The Empty Number Line in Dutch Second Grades: Realistic versus Gradual Program Design

Abstract

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...

Special education students’ use of indirect addition in solving subtraction problems up to 100—A proof of the didactical potential of an ignored procedure

“…Although researchers do not always use the same wording-for example, other expressions can be found in Klein, Beishuizen, & Treffers (1998) and Torbeyns, De Smedt et al, (2009)-there is broad agreement about the general meaning of these strategies. To use a splitting strategy, the subtraction problem is solved by decimally splitting both the minuend and the subtrahend and processing the tens and the ones separately (e.g., 54 − 31 is calculated as 50 − 30 and 4 − 1, with 23 as the final answer).…”

“…To contribute to this broad understanding of subtraction, students should be given more than just bare number problems. Several studies (Klein et al, 1998;Torbeyns et al, 2009b;Blöte et al, 2001;Van den Heuvel-Panhuizen, 1996) revealed that bare number problems hardly evoke the use of IA, which can be explained by the presence of the minus sign that emphasizes the "taking-away" action (Van den Heuvel-Panhuizen, 1996). Context problems, on the contrary, lack this operation symbol and therefore open up both interpretations of subtraction (Van den Heuvel-Panhuizen, 2005).…”

“…Several studies (Blöte, Van der Burg, & Klein, 2001;Klein et al, 1998;Torbeyns et al, 2009a, b) suggested that in such situations, students will hardly apply this procedure. However, these studies are challenged by other intervention studies that support the claim that, already in the first grades of primary mathematics education, students with a wide range of mathematical abilities can learn to solve subtraction problems flexibly by applying IA (Blöte et al, 2001;Fuson & Willis, 1988;Klein et al, 1998;Menne, 2001). For example, the study by Klein et al, (1998) revealed that the early introduction of flexible strategies or procedures, such as IA, helped improve students' scores.…”

“…Several studies (e.g., Blöte et al, 2001;Fuson & Willis, 1988;Klein et al, 1998;Menne, 2001;Torbeyns, De Smedt et al, 2009;) have indicated that subtraction problems that require crossing the ten and that have a small difference between the minuend and subtrahend (e.g., 62 − 58) may evoke the use of IA. However, IA could also be an efficient procedure for solving large-difference subtraction problems with a relatively small difference around the tens and requiring crossing the ten (Torbeyns et al, 2009a).…”

Place value without number sense: Exploring the need for mental mathematical skills assessment within the Annual National Assessments

“…So as the Realistic Mathematics Education (RME) literature suggests perhaps a better way to begin might be by modelling a numberline-based method of counting-on that keeps the first number "whole" as this method does transfer to subtraction. In RME this method is known as the "N10" procedure, and it is contrasted from the "1010" procedure which splits both numbers to be operated upon (Klein, Beishuizen & Treffers, 1998). We discuss this further below when reviewing related literature.…”