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Constructivist mathematics education for students withmild mental retardationEvelyn H. Kroesbergen & Johannes E. H. Van Luita Utrecht University , The Netherlandsb Department of General and Special Education , Utrecht University , 80140, 3508 TCUtrecht, The Netherlands E-mail:Published online: 20 Jun 2007.

To cite this article: Evelyn H. Kroesbergen & Johannes E. H. Van Luit (2005) Constructivist mathematics educationfor students with mild mental retardation, European Journal of Special Needs Education, 20:1, 107-116, DOI:10.1080/0885625042000319115

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European Journal of Special Needs EducationVol. 20, No. 1, February 2005, pp. 107–116

ISSN 0885–6257 (print)/ISSN 1469–591X (online)/05/01/0107–10© 2005 Taylor & Francis Group LtdDOI: 10.1080/0885625042000319115

SHORT REPORT

Constructivist mathematics education for students with mild mental retardation

Evelyn H. Kroesbergen

*

and Johannes E. H. Van Luit

Utrecht University, The Netherlands

Taylor and Francis LtdREJS200107.sgm10.1080/0885625042000319115European Journal of Special Needs Education0885-6257 (print)/1469-591X (online)Original Article2005Taylor & Francis Ltd201000000February 2005EvelynH.KroesbergenUtrecht UniversityDept. of General and Special Education801403508 TC UtrechtThe NetherlandsE.Kroesbergen@FSS.UU.NL

This study examined the effects of a constructivist mathematics intervention for students with mildmental retardation, as compared to direct instruction, which is often recommended for these chil-dren. A total of 69 students from elementary schools for special education participated in the study,which focused on multiplication learning. They received one of two kinds of mathematics interven-tion, guided or directed instruction. Multiplication automaticity and ability tests were administeredbefore and after the four-month training period. The results show that students in both conditionsimproved significantly during the training period. However, students who received directed instruc-tion showed greater improvement than students who had received guided instruction. These resultsshow that students with MMR can profit from constructivist instruction, although direct instructionseems more effective.

Keywords:

Constructivism; Direct instruction; Mathematics education; Mild mental retardation; Multiplication

Children with mild mental retardation (MMR) often have mathematics learning diffi-culties and need special attention to acquire basic maths skills (Geary, 1994).Researchers have documented specific mathematical deficiencies. In the domain ofcomputation, students may exhibit deficits in fact retrieval, problem conceptualiza-tion and use of effective calculation strategies (Rivera, 1997). They require morepractice to achieve automaticity (Goldman & Pellegrino, 1987). In addition, studentswith MMR usually have difficulties with the use of effective cognitive and meta-cognitive problem-solving strategies, various memory and retrieval processes, and

*Corresponding author. Department of General and Special Education, Utrecht University, POBox 80140, 3508 TC Utrecht, The Netherlands. Email: E.Kroesbergen@FSS.UU.NL

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E. H. Kroesbergen and J. E. H. Van Luit

generalization or transfer (Geary

et al

., 1991). When students cannot automaticallyretrieve basic facts, they have to rely on strategies (Kerkman & Siegler, 1997).Through repeated practice, they may become faster at the execution of a certain strat-egy and begin to associate problems with their solutions (Lin

et al

., 1994). However,when the students’ strategy use is inadequate, as in children with MMR, they willhave even more difficulty with reaching automatic mastery of basic facts.

Instruction should obviously take the particular difficulties of children with MMRinto account. Van Luit and Naglieri (1999) suggest that teaching step-by-step fromconcrete to abstract, working with materials to mental representations and providingtask-relevant examples can certainly help. These students need directed and detailedinstruction, explicit task analysis, and explicit instruction for automatization andgeneralization. This can be realized with direct instruction. The main characteristicof direct instruction is, in fact, that it is very directed. In practice, direct instruction isteacher-led, because the teacher provides systematic explicit instruction (Jones

et al

.,1997). New steps in the learning process are taught one at a time, based on thestudents’ progress. The lessons are generally built up following the same pattern (e.g.Archer & Isaacson, 1989). In the opening phase, the students’ attention is gained,previous lessons are reviewed and the goals of the lesson are stated. In the main partof the lesson, the teacher demonstrates how a particular task can be solved and thenthe students can work on the task, when necessary with the help of the teacher. Whenthe students appear to have sufficient understanding of the task, they are given moreof the same problems to practise independently. The teacher monitors the studentsduring such practice and provides feedback on completed tasks. Interventions inwhich students receive direct instruction have been frequently found to be very effec-tive (e.g. Van Luit, 1994; Harris

et al

., 1995; Wilson

et al

., 1996; Jitendra & Hoff,1996; Timmermans & Van Lieshout, 2003).

