Patterns of strengths and weaknesses in children's knowledge about fractions.

The purpose of this study was to explore individual patterns of strengths and weaknesses in children's mathematical knowledge about common fractions. Tasks that primarily measure either conceptual or procedural aspects of mathematical knowledge were assessed with the same children in their fourth- and fifth-grade years (N=181, 56% female and 44% male). Procedural knowledge was regressed on levels of conceptual knowledge, and vice versa, to obtain residual scores. Residual scores capture variability in each kind of math knowledge that is not shared with the other type of knowledge. Cluster analysis using residuals indicated four distinct knowledge profiles in fourth graders: (a) higher than expected conceptual knowledge and relatively lower procedural knowledge, (b) relatively lower conceptual knowledge and higher procedural knowledge, (c) lower concepts but expected levels of procedural knowledge, and (d) relatively higher than expected levels of both procedural and conceptual knowledge. In fifth grade, another cluster emerged that showed lower procedures but expected levels of conceptual knowledge. In general, students with relatively lower than expected conceptual knowledge showed poorer accuracy on measures used to form the clusters and also word problem setups and estimation of sums. Implications for explaining seemingly conflicting results from prior work across studies are discussed. 2011 Elsevier Inc. All rights reserved.