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Effectiveness of Explicit and Constructivist Mathematics Instruction for Low-Achieving
Students in the Netherlands
Author(s): Evelyn H. Kroesbergen, Johannes E. H. Van Luit and Cora J. M. Maas
Source:
The Elementary School Journal,
Vol. 104, No. 3 (Jan., 2004), pp. 233-251
Published by: University of Chicago Press
Stable URL: http://www.jstor.org/stable/3202951
Accessed: 17-11-2015 10:41 UTC
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Effectiveness of
Explicit and
Constructivist
Mathematics
Instruction for Low-
Achieving Students in
The Netherlands
Evelyn H. Kroesbergen
Johannes E. H. Van Luit
Cora J. M. Maas
Utrecht University
The Elementary
School
Journal
Volume 104, Number 3
? 2004 by The University of Chicago. All rights reserved.
0013-5984/2004/10403-0004$05.00
Abstract
In this study we compared
the effects of small-
group constructivist
and explicit
mathematics
in-
struction
in basic
multiplication
on low-achieving
students'
performance
and motivation.
A total of
265 students (aged 8-11 years) from 13 general
and 11 special elementary
schools for students
with learning
and/or behavior disorders
partici-
pated in the study.
The experimental groups re-
ceived 30 minutes of constructivist
or explicit
in-
struction
in groups
of 5 students twice
weekly for
5 months. Pre- and posttests
were conducted to
compare the effects on students' automaticity,
problem-solving, strategy
use, and motivation to
the performance
of a control
group
who followed
the regular
curriculum.
Results showed that the
math performance
of students
in the explicit
in-
struction condition
improved significantly
more
than that of students
in the constructivist condi-
tion, and the performance
of students in both
experimental
conditions improved significantly
more than that of students in the control
condi-
tion.
Only
a few effects on motivation were
found.
We therefore
concluded that recent reforms in
mathematics instruction requiring students to
construct
their
own knowledge may not be effec-
tive for low-achieving
students.
Mathematics instruction in elementary
schools has changed in the past 20 years. In
The Netherlands, these reforms are based
mainly on the didactic principles of Realis-
tic Mathematics Education (RME;
Freuden-
thal, 1991). The main idea of RME is that
instruction should be based on student con-
tributions, that students must actively par-
ticipate in the learning process to become
active learners and to construct personal-
ized knowledge. In other countries math in-
struction has also undergone changes com-
parable to the reforms proposed by the
National Council of Teachers of Mathemat-
ics (NCTM, 1989) in the United States.
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234 THE ELEMENTARY
SCHOOL
JOURNAL
These changes have led to instruction in
which students are expected to contribute
actively to mathematics lessons by explain-
ing their mathematical reasoning to each
other and constructing their own under-
standings of mathematical concepts. This
means that students must listen to the
teacher and their peers, be able to explain
their mathematical reasoning to others, and
thereby build their own mathematical
knowledge (Baxter, Woodward, & Olson,
2001). Research has shown such a
constructivist-based approach to be prom-
ising (Ginsburg-Block & Fantuzzo, 1998;
Gravemeijer et al., 1993), and positive ef-
fects have been found for both student per-
formance and motivation. Such constructiv-
ist instruction appears to motivate students
because they find it more pleasant and more
challenging to study in such a manner
(Ames & Ames, 1989).
In recent years, the question of whether
such constructivist-based mathematics in-
struction is as effective for low-achieving
students as for normally achieving students
has been raised (e.g., Woodward & Baxter,
1997). One assumption underlying the
above-mentioned reforms is that the new
mathematics instruction is effective for all
students and therefore for low achievers.
However, results of a few recently pub-
lished studies show that this assumption is
not always correct. For example, Baxter et
al. (2001) studied the response of low-
achieving third graders in five classrooms
to reform-based mathematics instruction
and concluded that the form and content of
instruction must be adapted to the needs of
low achievers before they can benefit from
reform-based instruction. In an earlier
study, Woodward and Baxter (1997) also
found an innovative curriculum to clearly
benefit average and above-average students
but to benefit students with learning dis-
abilities and low achievers only marginally.
In addition, a meta-analysis of 58 mathe-
matics interventions showed direct instruc-
tion to be more effective than constructivist
instruction for students with special needs
(Kroesbergen & Van Luit, 2003b). However,
in a recent study, constructivist instruction
was more effective than direct instruction
for low achievers (Kroesbergen & Van Luit,
2002).
Nevertheless, many researchers believe
that students with learning difficulties need
more direct and explicit instruction to learn
basic facts and problem-solving skills (Jiten-
dra & Hoff, 1996). Explicit instruction is also
one of the most popular methods for help-
ing learners acquire greater automaticity
(Bottge, 2001; Harris, Miller, & Mercer,
1995). Studies have shown that carefully
constructed explicit instruction is effective
for teaching computational skills (Carnine,
1997). Similarly, Jones, Wilson, and Bho-
jwani (1997) argued that, although the ef-
fectiveness of explicit instruction is ques-
tioned in current mathematics reforms,
students with learning disabilities and low
math achievers require explicit instruction
to learn math concepts, skills, and relation-
ships; in fact, the presentation of multiple
approaches and alternative strategies for
the computation and solution of problems
may only lead to confusion on the part of
such students. Furthermore, explicit in-
struction can increase the motivation of low
achievers in addition to facilitating their
performance because such instruction en-
ables them to handle difficult tasks and
thereby motivates them, in many cases, to
attempt new tasks (Ames & Ames, 1989).
A discrepancy thus exists between the
application of constructivist learning theo-
ries in general education, as promoted by the
current mathematics reforms, and the appli-
cation of more explicit instruction, as rec-
ommended for low achievers. In the present
study, we investigated this discrepancy fur-
ther, specifically, the role of students' contri-
butions in their learning (constructivist in-
struction vs. explicit instruction). To evaluate
the two teaching methods, we had to con-
sider how students learn, particularly low-
achieving students in comparison to nor-
mally achieving students. The two most
important goals of the current elementary
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MATHEMATICS INSTRUCTION 235
mathematics curriculum in The Netherlands
are the automatized mastery of basic opera-
tions and the acquisition of adequate
problem-solving strategies. Given that chil-
dren with learning difficulties generally
have less than adequate memory skills, con-
comitant storage and retrieval problems,
and limited development of the strategies
needed for successful problem solving (Ri-
vera, 1997), they tend to have difficulties in
the two aforementioned areas of mathemat-
ics. They frequently show motivational def-
icits as well (Mercer, 1997).