These recommendations, however, appear to be in clear opposition to internationaldevelopments in maths education that ask for a more constructivist-based instruction,in which the students’ own productions and constructions play a central role (e.g.NCTM, 1989, 2000). Students must actively participate in the learning process tobecome active learners. Instead of the teacher passing on mathematics knowledge insmall and basically meaningless parts, students have to play an important role in theconstruction of their own knowledge base (Cobb, 1994; Gravemeijer, 1997). Thechallenge of teaching from such a constructivist perspective is to create experiencesthat engage students and encourage them to discover new knowledge, both in regularand special education settings (Woodward & Montague, 2002). By working togetherand discussing possible solutions to a problem with one another, students developproblem-solving strategies, which they must explain and justify to one another. Suchlearning can be promoted by guided instruction, and this forms the focus of this study.

Research suggests that instruction based on constructivist principles leads to betterresults than more direct, traditional mathematics education (Cobb

et al

., 1991;Gravemeijer

et al

., 1993; Klein, 1998). And many researchers have observed thatlearning in such a manner is more motivating, exciting, and challenging (Ginsburg-Block & Fantuzzo, 1998). Students who learn to apply active learning strategies are

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Maths instruction for students with MMR

109

also expected to acquire more useful and transferable knowledge (Gabrys

et al

.,1993). Given the need to provide a general education curriculum to students withspecial needs, special educators have to investigate the possibilities to provide instruc-tion that is consistent with constructivist theories (Mercer

et al

., 1994; Woodward &Montague, 2002). Furthermore, the increase in the number of inclusion settingsresults in more students with special needs receiving the same instruction as theirnormally achieving peers. However, little empirical evidence to date is available onthe effects of this kind of maths instruction for students in special education in generaland students with MMR in particular.

The question that arises is whether teachers can ask students with MMR to activelycontribute to lessons by inventing new strategies. Asking for such a contribution actu-ally appears to deny the characteristic weaknesses of these children, namely deficits inknowledge generalization and connecting new information to old. Nevertheless, somerecent studies have shown that constructivist-based instruction can be effective forlow performers and students with specific learning disabilities (Woodward & Baxter,1997; Kroesbergen & Van Luit, 2002). However, students with MMR differ in theirinstructional needs from students with LD (Mastropieri & Scruggs, 1997). Ourresearch question is therefore whether guided instruction, which is based onconstructivist theory, is also appropriate for teaching students with MMR or ifdirected instruction is always needed for these children, instead of, or in addition to,guided instruction.

Method

Procedure and design

In order to study the effectiveness of guided instruction, a group intervention designappeared to be most powerful (Gersten

et al

., 2000). The group design involved twoconditions: the experimental group who received guided instruction or GI, and thecontrol group who received directed instruction or DI. The instruction during theintervention period was restricted to multiplication. Pre- and post-tests wereconducted to measure achievement in multiplication ability, automaticity and strategyuse. The participating students received 30 half-hour multiplication lessons across aperiod of four months. Research assistants trained and coached by the experimenterconducted the intervention. The lessons were conducted twice weekly in small groupsof four to five students each, at the time that the children would normally receivemaths instruction. On the other three days of the week, the children followed the regu-lar maths curriculum with the exception of the multiplication instruction.

