Knowledge of fractions & division predict success in algebra

When I first started writing kitchen table math, with Carolyn Johnston, Carolyn told me that fractions are the math cliff.

Yesterday, Glen left a link to a new study in Psychological Science confirming the critical importance of fractions -- and long division -- to a child's future success in algebra:

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.

and:

Early knowledge of whole-number division also was consistently related to later mathematics proficiency.

and:

The greater predictive power of knowledge of fractions and knowledge of division was not due to their generally predicting intellectual outcomes more accurately.

More from the article:

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.

[snip]

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 development and to improving mathematics education. If researchers can identify specific areas of mathematics that consistently predict 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.

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 development. 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 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 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 numbers. 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 countable, 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 children, 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 understanding 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 fractions arithmetic procedures (Hecht & Vagi, 2010; Siegler et al., 2011). Thus, we hypothesized that 10-year-olds’ knowledge 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.

Early Predictors of High School Mathematics AchievementRobert S. Siegler1, Greg J. Duncan2, Pamela E. Davis-Kean3,4, Kathryn Duckworth5, Amy Claessens6, Mimi Engel7, Maria Ines Susperreguy3,4, and Meichu Chen4Psychological Science 23(7) 691–697