The present study focused on the ac-
quisition of multiplication skills. Two im-
portant aspects of the acquisition of basic
multiplication can be distinguished: the
development of automaticity (i.e., direct re-
trieval from long-term memory) and the ad-
equate use of backup strategies for solving
problems that involve finding an unknown
quantity. The main instructional goal is au-
tomatized mastery of basic multiplication
facts, which can be reached with practice.
Until students have attained full automatic-
ity, however, they must rely on other strat-
egies to solve multiplication problems. For
the solution of a given problem, the student
must typically choose among several strat-
egies (Geary, Brown, & Samaranayake,
1991), and a gradual shift from the use of
counting and other backup strategies to-
ward the increased use of direct retrieval
with fewer retrieval errors is typically ob-
served (Lemaire & Siegler, 1995). The
backup strategies that students with diffi-
culties learning mathematics use often re-
semble the backup strategies that younger
but otherwise normally achieving students
use. In addition, the long-term memory rep-
resentations of students with difficulties
learning mathematics appear to develop
differently and more slowly than those of
normally achieving students (Bull & John-
ston, 1997). As a consequence, low achiev-
ers have more difficulty retrieving relevant
information from long-term memory and
also make more mistakes than their peers
when relying on retrieval strategies.
Research Questions and Hypotheses
The central question in the present study
was whether low-achieving mathematics
students benefit more from instruction that
requires them to contribute actively to les-
sons and to construct their own mathemat-
ical knowledge under the guidance of a
teacher (constructivist instruction) or from
instruction that is clearly structured and
presented by the teacher (explicit instruc-
tion). The constructivist instruction was de-
signed according to the constructivist ideas
promoted in RME (Freudenthal, 1991;
Gravemeijer, 1997). In the explicit instruc-
tion, which is based on reductionist princi-
ples (e.g., Mercer, 1997), the teacher shows
students how to solve problems by provid-
ing systematic explicit instruction. We in-
vestigated the effects of the two kinds of in-
struction on automaticity, problem-solving,
strategy use, and such motivational vari-
ables as goal orientation, self-concept, and
beliefs about mathematics. These were our
hypotheses:
1. Students who receive
explicit instruction
will improve
more in automaticity
than students
receiving constructivist instruction. Explicit
instruction provides unambiguous infor-
mation with regard to which strategy to use
to solve a multiplication problem. When the
teacher demonstrates the solution of prob-
lems, only correct strategies and solutions
are typically presented, in contrast to con-
structivist instruction, where strategies in-
troduced by peers could be incorrect. Given
that low achievers often fail to select the
most appropriate strategies, direct instruc-
tion may lead to improved strategy selec-
tion and a quicker transition from the use
of backup strategies to direct retrieval and
thus to automatized mastery of the basic
multiplication facts (Archer & Isaacson,
1989; Harris et al., 1995).
2a. Students who receive constructivist in-
struction will improve
more in problem solving
and (2b) will use more varied and efficient
problem-solving
strategies
than students receiv-
ing explicit instruction. We expected that
students who had to build their own under-
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236 THE ELEMENTARY
SCHOOL
JOURNAL
standing and concepts (constructivist in-
struction) would gain greater insight into
what they had learned and might therefore
be more willing than other students to use
the strategies they had developed (Cobb et
al., 1991; Gravemeijer et al., 1993). Further-
more, constructivist instruction might mo-
tivate and challenge students more than ex-
plicit instruction because students must
discover the relevant math facts and solu-
tion strategies in the case of constructivist
instruction (Ames & Ames, 1989; Ginsburg-
Block & Fantuzzo, 1998). Such instruction
may also, therefore, result in the more ade-
quate use of problem-solving strategies and
thereby in better performance on multipli-
cation tests.
3a. Students in both conditions will show
the same scores on self-concept;
however,
(3b)
students in the constructivist condition will
show higher task orientation, and students in
the explicit condition will show higher ego ori-
entation,
and (3c) the
former
will have
more
con-
structivist beliefs
and the latter more traditional
beliefs about mathematics.
A possible disad-
vantage of explicit instruction is that stu-
dents may lose their motivation because
they are not given opportunities to experi-
ment with their own ideas and solutions
(Weinert, Schrader, & Helmke, 1989). Con-
versely, Ames and Ames (1989) noted that
explicit instruction might encourage and
motivate low achievers more than construc-
tivist instruction because the latter might
discourage them in many cases. The focus
in explicit instruction on the provision of
clearly adequate strategies and correct an-
swers may foster the assumption that it is
important to do better than the rest of the
class by providing the best answers and the
best strategies (ego orientation; Nicholls,
1984). In contrast, constructivist instruction
focuses more on the task itself and what one
can learn (task orientation).
The following student variables were
included in the analyses, because of their
expected effects on the study outcomes:
(1) gender; boys are often found to perform
higher on math tests than girls (e.g., Holden,
1998; Vermeer, 1997); (2) IQ; which seems to
be correlated with math performance (e.g.,
Naglieri & Das, 1997, found correlations in
8- to 17-year-olds between math and IQ from
.67 to .72); (3) months of prior multiplication
instruction; students who have received in-
struction for a relatively long period but
have not yet reached an adequate level of
performance could be defined as slow learn-
ers and may have special needs; (4) general
math level; increased general math knowl-
edge will facilitate the learning of new
knowledge (Cobb, 1994; Gravemeijer, 1997);
and (5) type of education; students attending
special education generally show more com-
plex and severe learning disabilities than
their peers in regular education and thus
may need another kind of instruction (Car-
nine, 1997).
Method
Participants
Thirteen elementary schools for general
education and 11 schools for special edu-
cation participated. The schools for special
education included students with learning
and/or behavior disorders and students
with mild mental retardation. Students
were selected on the basis of low math per-
formance as measured by a national crite-
rion test that divides the population into
five categories of percentile scores (A: 75%-
100%; B: 50%-75%; C: 25%-50%; D: 10%-
25%; E: 0%-10%). Only students perform-
ing at the D or E levels were included in the
present study. Students also had to meet the
following criteria: able to count and add to
100 (in order to learn multiplication on the
basis of repeated addition) and show no
mastery of the multiplication facts up to 10
times 10. Two selection tests were admin-
istered to measure students' performance
on addition and multiplication.
