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The effect of manipulatives on mathematicsachievement across different learning styles

Zeynel Kablan

To cite this article: Zeynel Kablan (2016) The effect of manipulatives on mathematicsachievement across different learning styles, Educational Psychology, 36:2, 277-296, DOI:10.1080/01443410.2014.946889

To link to this article: http://dx.doi.org/10.1080/01443410.2014.946889

Published online: 13 Aug 2014.

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The effect of manipulatives on mathematics achievement acrossdifferent learning styles

Zeynel Kablan*

Educational Sciences, Kocaeli University, Kocaeli, Turkey

(Received 1 November 2013; final version received 27 May 2014)

The current study investigates the influence of manipulatives used in combinationwith traditional approaches to mathematics education and how varying amountsof time spent on manipulative use influence student achievement across differentlearning styles. Three learning environments were created that incorporated vary-ing proportions of traditional teaching approaches and manipulative methods. Inone of the learning environments, the teacher used strictly lecture- and exercise-based teaching activities, which are more conducive to abstract learning. Abstractlearners showed higher academic performance compared with concrete learners inthe environment where only traditional methods were used. For the other twoenvironments, which utilised varying combinations of manipulative tools andtraditional methods, the differences in the mathematics achievement levels amongstudents of varying learning styles were not statistically significant. The study alsoshowed that concrete learners demonstrated higher performance in mathematicswhen manipulatives were used than did their counterparts in the environmentwhere only abstract activities were used; however, in the third learning environ-ment, increasing the amount of manipulative use did not provide an extra benefitto concrete learners.

Keywords: learning environment; manipulative; concrete instruction; abstractinstruction; mathematics teaching; Kolb’s learning styles

Introduction

Learning styles are individual characteristics that describe a person’s preferences in alearning environment, and students’ learning styles significantly influence their aca-demic achievement (Biçer, 2010; Cheng, Cheng, & Chen, 2012; JilardiDamavandi,Mahyuddin, Elias, Daud, & Shabani, 2011; Kablan & Kaya, 2013; Kurbal, 2011;Özkan, Sungur, & Tekkaya, 2004). Studies on learning styles have highlighted thatdifferent teaching methods have varying effects on student academic performance instudents who have different learning styles (Davies, Rutledge, & Davies, 1997; Kvan& Jia, 2005; Tulbure, 2011). In mathematics education, traditional teaching strategiessuch as lecturing are commonly used (Dana-Picard, Kidron, Komar, & Steiner, 2006).However, traditional teaching strategies do not always accommodate all learningstyles (White, 2012). Hence, educators highlight the importance of tailoring instruc-tion to fit students’ learning styles (Bhatti & Bart, 2013). Davies and colleagues(1997) state that to meet the needs of students with different learning styles whenteaching, it is essential to incorporate different learning methods into mathematics

*Email: zeynel.kablan@kocaeli.edu.tr

© 2014 Taylor & Francis

Educational Psychology, 2016Vol. 36, No. 2, 277–296, http://dx.doi.org/10.1080/01443410.2014.946889

education. There is a need for further exploring the effects of different teachingstrategies on students with different learning styles in mathematics. This studyinvestigates the influence of concrete experience (CE) and traditional strategies on themathematics achievement of students with different learning styles. One of the promi-nent theories that explain the relationship between learning styles and academicachievement is David Kolb’s Experiential Learning Theory (ELT) (Kolb, 1984).

In Kolb’s ELT, there are four modes of effective learning: CE, reflective observa-tion (RO), abstract conceptualisation (AC) and active experimentation (AE). Kolbstated that an individual’s learning style is determined by the combination of thefour learning modes that a person prefers and that each mode is divided into one oftwo dimensions. The first dimension (AC-CE), also known as the abstract-concretedimension, reflects how we perceive and comprehend new information. When pre-sented with new situations, some people prefer to respond using concrete methodsthat engage their feelings and senses, while others prefer abstract methods thatinvolve thinking and analysing. The second dimension (AE-RO), the active-reflec-tive continuum, addresses how we process new information. Some people prefer totransform new information reflectively through observation, while others preferactive involvement by doing (Kolb, 1984; Sutliff & Baldwin, 2001). Learningmodes and learning styles that result from differing combinations of these modes aredisplayed in Figure 1.

