Building integrated number sense in adults and children: Comparing fractions-only training with cross-notation number line training.

Mounting evidence points to the predictive power of cross-notation rational number understanding (e.g., 2/5 vs.

Learning from errors versus explicit instruction in preparation for a test that counts.

Background: Although the generation of errors has been thought, traditionally, to impair learning, recent studies indicate that, under particular feedback conditions, the commission of errors may have a beneficial effect.

Aims: This study investigates the teaching strategies that facilitate learning from errors.

Materials And Methods: This 2-year study, involving two cohorts of ~88 students each, contrasted a learning-from-errors (LFE) with an explicit instruction (EI) teaching strategy in a multi-session implementation directed at improving student performance on the high-stakes New York State Algebra 1 Regents examination.

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

This article describes UMA (Unified Model of Arithmetic), a theory of children's arithmetic implemented as a computational model. UMA builds on FARRA (Fraction Arithmetic Reflects Rules and Associations; Braithwaite et al., 2017), a model of children's fraction arithmetic.

Developmental trajectories of numerical magnitude representations of fractions and decimals.

We examined the development of numerical magnitude representations of fractions and decimals from fourth to 12th grade. In Experiment 1, we assessed the rational number magnitude knowledge of 200 Chinese fourth, fifth, sixth, eighth, and 12th graders (92 girls and 108 boys) by presenting fraction and decimal magnitude comparison tasks as well as fraction and decimal 0-1 and 0-5 number line estimation tasks. Magnitude representations of decimals became accurate earlier, improved more rapidly, and reached a higher asymptotic accuracy than magnitude representations of fractions.

Why do we have three rational number notations? The importance of percentages.

The integrated theory of numerical development provides a unified approach to understanding numerical development, including acquisition of knowledge about whole numbers, fractions, decimals, percentages, negatives, and relations among all of these types of numbers (Siegler, Thompson, & Schneider, 2011). Although, considerable progress has been made toward many aspects of this integration (Siegler, Im, Schiller, Tian, & Braithwaite, 2020), the role of percentages has received much less attention than that of the other types of numbers. This chapter is an effort to redress this imbalance by reporting data on understanding of percentages and their relations to other types of numbers.