, relations between

elementary school childrens division knowledge and high school students mathematics

proficiency have not been documented, or even hypothesized, previously.

Siegler, et al., page 9

Regressions like those in Tables 1 and 2 place no constraints on the estimated

coefficients. To analyze whether early fractions and division knowledge was more strongly

predictive of high school mathematics achievement than was early knowledge of whole number

addition, subtraction, and multiplication, we re-estimated our regression models imposing an

equality constraint on the coefficients on the coefficients for fractions and division, as well as an

equality constraint on the coefficients for addition, subtraction, and multiplication. We then

tested whether the two sets of pooled coefficients differed from each other. The predictive

relation was stronger for fractions and division in both the U.K. data (F (1, 3664) =28.79, p <

.001) and the U.S. data (F (1, 558) = 9.72, p < .005).

The greater predictive power of fractions and division was not due to their generally

predicting intellectual outcomes more accurately. When the models shown in columns (2) and

(4) in Table 1 and 2 were applied to predicting high school students literacy (spelling and

vocabulary on the BCS, passage comprehension and working memory on the PSID-CDS),

estimated coefficients on early fractions and division knowledge did not differ from those on

addition, subtraction, and multiplication knowledge on either measure in the U.S. data and only

differed for vocabulary in the U.K. data, F (1, 3675) = 5.53, p < .05, Appendix S4).

Discussion

The present findings demonstrate that elementary school students knowledge of fractions

and division predicts their mathematics achievement in high school, above and beyond the

contributions of whole number ar...

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