This study examined the effects of conceptually based instruction on the development of students' procedural flexibility. Students from 10 2nd-grade classes (206 students) in 9 primary schools in the Netherlands participated in 1 of 2 different instruction programs: the Realistic Program Design (RPD) or the Gradual Program Design (GPD). The RPD teaches conceptual understanding along with procedural skills. The GPD emphasizes the acquisition of standard procedures. Students' use of and preference for procedures were assessed as well as their ability to solve problems in 2 different ways. The RPD led to a higher level of flexibility than did the GPD on all 3 measures, suggesting that the RPD was better in teaching the students conceptual knowledge. In both programs, conceptual understanding seemed to precede procedural skill. However, skill and understanding were closer related in the RPD than in the GPD group.
In this study we compare 2 experimental programs for teaching mental addition and subtraction in the Dutch 2nd grade (N = 275). The goal of both programs is greater flexibility in mental arithmetic through use of the empty number line as a new mental model. The programs differ in instructional design to enable comparison of 2 contrasting instructional concepts. The Realistic Program Design (RPD) stimulates flexible use of solution procedures from the beginning by using realistic context problems. The Gradual Program Design (GPD) has as its purpose a gradual increase of knowledge through initial emphasis on procedural computation followed by flexible problem solving. We found that whereas RPD pupils showed a more varied use of solution procedures than the GPD pupils, this variation did not influence the procedural competence of the pupils. The empty number line appears to be a very powerful model for the learning of addition and subtraction up to 100.Key Words: Addition, subtraction; Children's strategies; Early childhood, K-4; Early number learning; Mental arithmetic; Representations, modeling; Whole numbers Becker and Selter (1996) gave an overview of recent developments in mathematics education in elementary schools. Their central thesis was that "teaching is no longer seen as a treatment and learning as the effect. Learners are people who actively construct mathematics" (p. 511). Mental arithmetic plays a central role in this respect, stimulating not only conceptual understanding and procedural proficiency but also number sense and the understanding of number relations (McIntosh, Reys, & Reys, 1992). Many researchers in the field of mathematics education hold similar viewpoints (cf. Cobb, 1995;Freudenthal, 1991;Gravemeijer, 1994;Hiebert et al., 1996;Müller & Wittmann, 1995;Resnick, Bill, & Lesgold, 1992;Reys, Reys, Nohda, & Emori, 1995). In the Netherlands these ideas have been incorporated in the theoretical framework of Realistic Mathematics Education (RME) (Freudenthal, 1973(Freudenthal, , 1991Gravemeijer, 1994;Treffers, 1987Treffers, , 1991. RME came into prominence after a national evaluation study of mathematics education in the Netherlands in primary schools (Wijnstra, 1988) 1998, Vol. 29, No. 4, 443-464 Research reported in this article was supported by the Dutch National Science Foundation Grant 575-90-607. In the International Handbook on Mathematics Education, which Journal for Research in Mathematics EducationThis material may not be copied or distributed electronically or in any other format without written permission from NCTM.Copyright © 1998 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. pointed to an unacceptably low level of procedural competency in certain domains; for example, only 55% of the Dutch third graders were capable of solving the subtraction problem 64 -28 correctly. This study also revealed a generally low level of flexibility in using arithmetic strategies. As a result of this evaluation study, Treffers and De Moor (1990) published a...
In this study we compare 2 experimental programs for teaching mental addition and subtraction in the Dutch 2nd grade (N = 275). The goal of both programs is greater flexibility in mental arithmetic through use of the empty number line as a new mental model. The programs differ in instructional design to enable comparison of 2 contrasting instructional concepts. The Realistic Program Design (RPD) stimulates flexible use of solution procedures from the beginning by using realistic context problems. The Gradual Program Design (GPD) has as its purpose a gradual increase of knowledge through initial emphasis on procedural computation followed by flexible problem solving. We found that whereas RPD pupils showed a more varied use of solution procedures than the GPD pupils, this variation did not influence the procedural competence of the pupils. The empty number line appears to be a very powerful model for the learning of addition and subtraction up to 100.
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