Participants

A total of 69 students from elementary schools for special education participatedin the experiment, 48 boys and 21 girls. Dutch criteria for placement of children inspecial schools include evidence of (a) a disorder in one or more of the basic

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psychological processes; academic achievement significant below the student’s levelof intellectual functioning; learning problems that are not due primarily to otherhandicapping conditions; and the ineffectiveness of general educational alternativesin meeting the student’s educational needs, or (b) intellectual functioning (far)below average (IQ < 85); and academic functioning (far) below the level of‘normal’ children with the same age in elementary schools. For this study, studentswere selected from the second group, although only students with a score between1 and 2 SDs below the normative mean were included (mean IQ, 78.8;

sd

= 3.8).This group of children is referred to as ‘mild mentally retarded’. The mean age ofthe students was 10.3 years (

sd

= 1.2). The students also had to meet the follow-ing criteria: they had to be able to count and add to 100 (in order to learnmultiplication on the basis of repeated addition) and they had to score below the50% level on a test of multiplication facts up to ten times ten. The students wereassigned to either the directed instruction (

N

= 34) or the guided instructiongroup (

N

= 35). There were no significant differences between both experimentalgroups on the basis of IQ (

t

(67) = 0.690,

p

= 0.492), or age (

t

(67) = 1.676,

p

= 0.098).

Materials

At pre- and post-test, two multiplication tests were administered. The automaticitytest contains 40 multiplication problems up to 10

×

10. This test was aurally admin-istered. Students were asked to solve as many problems as possible within a two-minute period. The multiplication ability test is a paper-and-pencil test containing20 multiplication problems; most of them are context problems. Fifteen items arebelow 10

×

10; five items are ‘easy’ problems above 10

×

10 (e.g. 8

×

15 or 3

×

30).The answers are scored as right or wrong. Furthermore, the students were asked towrite down the strategies they used to solve the problems. For every student, thenumber of different strategies used was counted. The following strategies weredistinguished: (1) automatic; (2) reversal (5

×

6 = 6

×

5); (3) splitting at five or tenwith addition (8

×

7 = 5

×

7 + 3

×

7; 13

×

5 = 10

×

5 + 3

×

5); (4) splitting at five orten with subtraction (9

×

7 = 10

×

7

−

1

×

7); (5) doubling (6

×

4 = 3

×

4 + 3

×

4);(6) neighbour (8

×

9 = 9

×

9

−

1

×

9); (7) division; (8) reciting aloud or writing downthe multiplication table; (9) repeated addition; and (10) use of concrete materials ordrawing. Students could thus score between 1 and 10 on strategy diversity.

Furthermore, the adequacy of the strategies was calculated. To measure theadequacy of strategy selection, the strategies were assigned to one of the following fivecategories, based on the steps needed to solve the problem:

1. Repeated addition or counting.2. Marginally adequate solution requiring more than two steps (e.g. 4

×

8 = 8

×

4; 2

×

4 = 8; 8 + 8 = 16; 16 + 16 = 32).3. Semi-adequate solution requiring two steps (e.g. reversal and splitting, splitting

and adding: 4

×

6 = 6

×

4 = 5

×

4 + 4).

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4. Adequate solution requiring one step (e.g. reversing or splitting).5. Retrieval.

A mean efficacy score (range 1–5) was then calculated for each student.

Instructional programme

In this study, two interventions are compared: guided versus directed instruction. Forthe two experimental interventions, adjustments were made to the

MASTER trainingprogram

(Van Luit

et al

., 1993) (see also Van Luit & Naglieri, 1999; Kroesbergen

etal

., 2004), which is a remedial programme for multiplication and division. In thisstudy, only the multiplication part was used. The instruction was changed in order tocreate the two different conditions, and the working sheets were adjusted to the twoinstructional conditions. The experimental programmes consisted of a series of 25multiplication lessons (8 lessons on basic procedures; 11 lessons on multiplicationtables; and 6 lessons on ‘easy’ problems above 10

×

10). In each lesson, a new kindof task is introduced. Each of the series teaches new steps for the problem-solvingrelated to specific tasks. The emphasis in the lessons was on: (1) automated masteryof the multiplication facts, and (2) the use of back-up strategies.

Directed instruction.

The lessons in the DI condition always start with a repetition ofwhat was done in the preceding lesson. When the students show sufficient knowledgeof the preceding lesson, the teacher proceeds to the content of the actual lesson. Theteacher then introduces a new task and explains how to solve such a task. The empha-sis is on the explanation of the strategy to be used. When necessary, concrete materi-als are used to explain the task to the students. After the presentation of one or twoexamples, several tasks are next practised and discussed within the group. The groupinstruction is followed by an individual practice phase, in which the students canfamiliarize themselves with the kind of tasks involved and establish connections to themental solution of the problems.