Within each participating school, stu-
dents were selected according to the above-
mentioned criteria. They were then assigned
randomly to one of the two experimental
conditions or the control condition. In each
school, only one experimental condition was
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MATHEMATICS INSTRUCTION 237
implemented with one or two groups of five
students. Although all of the students who
participated in the study had difficulties
learning mathematics, the math learning
difficulties of students in special education
schools were more often accompanied by
additional learning and/or behavior prob-
lems than the learning difficulties of stu-
dents in regular schools. For this reason,
whether differences also existed in the in-
tervention outcomes for these two popula-
tions of students was also investigated.
A total of 283 students were initially se-
lected for the study. Due to missing data
and students moving during the school
year, 265 students were available for the
analyses (see Table 1). The study included
153 boys and 112 girls, with a mean age of
9.7 years. These students were in different
grades (grades 2 and 3 in regular education
schools and grades 2-6 in special education
schools) but had similar multiplication
skills, because this was a selection criterion.
Twenty-two percent of students had an eth-
nic minority background; these students
were equally divided across the three study
conditions, however. Analyses of variance
showed no differences across the three
groups for age or IQ but did reveal a sig-
nificant difference in the number of months
of multiplication instruction (F(2, 262) =
5.876, p = .003), in spite of the random as-
signment. Students in the El condition had
received 2.7 months less instruction on av-
erage when compared to students in the
other conditions.
Table 1 presents an overview of the rele-
vant variables for students in the general
and special schools. As expected, students
from the two types of schools differed sig-
nificantly with regard to age and IQ (as
measured with an individual test). Students
in the special education schools were older
(t(263) = 12.796, p <.001) and had a lower
IQ (t(263) = 9.722, p <.001) compared with
students in regular schools. The number of
boys and girls was not equally divided
across the two types of education (z2(1) =-
34.245, p <.001); more girls than boys were
found to meet the selection criteria in the
regular schools, whereas the reverse was
true for the special schools. The special edu-
cation group also contained fewer students
from an ethnic minority group than the gen-
eral education group (8% vs. 35%;
X2(1) =
31.444, p <.001).
Procedure and Design
To compare the effectiveness of the two
instructional methods, a group intervention
design appeared to be most powerful (Ger-
sten, Baker, & Lloyd, 2000). The design
involved two experimental conditions (con-
structivist instruction [CI] and explicit in-
struction [EI]) and a control condition (reg-
ular curriculum instruction). Pre-, post-, and
follow-up (3 months) tests were conducted
to measure multiplication automaticity,
problem solving, and strategy use. In ad-
dition, a motivation questionnaire was ad-
ministered before and after the training.
Students in the experimental conditions
received 30 multiplication lessons with a
duration of 30 minutes each outside the
classroom for 4-5 months. Research assis-
tants who were qualified to give instruction
to special students in small groups con-
ducted the intervention. They were inten-
sively trained and coached by one of the ex-
perimenters. Logbooks and videotapes
were used to check for similar implemen-
tation of the instructional programs. The
lessons were conducted twice weekly in
small groups of four to six students each at
the time that students would normally re-
ceive math instruction. On the other 3 days
of the week, students in the experimental
conditions followed the regular math cur-
riculum with the exception of multiplica-
tion instruction; the teachers were asked to
provide regular multiplication instruction
at those times when the experimental stu-
dents were out of the room and otherwise
to give the experimental students work-
sheets to work on during the presentation of
the regular multiplication instruction in the
class. Students in the control condition fol-
lowed the regular curriculum on all 5 days
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MATHEMATICS
INSTRUCTION 239
of the week, including the multiplication in-
struction that was part of that curriculum.
They received approximately the same
amount of multiplication instruction (60
minutes a week) as the experimental groups.
Information on the regular curriculum was
obtained via a teacher interview and the ad-
ministration of weekly questionnaires with
regard to the amount and content of the
mathematics instruction in general and mul-
tiplication instruction in particular.
Measures
To measure the effects of the interven-
tions, four multiplication tests were admin-
istered: two automaticity tests and two
problem-solving tests. Automaticity 1 and
Problem-Solving 1 contain relatively easy
tasks, and Automaticity 2 and Problem-
Solving 2 contain more difficult tasks. Au-
tomaticity 1 contains 40 multiplication
problems up to 10 x 10. This test is admin-
istered orally. Students are asked to solve as
many problems as possible within a 2-min-
ute period. They are allowed to skip items
that they do not know. Automaticity 2 con-
tains 30 multiplication problems of the type
a x b or b x a, with a being a number be-
tween 0 and 10, and b being a number be-
tween 10 and 20. Students are asked to write
the answers to as many problems as possi-
ble within 10 minutes. The answers on the
tests are then scored as correct or incorrect.
The problem-solving tests contain rich
multiplication problems taken from the test
items used to measure the mathematics lev-
els of students in schools for special edu-
cation in The Netherlands (see Kraemer,
Van der Schoot, & Engelen, 2000). Problem-
Solving 1 is a paper-and-pencil test with 20
items; 16 of the items are multiplication
tasks from the tables up to 12, and the other
four items are: 2 x 110, 3 x 50, 8 x 50, and
4 x 55. The test consists of 12 numerical
problems (e.g., 6 x 12) and eight brief story
problems (e.g., "In a box of chocolates are 4
rows of 7 chocolates each. How many choc-
olates does the box contain?"). Students are
given 20 minutes to complete the test.
Problem-Solving 2 is also a paper-and-
pencil test containing 20 multiplication con-
text problems. This test is more difficult
than Problem-Solving 1; in five of the prob-
lems both numbers are under 10; 13 are
problems with one number under 10 and
the other between 10 and 50; and the re-
maining two problems are even more diffi-
cult. Answers are scored as correct or incor-
rect, and the solution strategies each
student uses are recorded. Students were
asked to write their strategies on scratch pa-
per and to explain their solution strategies.
The strategies were divided into the follow-
ing categories: (1) automatized, (2) ade-
quate strategies requiring one additional
step to solve the problem, (3) semiadequate
strategies requiring two steps, (4) not very
adequate solutions requiring more than two
steps, and (5) repeated addition or count-
ing. Subsequently, a mean efficacy score
was calculated for each student, as well as
the number of strategies (such as splitting
at five, doubling, or reversal) each student
used (for a more detailed description, see
Kroesbergen & Van Luit, 2003b).