David Hunt and his colleagues identified four additional learning styles: north-erner, easterner, southerner and westerner (as cited in Kolb & Kolb, 2005). Thenortherner learning style integrates the RO and AE dialectics and specialises in con-crete experience. It combines the characteristics and abilities of the diverging andaccommodating styles. The southerner learning style combines elements from theassimilating and converging styles, is flexible with respect to the RO and AE

Concrete experience

Active experimentation

Abstract conceptualization

Reflective observation

DivergingAccommodating

Converging Assimilating

Northerners

EasternersWesterners

Southerners

Manipulatives

Lecture and exercise

Figure 1. Kolb’s learning styles (Adapted from Kolb, 1984).

278 Z. Kablan

dimensions, and specialises in AC (Kolb, 1984). In mathematics instruction, lecturesand exercises are the most common instructional methods used by teachers.

Lectures and exercises as AC

Researchers indicate that in mathematics instruction, traditional methods such as lec-turing are more well suited to the assimilating learning style (Gardner & Korth,1998; Jones, Reichard, & Mokhtari, 2003; Kolb & Kolb, 2009; Sharp, Harb, &Terry, 1997). Assimilators are usually rational and they are more interested inabstract concepts rather than in people. They prefer to work individually and avoidpractical activities (Bhatti & Bart, 2013; Kolb & Kolb, 2009). Some researchershave argued further that lecturing directly serves the AC learning mode, which isused by both assimilators and convergers (Healey, Kneale, & Bradbeer, 2005;Orhun, 2007; Sutliff & Baldwin, 2001; Svinicki & Dixon, 1987). Similar to assimi-lators, convergers rely on AC; however, rather than using RO, they prefer the AElearning mode. Therefore, while convergers do not prefer lectures for extended peri-ods of time, they do enjoy solving problems and making decisions by identifyinglogical solutions to issues or problems (Bhatti & Bart, 2013; Healey et al., 2005;Jones, Reichard, & Mokhtari, 2003; Kolb & Kolb, 2009). In addition to lecturing,exercise-based teaching is a prominent teaching method that is used in mathematicsclassrooms; it constitutes a crucial component of the problem-solving curriculum(Dana-Picard et al., 2006). In traditional mathematics teaching settings, the teachertransmits theoretical knowledge through lectures and provides application throughexercises (Abiodullah & Akbar, 2010; Shandomo & Zelkowski, 2008). Althoughsome exercises may be concrete, especially those given to younger children, thisstudy used exercises that are conducive to AC. The exercises used were either pro-vided in books or written in front of students and were completed by a teacher on ablackboard. This style of exercise administration is used frequently in mathematicseducation (Dana-Picard et al., 2006). With the primary purpose of this approachbeing problem solving and practicing, exercise-based techniques used in mathemat-ics classrooms are conducive to the AC and converging learning styles (Cox, 2013;Healey et al., 2005). Related literature has also described the teaching and learningactivities that are preferred by accommodators and divergers.

Manipulatives as a form of concrete experience

Individuals who are inclined towards the northern learning styles, both accommodat-ing and diverging, prefer learning through concrete experience. Individuals who pre-fer the accommodating learning style use concrete experience and AE as theirdominant learning modes. They use their feelings in decision-making process andthey prefer group work (Bhatti & Bart, 2013; Gurpinar, Bati, & Tetik, 2011; Kolb &Kolb, 2005). A common approach used that is geared towards this learning style is‘hands-on’ learning (Coker, 2000; Davies et al., 1997; Healey et al., 2005; Holley &Jenkins, 1993; Jones et al., 2003; Kolb & Kolb, 2009; Sutliff & Baldwin, 2001).The other northern group, the divergers, who learn best through feeling and reflect-ing, first attempt to comprehend new information through concrete experienceand then later prefer observing and reflecting on their experiences (Kolb, 1984; Kolb& Kolb, 2009). Although divergers do not prefer active learning as much as

Educational Psychology 279

accommodators do, it is believed that concrete, hands-on methods will providereflective thinking opportunities for these learners.

As a form of hands-on learning, manipulatives in mathematics classrooms areconsidered effective instructional tools. Manipulatives are physical and concretematerials such as cubes, geometry boards, pattern blocks and other everyday objectsthat help students explore and develop an understanding of mathematical ideas andconcepts (Boggan, Harper, & Whitmire, 2010; Bouck & Flanagan, 2010; Clements& McMillen, 1996; Durmuş & Karakirik, 2006).