That explicit teaching of new strategies is intended to help students expand theirstrategy repertoires. When a new strategy is taught, it is always the teacher who tellsthe children how and when to apply the strategy. The children are then instructed tofollow the example of the teacher. In the DI condition, there is little room for inputfrom the children themselves (i.e. the children must follow the procedures theteacher teaches them). When a child applies a strategy that has not been taught, theteacher may state that the particular strategy is certainly a possible strategy for solv-ing the problem in question but that they are being taught a different strategy andthen ask the child to apply the strategy being taught. Students in the directedinstruction condition learn to work with a strategy decision sheet to help themchoose the most appropriate strategy. The strategies taught in this condition are:reversal, splitting at five or ten, neighbour problem, doubling, saying the table andrepeated addition.

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Guided instruction.

The lessons in the GI instruction condition also start with areview of the previous lesson. What the students do and say in this phase is then takenas the starting-point for the actual lesson. When it appears that the students do notfully understand the tasks discussed in the previous lesson, the teacher will focusagain on these tasks. Otherwise, the next topic is introduced (e.g. ‘Today we are goingto practise with the table of 3’). During guided instruction, the discussion is alwayscentred on the contributions of the children themselves, which means that topics andstrategies other than those that the teacher has in mind may sometimes be addressed.Just as in the DI condition, the GI condition contains an introductory phase, a grouppractice phase and an individual practice phase. However, in the GI condition greaterattention is devoted to the discussion of possible solution procedures and strategiesthan in the DI condition.

In the guided instruction condition much more attention is given to the individualcontributions of the students. The teacher presents a problem and the childrenactively search for possible solutions. The teacher supports the learning process byasking questions and promoting discussion between students. The teacher canencourage the discovery of new strategies by offering additional and/or more difficultproblems, but never demonstrates a particular strategy. As a consequence, when thechildren do not discover a particular strategy, the strategy will not be discussed withinthe group. The teacher does, however, structure the discussions during the lessons byhelping students classify various strategies and posing questions about the usefulnessof particular strategies, for example.

Results

In this study, the effects of guided instruction (GI) were compared with the effects ofdirected instruction (DI) to teach MMR students multiplication. First, the effects ofboth instructional conditions on the students’ automatic knowledge of the basicmultiplication facts were investigated. In Table 1 the mean pre- and post-test scores

Table 1. Means, standard deviations and effect sizes for the automaticity test (max = 40)

Condition Pre-test Post-test Effect size

1

t p

DI 15.2 (7.5) 22.9 (6.7) 1.08 8.198 <0.001GI 16.5 (5.0) 22.9 (6.8) 1.08 7.246 <0.001

1

d

= (

M

post –

M

pre

)/

σ

pooled

.

Table 2. Means, standard deviations and effect sizes for multiplication ability test (max = 20)

Condition Pre-test Post-test Effect size

t p

DI 7.5 (4.7) 13.2 (5.3) 1.14 7.415 <0.001GI 8.0 (3.9) 11.3 (4.9) 0.75 4.328 <0.001

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on the automaticity test are presented. No difference was found at pre-test. Paired-samples

t

-tests show significant improvement for both the guided and the directedinstruction groups. However, no differences in improvement were found between thetwo conditions,

t

(67) = 0.844,

p

= 0.402. Students who had received guided instruc-tion improved as much as the students who had received directed instruction.

Next, the effects on the students’ multiplication ability were studied. The resultsare shown in Table 2. Although both groups did not differ at pre-test, and both didimprove significantly, the directed instruction group made more improvement thanthe guided instruction group,

t

(67) = 2.102,

p

= 0.039.Finally, the use of strategies by students was studied. Table 3 shows the students’

strategy diversity, and Table 4 the students’ mean efficacy scores. It was found thatstudents in both the guided and the directed instruction condition used more strate-gies at post-test than at pre-test. However, no differences between the conditionswere found (

p

>0.5). Furthermore, the students in the DI condition improved signif-icantly in strategy adequacy, while this improvement was not found for the GI condi-tion. The DI group showed significantly more improvement in strategy adequacythan the GI group,

t

(67) = 2.345,

p

= 0.022.To summarize, the two groups did not differ in automaticity or strategy diversity;

however, the DI group improved more in multiplication ability and strategy adequacyas compared with the GI group.

