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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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