To measure motivational variables, we
administered the MMQ 8-11 (Motivation
Mathematics Questionnaire; Vermeer &
Seegers, 2002). This questionnaire contains
40 items and consists of seven scales. Two
scales measure goal orientation: ego orien-
tation (focus on performing better than
one's peers and showing how well one can
perform a task) and task orientation (focus
on learning new skills and increased under-
standing). Two scales measure beliefs about
mathematics: traditional beliefs (in keeping
with explicit instruction) and constructivist
beliefs (in keeping with constructivist in-
struction). Two scales measure attribution
style: external attributions (failure and suc-
cess are ascribed to external factors, such as
task difficulty) and effort attributions (fail-
ure and success are ascribed to the amount
of effort exerted). The final scale provides
information on the student's self-concept of
his or her mathematical abilities. Items are
presented in the form of statements (see Ap-
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240 THE ELEMENTARY
SCHOOL
JOURNAL
pendix), and students are asked to indicate
the extent to which they agree with a par-
ticular statement (1 = "not at all" and 5 =
"completely").
Instructional Programs
For the present study, adjustments were
made to the multiplication part of the MAS-
TER Training Program (Van Luit, Kaskens,
& Van der Krol, 1993; see also Van Luit &
Naglieri, 1999), a remedial program for
multiplication and division. We constructed
different versions of the program for each
experimental condition. The instructions
were changed to create the two conditions,
with the worksheets also adjusted to the in-
structional condition. The instructional pro-
gram contains three series of lessons: (1) ba-
sic procedures, (2) multiplication tables,
and (3) "easy" problems over 10 x 10. In
each lesson, a new kind of task is intro-
duced. Each series teaches new steps for
problem solving related to a specific task.
The teacher keeps students' existing knowl-
edge in mind and proceeds at a pace that
appears to fit students.
Constructivist instruction. Lessons in
the CI condition start with a review of the
previous lesson. What students do and say
in this phase is then taken as the starting
point for the current lesson. The teacher
states the topic of the lesson (e.g., "Today
we are going to practice with the table of
6"). However, the discussion then centers
on students' contributions, and topics and
strategies other than the ones the teacher
had in mind may be discussed. The intro-
ductory phase is followed by a group prac-
tice phase and then an individual practice
phase. Considerable attention is paid to the
discussion of possible solution procedures
and strategies.
In the CI condition, considerable time is
also provided for the contributions of indi-
vidual students. The teacher generally pres-
ents a problem and then students search for
a possible solution and thus construct their
own mathematical knowledge (Freuden-
thal, 1991). If necessary, students can use
materials or manipulatives that they prefer
to solve the problem. The teacher can en-
courage the discovery of new strategies by
offering additional and/or more difficult
problems. She/he supports student learn-
ing by asking questions and promoting dis-
cussion among students, so that students
can learn from each other's strategies
(Cobb, 1994). The teacher never demon-
strates the use of a particular strategy. As a
consequence, if students do not discover a
strategy on their own, it is not discussed
within the group. The teacher does, how-
ever, structure discussions during lessons,
for example, by helping students classify
strategies and posing questions about the
usefulness of particular strategies.
Explicit instruction. Lessons in the El
condition follow the same pattern. After a
review of what was done in the preceding
lesson, the teacher introduces the topic of
the current lesson (e.g., the table of 6), ex-
plains how to solve the problem in ques-
tion, and gives an example of a good solu-
tion strategy with the aid, when necessary,
of materials such as blocks or a number line.
For example, when discussing the table of
6, the teacher mentions the multiplication
facts with 6 and discusses how to solve
these problems. Next all students practice
several problems from the table in the
group and then work on their own. All
problems are discussed after their comple-
tion. During the practice phase, students fa-
miliarize themselves with the different
kinds of problems and learn what solution
strategies are appropriate for particular
kinds of problems.
In the explicit condition, the teacher
gives direct instruction, that is, the teacher
always tells students how and when to ap-
ply a new strategy. Students are then in-
structed to follow the example of the
teacher. The El program also promotes the
use of student self-instruction and teacher
modeling. There is little room for contribu-
tions from students (i.e., students must fol-
low the procedures the teacher teaches
them). When a student applies a strategy
JANUARY 2004
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MATHEMATICS INSTRUCTION 241
that has not been taught, the teacher may
state that the strategy can be used to solve
the problem in question but that the group
is currently working with a different strat-
egy and then request that the students use
that strategy.
Control condition. Students in the con-
trol condition received instruction based on
the regular curriculum used in the school.
Because these students also had math learn-
ing difficulties, however, their teachers gen-
erally gave them more attention and extra
instruction when compared to their peers.
It is important to note that the control con-
dition was not a no-intervention condition
but a control condition insofar as the usual
form of instruction was implemented.
Considerable variance was found in the
manner in which the lessons in the control
condition were taught due to the variety of
teaching methods used in the participating
schools. Although the majority of the
schools used an RME teaching method,
many teachers reported adjusting instruc-
tion to the needs of students by making it
more explicit and, consequently, less con-
structivist. The control condition thus in-
cluded a mixture of instructional methods
with instruction differing across schools but
always falling somewhere between construc-
tivist and explicit. The regular lessons con-
tained both instruction and practice phases.
Multiplication was part of the regular curric-
ulum, and almost every lesson involved at-
tention to multiplication skills. The amount
of time spent on multiplication was approx-
imately 1 hour a week. The size of the in-
structional groups in the control condition
also varied considerably from small groups
of three to five students working at the same
level to class-based instruction of 25-30 stu-
dents. Instructional groups in the special
education schools were generally smaller
(3-15 students) than groups in the general
education schools (12-30 students).
Examples of Experimental Lessons
Based on the video recordings of lessons
2 (learning that multiplication is the same
as repeated addition), 8 (using different
strategies), and 14 (the multiplication table
of 4), we describe instruction and learning
of one group of students receiving CI and
one group receiving EI. Each group had
four regular education students, and, at the
start of the intervention, both groups had
received about 12 months of multiplication
instruction. The IQs of students in the CI
group ranged from 78 to 116; for the El
group, they ranged from 92 to 98.