Discussion and conclusion

In the present study, the effectiveness of instruction based on students’ own contri-butions (guided instruction) was investigated and compared with the effectiveness ofexplicit instruction (directed instruction) for teaching multiplication to students withMMR. The two experimental groups were found to differ significantly from eachother on two of the four measures, multiplication ability and strategy adequacy, withthe DI group showing greater improvement than the GI group. These findings

Table 3. Means, standard deviations and effect sizes for amount of strategies used (max = 10)

Condition Pre-test Post-test Effect size

t p

DI 3.1 (1.1) 3.7 (1.3) 0.50 2.287 0.029GI 3.1 (1.6) 3.7 (1.5) 0.39 2.166 0.037

Table 4. Means, standard deviations and effect sizes for strategy adequacy

1

Condition Pre-test Post-test Effect size

t p

DI 2.2 (0.9) 3.0 (1.0) 0.84 4.560 <0.001GI 2.6 (1.0) 2.8 (1.0) 0.20 0.948 0.350

1

Ranging from 1 (marginally adequate) to 5 (very adequate).

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E. H. Kroesbergen and J. E. H. Van Luit

support the results of other research that students in special education profit mostfrom directed instruction (e.g. Swanson & Hoskyn, 1998; Xin & Jitendra, 1999;Kroesbergen & Van Luit, 2003; Timmermans & Van Lieshout, 2003). The studentsin the DI group have adequately learned to apply the strategies modelled by theteacher. Obviously, this is more effective for students with MMR than developingtheir own strategies, as promoted in general education teaching methods. The strat-egy adequacy of students in the GI condition did not improve during the trainingperiod; furthermore, they made more mistakes in the multiplication test than theirpeers in the DI condition.

Nevertheless, the groups did not differ on the other two measures, automaticity andstrategy diversity. It seems that automatic mastery of the multiplication facts can bereached with both instructional methods and is less dependent on the student’s strat-egy use than is often thought (e.g. Lin

et al

., 1994). Although students in the DIcondition could apply more adequate strategies, this did not result in a more auto-matic knowledge base of multiplication facts. Furthermore, no differences were foundbetween the groups in the mean amount of different strategies used. Both groups usedon average only 0.6 strategies more at post-test than at pre-test. It is difficult for thesestudents to acquire new knowledge in a short period. However, students in bothconditions did improve. Apparently, the students in the DI condition have learned aspecific set of strategies, which they then could apply adequately. However, thestudents in the GI condition, who were allowed to use as many different strategies asthey wanted to, did not use more different strategies. It seems that most of the MMRstudents in both conditions learned only a few (three to four) strategies and that thisset was appropriate to make the test. This is consistent with the theory that MMRstudents have difficulties with handling a large repertoire of strategies and that theybenefit most from instruction that involves the explicit teaching of a relatively smallbut adequate repertoire of strategies (Jones

et al

., 1997; Van Luit & Naglieri, 1999).Although students in the GI group showed less improvement than their DI peers,

it is noticeable that they nevertheless did improve significantly during the trainingperiod, with effect sizes of

d

= 0.75 and

d

= 1.08 on the multiplication tests. Appar-ently, students with MMR are also able to build their own mathematical knowledgeand, in contrast to what, for example, Carnine (1997) and Jones

et al

. (1997) haveargued, do not necessarily need explicit instruction. Moreover, the fact that they didnot improve as much as the students in the DI group, may even be explained by thefact that they were not used to such a guided instruction, because their teachersgenerally taught them in a more or less directive way. And it is difficult, especially forstudents with MMR, to make the switch from learning via directed instruction tolearning via guided instruction (Woodward & Montague, 2002). In this light, theseresults are promising for giving special students constructivist-based instruction.However, further research is needed before stronger conclusions could be drawn. Astarting-point is given by efforts to distinguish implicit from explicit constructivism,and attempts to describe a more structured variant of constructivism that providesskill development and guided practice (Woodward & Montague, 2002). Futureresearch should be focused on making adaptations to constructivist instruction to

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make it more suitable for students with MMR. This would make instruction for thesestudents less different from the general education instruction, which is especiallyadvisable with regard to inclusive settings.

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