Study significance

This experimental study aims to expand the literature on learning styles by examin-ing the influence of lecture- and manipulative-based instruction on mathematicsachievement in students with different learning styles. Previous research shows thatthe influence of learning styles on academic achievement varies by subject area(Jones et al., 2003). In the disciplines of science and mathematics, either convergers(Biçer, 2010; Davies et al., 1997; Kurbal, 2011) or assimilators were found to bemore successful (Özkan et al., 2004). In some cases, both were equally successfulcompared with accommodators and divergers (JilardiDamavandi et al., 2011; Kablan& Kaya, 2013). Convergers and assimilators possess stronger AC skills than do ac-commodators and divergers. Studies investigating the relationship between learningmodes and academic performance across different subjects have reported that ACscores are positively correlated with achievement (Arslan & Babadoğan, 2005;Boyatzis & Mainemelis, 2011; Lynch, Woelfl, Steele, & Hanssen, 1998; Kurbal,2011; Newland & Woelfl, 1992).

Several factors might explain the greater level of success achieved by southernersin mathematics compared with that of northerners. One of these factors relates to thefact that individuals with mathematical thinking ability typically rely on abstract con-cepts (Barmeyer, 2004; Kolb & Wolfe, 1981), which is compatible with southernerswho excel in AC. Another factor relates to the level of compatibility between mathe-matics instructors’ teaching styles and students’ learning styles (JilardiDamavandiet al., 2011). Accordingly, research shows that when students’ learning styles arecompatible with the curriculum design, teaching methods or characteristics of thelearning environment, better learning occurs compared with situations in which thesefactors are not aligned (Claxton & Murrell, 1987; Letele, Alexander, & Swanepoel,2013; Lovelace, 2005; Onwuegbuzie & Daley, 1998). Previous literature indicatesthat lectures and exercises are the preferred modes of instruction for AC. When thesestrategies are used in mathematics instruction, it is inevitable that southerners will bemore successful at this subject. However, previous studies on this issue have tendedto be descriptive, and experimental studies that test these arguments have not yet beenconducted. Further studies are needed to prove that the matching instruction withlearning styles yields better learning (Pashler, McDaniel, Rohrer, & Bjork, 2008)

The literature indicates that students who prefer northern learning styles tend tobe at a disadvantage when studying mathematics because these students require con-crete experiences and manipulatives to fully understand concepts. This study aims totest whether it is possible to close the achievement gap between southerners andnortherners by incorporating manipulatives into traditional lecture methods. Thefindings of this experimental study may provide educators with clues for strategiesto eliminate disadvantages among students of mathematics.

280 Z. Kablan

The current study explores the influence of manipulatives on mathematicslearning outcomes for southerners and northerners. Previous research on learningstyles has shown that students with different learning styles show varying levels ofperformance when certain methods are implemented (Davies et al., 1997; Kvan &Jia, 2005; Tulbure, 2011). Researchers emphasise that in classrooms where multipleteaching approaches are used, there will always be students whose needs are not metwhen using any given approach (Hayes & Allinson, 1996; Rush & Moore, 1991).Consequently, while matching teaching styles to learning styles in classrooms isideal for some learning situations, this strategy cannot be generalised to every learn-ing setting (Doyle & Rutherford, 1984; Gilakjani, 2012; Moallem, 2007). Moreover,Boyatzis and Mainemelis (2011) state that active learning strategies in traditionalclassrooms may not benefit learners who do not prefer or are not accustomed tothese activities. Although the positive impacts of manipulatives on mathematicslearning have been emphasised in previous studies (Clements, 1999; Olkun, 2003;Sowell, 1989; Suydam & Higgins, 1977), the question of whether students with dif-ferent learning styles benefit from manipulatives equally has not yet been addressed.This study also contributes to the field by investigating how varying the length oftime spent on manipulative learning tools may influence mathematics achievementlevels across students of different learning styles. To improve mathematics learningoutcomes, the use of concrete experiences alongside abstract learning methods isrecommended. However, introducing more concrete experiences into learning envi-ronments results in less time for AC. As many mathematics educators have previ-ously proposed, AC is a key component of mathematics education, and concreteexperiences are tools for achieving AC (Barmeyer, 2004; Kolb & Wolfe, 1981).

Purpose of the study

The primary purpose of this study is to investigate the impact of manipulative use inconjunction with traditional lecturing and exercise methods on mathematics achieve-ment in students with different learning styles. This study not only tests the effect ofmanipulatives (concrete experiences) on mathematics achievement for student withdifferent learning styles but also examines how varying the length of time spent onconcrete experiences influences achievement. To address these issues, three learningenvironments were created that include varying proportions of traditional teachingapproaches and manipulative strategies.