Constructivist instruction (lessons 2, 8,
and 14). Lesson 2 starts with a repetition of
the content of the preceding lesson. In re-
sponse to the teacher, one student mentions
that the group practiced making "table
sums" and demonstrates this by laying
down four groups of two blocks and saying
"4 times 2, that makes 8." The teacher re-
sponds by saying: "Very good. There are
four groups of blocks, and two blocks in
every group.., that makes 4 x 2." They
continue with this kind of practice, with the
teacher asking students to cite several prob-
lems matching the situation. For example,
when the teacher asks about four bikes with
two wheels each, one can say 2 + 2 + 2 +
2, 4 x 2, or 2 x 4. The teacher asks more
questions to prompt students to provide
other solutions. Sometimes students correct
each other's mistakes. In the case of three
groups of five blocks lying on the table, for
example, one student says, "5 plus 5 minus
5." A second student immediately responds
by telling him that this is wrong. When the
teacher asks for an explanation, this stu-
dents says, "You have to do 5 + 5 + 5 or
you won't have 15 blocks." In this same les-
son, a student already uses the doubling
strategy. Students are working on the fol-
lowing workbook problem: "How many
wheels do two cars have?" Note that the
problems and not the answers are central
during the first lessons. The "answer" to
this problem is 2 x 4 or 4 + 4. Now the
problem is: "How many wheels do four cars
have?" One student answers, "2 x 8, be-
cause 8 is 2 x 4, and then double ... that
makes 16." The teacher then demonstrates,
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242 THE ELEMENTARY SCHOOL
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by moving the blocks, what the student has
done and thus the doubling strategy. Stu-
dents also demonstrate in this lesson that
they understand that multiplication prob-
lems can be reversed.
In the eighth lesson, the teacher asks
more specifically about the reversibility of
multiplication. By asking how many 5 x 3
makes and how many 3 x 5 makes, the
teacher helps students see that the answer is
the same for both problems. The teacher then
asks whether this is always true and why.
After discussion of the fact that multiplica-
tion problems can always be reversed, the
discussion moves to the application of such
reversibility. Students initially have some
difficulties with the idea but decide in the
end that when one has to calculate 4 x 9 and
does not know the table of 9 but does know
the table of 4, one might as well reverse the
problem and find the answer to 9 x 4. After
this lesson, lessons focus on strategy use.
Which strategies are discussed depends on
the contributions of students.
The fourteenth lesson starts with the
teacher telling students what the topic is
and asking them if they already know one
or more problems from the table of four.
The discussion then proceeds as follows
(S =student, T =teacher):
Sl: 1 x 4 = 4
S2: and 0 x 4 = 0.
S3:
I don't know for sure,
but I think that
4 x 4 = 16.
S2:
Yes,
that's
right.
S4: 9 X 4 = 36
S1: 10 x 4 = 40
S2:
7 x 4 = 24
S3: No, that's wrong. That's
28, because
6 x 4 = 24.
SI: Or 7 x 3 = 21, and 7 makes 28.
T:
So you can
also use the table of 3, when
you don't know the answer...
S2: But how do you know that you have
to add 7?
33: Because
it's about the table of 7.
T: Look what S1 did. When you don't
know7 x 4, you can do 7 x 3 and 7
? 1, that equals 7 x 4.
33: Yes,
because
it is in fact the table of 7.
S4:
Or 8 x 4 = 32, minus 4 is 28.
As can be seen, students already use dif-
ferent strategies to solve the same problem.
The teacher's role is small; the students
clearly correct and help each other. The
strategies used in this lesson are: neighbor
problem (7 X 4= 8 X 4 - 1 X 4), reversal
(7 X 3 = 3 X 7), splitting at five (6 x 4 =
5 x 4 + 1 x 4) or 10 (9 x 4 = 10 x 4 -
1 X 4), and doubling (6 x 4 6 x 2 + 6
X 2). Students also clearly discuss the use
of different strategies with each other.
Explicit instruction (lessons 2, 8, and
14). In lesson 2, the teacher first repeats
what the group did in the preceding lesson:
"Do you remember what we did the last
time? We made groups with blocks, for ex-
ample, three groups of three blocks, like
this, and then we said that the matching
sum was 3 x 3." The group continues with
such tasks. For example, the teacher asks
students to lay down two groups of three
blocks and asks which multiplication prob-
lem this represents. One student answers
2 X 3, and another answers 3 x 2. The
teacher then responds that 3 x 2 is not the
same as 2 x 3 and that one can also say
3 + 3. The group continues with such tasks
until all students can state both the multi-
plication and the addition problem in re-
sponse to a given task.
The group also discusses the workbook
problem involving the wheels of a car, al-
though the problem is more structured than
in CI ("the addition is _ +_, the multipli-
cation is X "). After students have seen
that two cars have 4 + 4 wheels, the teacher
asks them to depict the problem using
blocks. When students do not know how to
do this, the teacher shows them how and
explains that "This is one car with four
wheels and this is another car with four
wheels; the addition sum is 4 + 4." She also
explains why the multiplication sum is 2 x
4. She then proceeds to the next question:
What is the addition sum for four cars?
When a student says 4 + 4, the teacher asks
how many more should be added. Another
student answers 4 x 4. However, this is not
correct because the teacher asked for addi-
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MATHEMATICS INSTRUCTION 243
tion. An answer of 8 + 8 is also not correct.
The teacher continues systematically until a
student provides the correct answer, or, if
the answer is not provided by one of the
students, she gives the answer herself. The
only strategy for multiplication used in this
lesson is repeated addition.
In the eighth lesson, the teacher explains
the reversal strategy. She demonstrates, us-
ing the blocks, that both 3 x 4 = 12 and
4 x 3= 12. She explains that one can always
reverse a multiplication problem. After stu-
dents have practiced with such reversibility,
the teacher continues with an explanation of
the "strategy decision sheet." The sheet pre-
sented in this lesson is relatively simple; each
time students learn a new strategy, it is
added to the sheet in the form of a possible
new step in solving a problem. The sheet
handed out in this lesson contains the fol-
lowing questions: "What is the multiplica-
tion problem?" "Do I know the answer di-
rectly?" "Do I know the answer if I reverse
the problem?" "Do I know the answer if I
say the multiplication table aloud?" and "Do
I know the answer if I do long addition?"
Later, the page will be expanded to include
the following strategies: splitting by 5 or 10,
using a neighbor problem, and doubling.
The strategy decision sheet helps students
learn to use different strategies.
The topic of the fourteenth lesson is the
table of 4. The teacher lays down strips of
paper with four presents printed on each
strip. As she lays down a strip, she asks:
"How many presents do I have when there
are 1 (2, 3, 4, ... 10) rows?" Students then
answer in turn. When they have reached
5 x 4= 20, she adds, "Yes, this is an im-
portant one." The student who is supposed
to respond 8 x 4 does not know the answer,
so the teacher helps by saying, "We
just said
that 7 x 4 makes 28, so what is 8 x 4?" The
student then counts on until he reaches 32.
When students proceed to work individu-
ally in their workbooks, the teacher explains
the strategy decision sheet once again, tell-
ing students that they can use this sheet to
solve difficult problems and that, by doing
this, they can learn to use a variety of strat-
egies.
