The Relationship Between Fraction Magnitude Knowledge and Pre-Algebra Learning

• In Event: Bridging Research and Practice: Findings From the National Center on Cognition and Mathematics Instruction

Abstract

Two quasi-experimental classroom studies were conducted with the following purposes: 1) To assess the influence of supplementing traditional problem-solving workbooks with a combination of correct and incorrect worked examples on middle school students’ pre-algebra knowledge and 2) To explore the relationship between fraction and whole number knowledge and learning of conceptual and procedural pre-algebra knowledge. In study 1, sixth-grade students (N=48) in two ethnically diverse public school classrooms completed whole number and fraction magnitude tasks as well as an arithmetic test before and after working with either a traditional Connected Mathematics workbook or an experimental workbook which was supplemented with worked examples. The pre-algebra assessment measured procedural and conceptual knowledge of ratios, decimals, and percents. In the whole number magnitude task, participants were asked to place 12 whole numbers on number lines with endpoints of 0 and 6257 at pre- and post; the fraction magnitude task was identical except for all target numbers were fractions and the endpoints were 0 and 1.

Results from Study 1 suggest that the experimental workbook was no more effective at increasing conceptual and procedural scores than the control workbooks. Whole number magnitude knowledge did not predict conceptual or procedural learning. However, fraction magnitude knowledge, as represented by students’R_Lin^2 , was a significant and unique predictor of conceptual knowledge at post-test, controlling for pre-test. That is, students with stronger fraction magnitude knowledge at the start of the study strengthened their understanding of the meanings of the features of the problems over the course of the study. This relationship was not found for procedural skill.

A follow-up study was conducted with a larger sample for purposes of replication. In Study 2, sixth-grade students (N=108) in six ethnically diverse public school classrooms followed the same procedures used in Study 1. The experimental workbooks were not more effective at increasing learning than the control workbooks. Again, whole number magnitude knowledge did not predict conceptual or procedural learning. As in study 1, students’ fraction magnitude knowledge at pre-test demonstrated predictive power. However, with the larger sample size in Study 2, the nature of the relationship differed. Results indicated a linear relationship between students’ fraction magnitude knowledge at the start of the study and their procedural learning over the course of the study. The relationship between students’ fraction magnitude knowledge at pre-test and conceptual learning was cubic, indicating that fraction magnitude knowledge is particularly predictive at the more extreme ends of the range of values. Relative contributions of fraction magnitude knowledge for particular types of fractions, such as unit fractions, are also explored. Implications for both theory and practice are discussed.

Authors

• Christina Barbieri, University of Delaware

• Julie L. Booth, Temple University