Individual differences in the development of children’s arithmetic fluency from grades 2 to 3.

Abstract

In the present research, we provide empirical evidence for the process of symbolic integration of number associations, focusing on the development of simple addition (e.g., 5 þ 3 = 8), subtraction (e.g., 5 -3 = 2), and multiplication (e.g., 5 3 3 = 15). Canadian children were assessed twice, in Grade 2 and Grade 3 (N = 244; 55% girls). All families were English-speaking, and parent education levels ranged from high school to postgraduate, with a median of community college. In Grade 2, children completed general cognitive tasks (i.e., receptive vocabulary, working memory, nonverbal reasoning, and inhibitory control). In both grades, children completed single-digit addition and complementary subtraction problems. In Grade 3, they completed single-digit multiplication problems and measures of applied mathematics, specifically, word-problem solving, algebra, and measurement. We found that addition and subtraction were reciprocally related (controlling for cognitive skills). Subtraction fluency predicted multiplication in Grade 3, whereas addition fluency did not. In Grade 3, both subtraction and multiplication fluency were predictors of applied mathematics, with multiplication partially mediating the relation between subtraction and applied mathematics performance. These findings support the view that learning arithmetic associations is a hierarchical process. As students practice each new skill, individual differences reflect the integration of the novel component into the developing associative network.

The hierarchical relations among mathematical competencies: From fundamental numeracy to complex mathematical skills.

“…Presumably, the relations among the mathematical skills will show complex and dynamic patterns over time that are not reflected in the current data, which reflect performance of adults after 10–15 years of experience with mathematics. In support of the hierarchical model, Xu et al (2021) found that for children in Grades 2–3, addition and subtraction become integrated, and later-learned subtraction knowledge predicted multiplication in Grade 3. However, longitudinal designs that cover longer learning trajectories are necessary to explore the causal links among the skills and to capture interactions among the different layers during the acquisition process.…”

“…According to the HSI model, students’ acquisition of mathematics knowledge involves constructing a unified knowledge system, starting with fundamental numeracy skills, and progressing to more complex mathematical knowledge (Xu et al, 2019). Moreover, the HSI model posits that knowledge acquisition and efficiency are fully integrated when the inner layers of knowledge are consolidated before moving towards the outer layers (Xu et al, 2021). Accordingly, consistent with other research (Lyons & Beilock, 2011; Reynvoet & Sasanguie, 2016; Sasanguie et al, 2017; Vos et al, 2017; Xu et al, 2019), we found strong correlations among the ordinal and arithmetic measures.…”

A unified model of arithmetic with whole numbers, fractions, and decimals.

“…Longitudinal studies of children have consistently found predictive relations between early individual differences in basic arithmetic skills and later differences in more advanced arithmetic skills. For example, single-digit addition and subtraction fluency in second and third grades predict single-digit multiplication in third and fourth grades (Jordan et al, 2003; Xu et al, 2021). Similarly, single-digit whole number addition fluency in third grade predicts fraction addition and subtraction in fourth grade (Jordan et al, 2013) and fraction arithmetic with all four operations in sixth grade (Hansen et al, 2017).…”

How does the modern home environment impact children's mathematics knowledge? Evidence from Canadian elementary children's digital home numeracy practice (DHNP)

“…As soon as children develop numeration knowledge, they understand the operations of addition and subtraction which require conception of additive formation and part versus whole relations of numbers (see Xu et al, 2021). Mental computation skills are a part of number operation strand and are defined as the ability to do mathematics calculations on mind without depending on pencil and paper (Ontario Curriculum, 2020).…”