The use of conceptual and procedural knowledge in the learning of concepts and multiplication of fractions in grades 4 and 5.

Abstract

National tests reveal low achievement on applied and word problems with fractions. In computation children view a fraction as separate digits instead of as a single quantity. In this study it was hypothesized that if fraction instruction emphasized treating fractions as part/whole quantities and reasoning with those quantities, children would do better with applied problems and would make fewer computational errors. An instructional sequence was designed to unfold in three stages (1) the learning of fraction concepts and the development of a quantitative notion of fractions, (2) reasoning with fraction quantities in applied situations and (3) connecting quantitative reasoning to algorithmic procedures. Fraction concepts were introduced as parts of wholes in four representations: model/diagram, oral language, real life objects and symbols. Fraction multiplication was then introduced as a part of a part application, e.g. one-half of one-fourth of a pizza. The usual algorithm was to be inferred by the students from worked applications. Two intact classes, a fourth and a fifth grade, were chosen as the experimental group and a fourth and fifth grade from the same district served as a control group. Instruction occurred over 25 days. Pretests, posttests and retention tests were administered and twelve children from the experimental group were interviewed twice during the experiment. Experimental and control groups had virtually the same achievement in computation, even though the experimental group had no direct teaching of the algorithm. The experimental group was substantially higher in applications and estimation. Experimental instruction was particularly effective in establishing part/whole interpretations of fractions. Children did reason with fraction quantities in applied situations although they did not apply this reasoning in computational situations. Error analysis revealed that story problem reasoning and computational situations are two different domains. In story problem reasoning fraction quantities were held intact and errors were mostly perceptual in nature or were the result of losing track of quantities in the solution process. The results give promise of teaching operations on fractions from a quantitative perspective, resulting in much better problem solving ability without a loss of the usual computational expectations.