Data Analysis
The data analyses were conducted using
the MLwiN software (Rasbash et al., 2000).
Given the hierarchical structure of the data,
we used multilevel modeling. In general,
observed differences in outcome variables
can be due to individual and/or group
characteristics. However, students in our
study often came from the same class and
thus had identical scores for such group
characteristics as group size or experience
of the teacher, so that the assumption of in-
dependence was violated. Multilevel mod-
eling "corrects" for such dependence. The
total amount of variance is thus split into
student and class variance, for example.
The amount of variance in outcome mea-
sures explained by the variables at the dif-
ferent levels (e.g., the individual, class, and
school levels) is then analyzed.
A first step in multilevel modeling is to
analyze the amount of variance at the dif-
ferent levels (with the "empty model" con-
taining no explanatory variables). These
analyses called for a two-level model (stu-
dent and class) for all of the tests adminis-
tered because the amount of variance was
significant for both levels (p < .05). This
means that students in the same class were
more alike than students in different classes.
Analyses based on a model with an addi-
tional school level showed no significant
variance in any of the outcome variables at
that level. Note that the second level con-
sisted of the class variables and not the ex-
perimental intervention groups. We did this
because students spent most of their time in
their regular classes and only about an hour
a week in the intervention groups.
First, we conducted multivariate multi-
level analyses on the posttest multiplication
scores, follow-up multiplication scores, and
posttreatment motivation scores. There-
after, we performed univariate multilevel
analyses for the outcome measures. For
each outcome, which variables included in
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244 THE ELEMENTARY
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the study accounted for differences in out-
comes was examined to determine the "ex-
planatory model" (the model containing
only the significant variables). Note that
variables were entered simultaneously in
the model, which implies that the beta co-
efficients depend on the other variables in
the model but that the order of variables is
irrelevant. We tested the effects of the fol-
lowing independent variables on the de-
pendent variables automaticity 1 and 2,
problem-solving 1 and 2, and strategy use:
pretest score, instructional condition (CI, EI,
or control), number of months of previous
multiplication instruction, gender, general
math level, IQ, and type of education (gen-
eral or special). For the motivational vari-
ables, the effects of improvement in auto-
maticity and problem solving were added.
The variable condition was included in the
model as a dummy variable. Third, we
tested whether the relation between depen-
dent and independent variables varied be-
tween the experimental and control groups.
Because none of the slopes showed a sig-
nificant variance, these analyses are not re-
ported here. The outcomes of the analyses
are presented using standardized beta co-
efficients.
Results
Multiplication Tests
First, the effects on automaticity and
problem solving were tested. In Table 2, the
means and standard deviations of pre- and
posttest scores for the four multiplication
measures are reported for the treatment and
control groups combined. Paired-samples t
tests showed significant improvement on
all four tests. Multivariate multilevel anal-
yses showed significant variance in the four
outcome measures (p= 0), which justified
separate univariate tests. Univariate analy-
ses for the multiplication tests showed most
of the variance to be located at the student
level (72%, 60%,
58%,
and 56%,
respectively).
Which independent variables contributed
significantly to the amount of explained var-
iance was considered next. Table
3 shows the
beta coefficients for these variables and the
amount of variance explained by these vari-
ables at the student and class level. Variables
with no accompanying beta coefficient were
not included in the model because they were
not significant. Two nonsignificant (n.s.) ef-
fects were included in the model because the
variable condition was entered as a dummy
variable, and it is not possible to include one
experimental condition without including
the other, and because the amount of vari-
ance changed significantly when the vari-
ables Explicit Instruction (EI) and Construc-
tivist Instruction (CI) were both excluded.
The univariate multilevel analyses
showed a significant effect of pretest score,
general math level, and instructional con-
dition on all four multiplication tests. Stu-
dents with higher pretest scores attained
higher posttest scores, which is not surpris-
ing, and students with higher general math
scores at pretest improved more than stu-
dents with lower general math scores. Fur-
thermore, students in the two experimental
conditions performed better than students in
the control condition. Of special interest
were the effects of the two experimental con-
ditions. The automaticity tests showed no
differences between the groups receiving
constructivist and explicit instruction. How-
ever, on the problem-solving tests students
receiving El scored higher than students in
the CI condition at posttest (p < .05). Stu-
dents in special education performed lower
than students in general education on the
Automaticity 1 and Problem-Solving 1 tests.
Also, previous multiplication instruction af-
fected scores on the Automaticity 1 and
Problem-Solving 2 tests. Finally, a significant
effect of gender was found on the Problem-
Solving 1 test, with boys outperforming
girls.
Follow-up Tests
To investigate whether the acquired
knowledge remained after a retention pe-
riod, follow-up tests were administered 3
months after the posttest, after the summer
holidays. Thirty-one students had changed
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MATHEMATICS INSTRUCTION 245
TABLE
2. Means and Standard Deviations of all Measures on Pre- and Posttest
Pretest Posttest
Test Range of Possible Scores M SD M SD t p
Automaticity:
Test 1 0-40 17.4 7.3 25.8 7.5 19.303 <.001
Test 2 0-30 6.7 5.8 12.9 8.2 15.407 <.001
Problem solving:
Test 1 0-20 10.2 5.4 14.8 3.9 16.335 <.001
Test 2 0-20 8.4 5.2 11.8 5.2 13.610 <.001
Strategy use:
Number 0-10 2.92 1.43 3.48 1.33 4.530 <.001
Efficacy 0-5 3.70 1.00 3.10 0.98 6.861 <.001
Motivation:
Ego orientation 1-5 3.97 .91 3.84 1.01 2.247 .025
Task orientation 1-5 4.09 .68 4.25 .72 - 3.207 .002
Constructivist beliefs 1-5 3.18 1.14 3.42 1.02 -3.334 <.001
Traditional beliefs 1-5 4.03 .76 3.82 .83 3.639 <.001
Effort attributions 1-5 4.28 .67 4.16 .81 2.217 .027
External attributions 1-5 3.84 .84 3.61 .96 4.070 <.001
Self-concept 1-5 3.56 .81 3.59 .77 - .772 .441
TABLE
3. Beta Coefficients for Independent Variables Significantly Related to Multiplication
Test Scores, and Amount of Variance Explained
Automaticity Problem Solving
Independent Variable Test 1 Test 2 Test 1 Test 2
Pretest score .54 .49 .33 .55
Constructivist instruction .16 .19 .15 .08(n.s.)
