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Topic: Research Connected to the Classroom - An Interesting Article in
the JRME

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Jerry P. Becker

Posts: 16,576
Registered: 12/3/04
Research Connected to the Classroom - An Interesting Article in
the JRME

Posted: Jul 25, 1998 4:26 PM
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att1.dat (4.1 K)

For information ...

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

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)

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