For the current study, the traditional teaching approach and manipulativeapproach were not treated as mutually exclusive of one another; instead, they werecombined in some of the learning environments. In one of the learning environ-ments, the teacher only used lecture- and exercise-based teaching activities, whichare more conducive to AC. This approach, also known as the traditional teachingapproach, is commonly used in mathematics classrooms. Despite the emphasis onmanipulative-based instruction in recent education reform, teachers continue to pre-fer traditional methods such as lectures in mathematics teaching settings (Levenson,Tsamir, & Tirosh, 2010). Gilakjani (2012) states that teachers not only are used tothis approach but also tend to believe that their students will learn more successfullywhen traditional methods are used.

In another learning environment, the teacher divided his teaching time equallybetween lectures/exercises and manipulatives. Thus, the effect of manipulative use

Educational Psychology 281

alongside the traditional approach on mathematics achievement in students withdifferent learning styles was assessed.

To test how varying the length of time spent on concrete experiences influencesachievement, a third learning environment with less time dedicated to manipulativeuse than abstract activities (70% lectures/exercises and 30% manipulatives) was alsocreated.

As a whole, this study seeks to answer the following research questions:

(1) Do the mathematics achievement levels differ across the different learningstyles within the same learning environment?

(2) Do the mathematics achievement levels differ among the different learningenvironments?

(3) Is there an interaction effect between the learning environment and thelearning styles on students’ mathematics achievement levels?

Investigating the relationships among learning styles, instructional methods andlearning environments is critical to the quality of instructional design (Liu & Reed,1994). The results of this study will provide strategies for accommodating differentlearning styles in classrooms and suggestions for future instructional design studiesin the discipline of mathematics education.

Method

Two experiments were conducted for this study. Experiment 1 employed a single-group design, and experiment 2 used a comparative group design. To control for the‘teacher effect’ and prevent variations in teaching styles, the same teacher taught inall three learning environments. For this reason, a large sample size could not beused. This led the researcher to conduct two separate experiments. The first experi-ment involved a single-group design in which each of the four groups received allthree treatments sequentially. This design provided a sample size advantage. In thefirst experiment, the learning differences across the four learning style groups wereexamined for the different learning environments.

Because the second experiment did not have a sample size advantage, learningstyles were investigated as a comparison between the northern and southern learningstyles. Despite this limitation, the second experiment allowed for an examination ofthe impact of learning environments and learning styles, and the interaction effectsof these two variables on mathematics achievement, which could not be performedin the first experiment.

Experiment 1

A learning environment is defined in this study as a classroom where different pro-portions of concrete and abstract teaching strategies are utilised. Three learning envi-ronments were designed that employed varying proportions of abstract and concreteactivities. Students in experiment 1 were exposed to all three learning environmentsat different times. The experiment lasted a total of 12 h, during which four hourswere consecutively spent in each learning environment. In the first learning environ-ment, 50% of the instruction provided was abstract and 50% was concrete in theform of manipulative use. In the second learning environment, the instruction given

282 Z. Kablan

was 70% abstract and 30% concrete. Finally, in the third learning environment,100% of the instruction provided was abstract in the form of lectures and exercises.

Participants

A total of 101 seventh-grade students in four different classrooms participated in thefirst experiment. There were 24–26 students in each cohort. With respect to learningstyles, 33 (32.7%) students were diverging, 24 (23.8%) were assimilating, 21 (20.8%)were converging and 23 (22.8%) were accommodating. A primary school with stand-ardised test scores above the national average and low dropout and absentee rates wasselected for the study. The teacher who voluntarily participated in the study possessed15 years of teaching experience and was rated highly by the school administration.

Design and treatment

The independent variables of the first experiment were the four different learningstyles (diverging, assimilating, converging and accommodating), and the dependentvariable was the mathematics achievement post-test for each learning environment.Differences in achievement levels according to the learning styles were examinedacross the three learning environments.

Figure 2 shows the design of the first experiment. Students in each learning envi-ronment were given a pre-test before the lesson to measure their achievement levels.No significant differences in the pre-test scores were found across the different learn-ing styles (Pretest 1, F = 1.04, p > .05; Pretest 2, F = 1.69, p > .05; Pretest 3,F = .77, p > .05). Because the students did not have any prior knowledge on thetopic, no differences in the pre-test scores were found between students with differ-ent learning styles. After the lesson was completed in each learning environment,the students were given the post-test. The post-test scores were compared across thestudent groups according to Kolb’s four learning styles.

