The current issue of the Journal for Research in Mathematics Education (JRME) has a significant manuscript that reports on Dutch research in elementary school mathematics: "The Empty Number Line in Dutch Second Grades: Realistic Versus Gradual Program Design" -- Volume 29, No. 4, 1998, pp. 443-464. The authors are Anton S. Klein and Meindert Beishuizen of Leiden University (The Netherlands) and Adri Treffers, Utrecht University (The Netherlands).
Abstract of the Article:
In this study we compare 2 experimental programs for teaching mental additon 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. ---------------------------- Regarding the Empty Number Line, the researchers (pp. 461-62) write the following:
At the end of Grade 2 the success of both the RPD and GPD pupils on the difficult subtraction problems in the National Arithmetic Test confirmed that the empty number line is a powerful mode for instruction. Pupils' work as well as incidental classroom observations and teachers' experiences provided the following additional clues for the interpretation of its success: During the first half-year in both programs, the modeling function of the number line supported both procedural operations and problem representation. Evoking children's own mental activity was also a significant function, and this activity became stronger during the second half-year. In this latter period teachers using the Realistic Program Design, more than those using the Gradual Program Design, created a classroom climate of interactive teaching and discussion about children's different solutions. Students' verbal labeling of strategies and procedures proved to be very useful in this climate. Raising the level of the students' activities from their using the number-line model to their carrying out mental solution steps occurred more frequently, for instance, in the children's productions on successive tests.
Additional analysis and discussion of these data supported the conclusion that the RPD pupils made better use then the GPD pupils of the different functions of the empty number line (Klein, 1998). In particular, the RPD holds promise for future instruction, and all experimental schools have volunteered to continue with the empty-number-line program. More fine-grained analyses of the weaker pupils' progress supported the positive outcomes for use of the RPD with these students (Klein, 1998). These results gave us a more complete view of the implications of our research for the learning of addition and subtraction up to 100 in primary school ------------------------------ The researchers also commented (p. 460) on an External Criterion Test: The results on the National Arithmetic Test at the end of Grade 2 showed that problems like 64 - 28 were correctly solved by 78% of both the RPD and GPD pupils. Compared to the 55% correct for the Dutch third graders, reported in 1988 by Wijnstra, this is a remarkable increase.
****************************************************************** Jerry P. Becker Dept. of Curriculum & Instruction Southern Illinois University Carbondale, IL 62901-4610 USA Fax: (618)453-4244 Phone: (618)453-4241 (office) E-mail: JBECKER@SIU.EDU