Using the Concrete-Representational-Abstract Sequence to Teach Conceptual Understanding of Place Value,Rounding, and Expanded Notation

This study investigated the effects of a place value intervention with third-grade students with learning disabilities. The intervention added content to a research-based intervention using the concrete-representational-abstract (CRA) sequence. The added content reflected current mathematics standards for third grade. Students’ place value understanding was measured using probes in which students had to identify the value of each digit within a three-digit number, round three-digit numbers to the nearest 10 or 100, and write three equations showing an expanded form of a three-digit number. A single-case, multiple-probe-across-students design showed a functional relation between CRA and completion of items requiring place value understanding. Students completed a generalization task by estimating the sum of a given equation.     

The achievement gap: The impact of between-class attainment grouping on pupil attainment and educational equity over time

Despite extensive research on attainment grouping, the impact of attainment grouping on pupil attainment remains poorly understood and contested. This paper presents evidence from a study conducted with 2944 12–13 year olds, from 76 schools in England, who were allocated to between-class attainment groups (‘setting’) in English and mathematics over the first 2 years of secondary schooling. After controlling for prior attainment, pupils in the top set performed significantly better than pupils in the middle and bottom sets in both English and mathematics. The findings indicate a widening gap in attainment, especially in the case of English. Findings, especially in the case of mathematics, provide more evidence of a relative benefit for pupils placed in top sets than a relative detriment for those in bottom sets.