Explicit instruction .11(n.s.) .21 .32 .21
Prior instruction .16 .13
Gender .11
General math level .12 .19 .43 .30
IQ
Type of school .33 .17
Explained Variance (%)
Class level 45 67 62 79
Student level 38 28 43 38
NOTE.-Because the intervention was conducted in small groups and not in the regular
classes, corrections were made for the different group and class sizes.
schools due to transfers or entering second-
ary education. Sixteen of these students
were tested in their new schools, but it was
not possible to test the other 15. In these
analyses we only included the 250 students
who participated at all three measurement
points. Table 4 shows the scores for this
group. The students generally performed
the same or slightly lower at follow-up as
at posttest. Only Automaticity 2 and
Problem-Solving 2 tests showed a signifi-
cant decline. Multivariate analyses showed
significant multilevel variance (p =.011).
However, the univariate tests showed none
of the independent student variables to
cause the differences in improvement in the
period of experimental training.
Strategy Use
In addition to the multiplication scores,
students' strategy use was recorded. Table
2 presents an overview of strategy use on
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246 THE ELEMENTARY SCHOOL
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the pre- and posttests. At posttest, students
used more strategies, and the strategies they
used were in general more adequate than at
pretest. Results of the multilevel analysis
(Table 5) showed a significant effect of con-
dition. Students who received explicit in-
struction showed better strategy use than
control students: they used more diverse
strategies, and their strategy use was more
adequate. However, no significant differ-
ences between the experimental conditions
were found (p > .05). Students in general
education usually used more strategies than
special education students. In addition,
strategy efficacy was higher for students
with more prior instruction in multiplica-
tion, higher general math level, with higher
IQs, and for boys.
Motivation
Table 2 gives an overview of the pre- and
posttest scores on the motivation question-
naire. On average, students showed signifi-
cant changes from pretest to posttest. They
became less ego oriented and more task ori-
ented, and their beliefs about mathematics
also became more in accordance with a con-
structivist view of learning. Furthermore,
their external attributions and effort attri-
butions decreased. None of these changes
was accompanied by changes in self-concept
about mathematics, however.
The multivariate multilevel analysis was
significant (p < .001), which justified further
univariate analyses. These analyses showed
that the largest part of the variance in the
measures of motivation occured at the stu-
dent level (from 61% to 81%). According to
our expectations, the constructivist condition
was less ego oriented than the other two con-
ditions (p= .005). The only other motivation
scale influenced by condition was traditional
beliefs. The explicit instruction group had
less traditional beliefs regarding mathemat-
TABLE 4. Means and Standard Deviations for Multiplication Test Scores (N = 250)
Follow-up
Pretest Posttest Test
Test M SD M SD M SD t p
Automaticity:
Test 1 17.2 7.1 25.7 7.5 25.8 7.6 .410 .682
Test 2 6.7 5.8 12.8 8.2 12.0 7.9 - 2.176 .031
Problem solving:
Test 1 10.1 5.4 14.7 3.9 14.8 3.9 .345 .731
Test 2 8.2 5.2 11.6 5.2 11.0 4.8 - 2.410 .017
TABLE 5. Beta Coefficients for Variables Significantly Related to Strategy Use, and Amount of
Variance Explained
Independent Variable Number of Strategies Efficacy
Pretest score .32 .28
Constructivist instruction .10(n.s.) -.09(n.s.)
Explicit instruction .16 -.18
Prior instruction -.16
Gender .11
General math level -.18
IQ -.26
Type of school .15
Explained Variance (%)
Class level 36.7 46.7
Student level 9.3 23.9
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MATHEMATICS
INSTRUCTION 247
ics than the control group. However, stu-
dents in the two experimental conditions did
not differ from each other on this scale (p >
.05). A number of other variables explained
part of the variance in the motivation out-
comes. Girls had lower scores on self-
concept in math than boys. Also, we found
a negative relation between IQ score and
the motivation scales of ego orientation,
constructivist beliefs, and external attribu-
tions. Finally, students who improved more
on the problem-solving tests showed higher
scale scores for task orientation, traditional
beliefs, external attributions, and effort at-
tributions when compared to students who
improved less on the same tests. However,
there was a negative correlation between
improvement on the automaticity tests and
task orientation. No effect of previous mul-
tiplication instruction or type of education
(regular vs. special) was found for any of
the motivation scales.
Discussion
Few studies have examined the effective-
ness of recent mathematics instructional
changes for low-achieving students (e.g.,
Baxter et al., 2001; Kroesbergen & Van Luit,
2002). A central characteristic of these re-
forms is an emphasis on students' contribu-
tions to their learning (Freudenthal, 1991). In
the present study, we compared the effec-
tiveness of constructivist and explicit in-
struction, which is generally recommended
for low achievers (e.g., Bottge, 2001;
Carnine,
1997; Mercer, 1997).
When compared to regular instruction,
the CI intervention was effective. Students
in the CI condition improved more on au-
tomaticity (both under and above 10 x 10)
and problem solving; they also became less
ego oriented and showed a decline in tra-
ditional beliefs regarding mathematics after
the intervention. However, there were no
differences in strategy use between CI and
control groups. Apparently, low achievers
are able to build their own mathematical
knowledge and-in contrast to what, for
example, Carnine (1997), Jones et al. (1997),
and Mercer (1997) have argued-do not
need explicit instruction all the time. Con-
structivist instruction also positively influ-
enced students' motivation, although the
differences between CI and control students
were small. The period of intervention (30
lessons) may have been too short to estab-
lish significant differences in motivation.
Explicit instruction was also effective
when compared to regular instruction. Stu-
dents receiving El showed greater improve-
ment in automaticity and problem solving,
accompanied by a decline in traditional be-
liefs regarding mathematics. In addition,
they showed more diversity and greater ad-
equacy in strategy use than control stu-
dents. In interpreting these results, how-
ever, one should keep in mind that the
instructional groups in the control condi-
TABLE
6. Beta Coefficients for Variables Significantly Related to Motivation, and Amount of Variance Explained
Independent Variable Ego Task Constructivist Traditional Effort External Self
Explicit instruction .04 - .17
Constructivist instruction - .14 - .10
IQ -.17 -.16 -.16
Pretest .49 .30 .32 .22 .38 .42 .51
Gender -.22
Automaticity - .14
Problem solving .12 .18 .12 .18
Explained variance (%)
Class level 53 36 42 32 36 84 74
Student level 27 18 12 5 12 16 29
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248 THE ELEMENTARY
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tion were larger than those in the experi-
mental conditions. Although the total
amount of instruction was the same across
the three conditions, students in the gener-
ally smaller groups within the experimental
conditions received more intensive training,
which probably influenced the results.
Our main question in this study was
whether low-achieving students benefit
more from constructivist or from explicit in-
struction. Our first hypothesis was not con-
firmed; students who received explicit in-
struction did not differ in improvement in
automaticity from the students who re-
ceived constructivist instruction. Also, the
second hypothesis could not be confirmed.
Although constructivist instruction was
clearly effective, explicit instruction was
more effective at increasing students' ability
to solve multiplication problems. This find-
ing supports results of other research (Bax-
ter et al., 2001; Woodward & Baxter, 1997),
although the advantage for El was not large
in the present study and students from the
two experimental groups did not differ in
strategy use. These results are surprising
because we expected students in the El con-
dition to attain higher levels of automaticity
and differences in favor of constructivist in-
struction to occur on the problem-solving
tests and strategy use (Kroesbergen & Van
Luit, 2003a). Both instructional methods
proved equally effective for the training of
automaticity, and explicit instruction was
more effective than constructivist instruc-
tion for improving students' problem solv-
ing. These findings confirm the assumption
that low achievers, in comparison with nor-
mally achieving students, benefit more
from instruction that involves explicit
teaching of a small but adequate repertoire
of strategies and how and when to apply
them (Jones et al., 1997; Mercer, 1997). Sup-
plying low achievers with opportunities to
discover their own strategies and problem
solutions, as mathematics reforms have
suggested (NCTM, 1989), also appears to be
effective, but not as effective as direct in-
struction. One possible explanation for the
less-than-optimal results obtained in the CI
condition is that students in this condition
experienced both correct and incorrect so-
lutions, which could produce confusion for
low achievers (Jones et al., 1997). In the El
condition, only correct solutions were pre-
sented and little confusion could therefore
arise.
Although there were differences in out-
comes between instructional conditions at
posttest, no changes were found 3 months
following training. Although students' per-
formance often declines after training has
ended (e.g., Case, Harris, & Graham, 1992),
the students in our study barely declined,
although they did not improve. As we
noted, the 6-week summer vacation oc-
curred between the posttest and follow-up
test, which may explain why students' per-
formance essentially remained unchanged.
These findings also suggest that continued
training should be provided for students
with special instructional needs and that
teachers should thus not stop once students
have learned a particular concept or skill.
The results of the motivation question-
naire both support and contradict our re-
search hypotheses. The students in both
conditions showed the same scores on self-
concept (hypothesis 3a). Furthermore, the
El condition produced higher ego orienta-
tion scores, confirming hypothesis 3b. The
focus on providing adequate strategies and
correct answers in this condition may have
led students to assume that it is important
to do better than others by providing the
best answers and strategies (Nicholls, 1984).
A focus on the task itself and what one can
learn is presumably better for learning.
Nevertheless, the motivation questionnaire
responses showed no other differences be-
tween instructional conditions, indicating
that explicit instruction was not more mo-
tivating for low performers than other
forms of instruction (Ames & Ames, 1989).
The finding of no differences between El
and CI students regarding beliefs about
mathematics contradicts hypothesis 3c. In
keeping with the instructional principles
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used in El and CI, we expected CI students
to develop constructivist beliefs and El stu-
dents to develop more traditional beliefs.
One explanation for the absence of differ-
ences may be that the motivation question-
naire addressed mathematics in general and
not multiplication (i.e., the topic of the in-
tervention) in particular. Students' beliefs
may have reflected their experiences with
the general math curriculum and not the in-
tervention itself. In other words, their gen-
eral attitudes and beliefs regarding math
did not change as a result of the interven-
tions.
In addition to the instructional interven-
tions themselves, other variables influ-
enced, or sometimes unexpectedly, did not
influence students' scores on the multipli-
cation tests. For example, students in regu-
lar schools improved more than students in
special schools. This is not surprising, be-
cause students in special education tend to
have more difficulty learning than students
in general education. However, it is sur-
prising that performance on two of the four
multiplication tests showed no effects of
type of education. Thus, differences be-
tween students in special versus general
education may not be as great as educators
think, and the two groups of students may
often have similar instructional needs (Ka-
vale & Forness, 1992). Moreover, few effects
of intelligence were found, although a rela-
tively strong association between intelli-
gence and mathematical ability is normally
assumed (Naglieri, 2001). However, IQ was
related to efficacy of strategy use. Further
research should be conducted to gain a bet-
ter picture of this relation. We also found
that students with more prior multiplica-
tion instruction improved more during the
interventions. Furthermore, general math
level was an important predictor of the in-
tervention outcomes, with students initially
showing a higher math level benefiting
most from the intervention. This finding
may reflect a generally higher learning abil-
ity on the part of students with higher gen-
eral math scores, or it may support the idea
that certain math skills are prerequisites for
mastery of other skills. A final conclusion
concerns the differences between boys and
girls. Boys performed better on one of the
multiplication tests, used more adequate
strategies, and thought that they were better
at mathematics than girls. That boys per-
form better than girls in mathematics is con-
gruent with other research (e.g., Vermeer,
1997). However, we could not find a good
explanation for this difference.
To conclude, explicit math instruction
was more effective than constructivist in-
struction, although the latter was still more
effective than regular math instruction for
the low achievers in this study. Our results
indicate that recent reforms in mathematics
instruction may not be based on the most
adequate instructional principles for low
achievers. These students have special
needs, to which their instruction should be
adapted.
Appendix
TABLE
Al. Examples of Motivational Scale Items
Scale Example
Ego orientation I like it when I am the only one who knows the answer to a problem.
Task orientation I like it when I can solve a problem that I could not solve before.
Traditional beliefs In math, it is important to remember how to solve a certain problem.
Constructivist beliefs In math, it is important to understand how problems can be solved.
External attributions If I do better than usual, it's because the problems were easier.
Effort attributions If I do better than usual, it's because I tried harder.
Self-concept How good are you in mathematics?
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250 THE ELEMENTARY
SCHOOL
JOURNAL
Note
The funding of this project by the Dutch Or-
ganization for Scientific Research is gratefully ac-
knowledged. The research was supported by
Grant No. 575-36-002 from the Social Science Re-
search Council of this organization. We also ap-
preciate the comments provided by M. Jong-
mans and A. Vermeer on an earlier draft of this
article.
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