As shown in Figure 2, the first learning environment divided 160 min of instruc-tion (four lessons) equally between abstract and concrete activities and was calledthe ‘50% Abstract – 50% Concrete’ environment. As shown in Table 1, in the firstlesson provided in learning environment 1, the teacher gave a lecture on polygonsduring the first 10 min, and students engaged in concrete activities over the

Groups A-B-C-D

Abstract-Concrete

(50%-50%) Pre

test

1

Lea

rnin

g E

nvir

onm

ent

Com

parison based on four learnin g styles

Abstract-Concrete

(70%-30%) Pre

test

2

Abstract (100%)

Pret

est 3

Post

test

1

Post

test

2

Post

test

3

Groups A-B-C-D

Groups A-B-C-D

(1)

(2)

(3)

Figure 2. Repeated measures design.

Educational Psychology 283

remaining 30 min. These activities required the students to use an isometricgeoboard to examine the relationships among polygons, diagonalism and triangles.In lesson 2, students solved exercise questions, which can be described as abstractinstruction. For this activity, the teacher wrote the questions on a blackboard andallowed students to solve them on their own. The instructor then lectured the stu-dents on the solutions and wrote the answers on the board. In the second half of thislesson, students learned about concave and convex polygons using geometry boardsand rubber bands as manipulatives.

As shown in Table 1, the second learning environment involved less time dedi-cated to manipulatives compared with lecture- and exercise-based instruction. In thisenvironment, 70% of the instructional time was spent on abstract teaching, and 30%of the time was spent on concrete teaching. For example, for the first lesson of learn-ing environment 2, the teacher provided a lecture on patterns for the first 15 min,and over the subsequent 25 min, students created patterns using pattern blocks asmanipulatives. For the third learning environment, the entire 160 min of instruc-tional time was dedicated to abstract teaching. For example, for the first lesson pro-vided in learning environment 3, the entire 40-min lesson was spent on lectures andexercises on the properties of a circle. Two other mathematics teachers who teach atthe seventh-grade level were consulted when the learning activity durations werebeing allocated.

Fidelity of implementation

A group of specialists, two of whom were university instructors and three of whomwere mathematics teachers, developed the lesson plans used according to thedescriptions of abstract and concrete learning activities provided in the literature.These abstract and concrete activities were piloted on a smaller group prior to thestudy, and some adjustments were made. Each lesson was videotaped, and these vid-eos were reviewed by the university instructors to ensure that the activities wereadministered according to the lesson plan. Based on the evaluations, it wasconcluded that the lesson plans adhered to theoretical principles and that the timeperiods allotted to each activity matched the lesson plan guidelines.

Research instrument

Student learning styles were determined using Kolb’s Learning Style Inventory(LSI)-Version 3 (Kolb, 1999). The Turkish adaptation of Version 3 was produced by

Table 1. Duration of concrete and abstract activities in each learning environment.

Learning environment1

Learning environment2

Learning environment3

Lesson Abstract Concrete Abstract Concrete Abstract Concrete

1 10′ 30′ 15′ 25′ 40′ –2 20′ 20′ 40′ – 40′ –3 10′ 30′ 15′ 25′ 40′ –4 40′ – 40′ 40′ –Total time 80′ (50%) 80′ (50%) 110′ (70%) 50′ (30%) 160′ (100%) –

284 Z. Kablan

Gencel (2006). Permission to use the inventory was received from Kolb and hiscolleagues and from Gencel. To measure mathematics achievement, three achieve-ment tests were developed for the three learning environments. The test questionswere written in a multiple-choice format and were based on the objectives of themathematics curriculum outlined by the Turkish Ministry of Education. With respectto cognitive processes, students were expected to use understanding, analysing andapplying skills as they are described in Bloom’s revised taxonomy.

Validity and reliability

After applying the Turkish adaptation of Kolb’s Learning Style Inventory-Version 3,the reliability coefficients were found to be between .71 and .84 (Gencel, 2006). Theachievement test was piloted with eighth-grade students enrolled at the same school.Item difficulties, item discriminations and reliability were computed. Accordingly,for each learning objective, items with high discrimination and moderate difficultyvalues remained in the test. The KR-20 values for the tests were .75, .77 and .72,respectively. The average difficulty levels of the tests were .45, .46 and .50, respec-tively. Finally, the average discrimination values of the tests were .44, .43 and .40,respectively. These values were within the acceptable intervals.

Data analysis

Statistical analysis of this study was conducted using SPSS 18. The differences inthe mathematics scores across students of Kolb’s four learning styles were examinedusing a one-way analysis of variance (ANOVA). Three separate ANOVA calcula-tions were completed for each learning environment.

Results of experiment 1

Before testing for significant differences among the groups, the data were tested fornormality. The Kolmogorov–Smirnov test results reveal that the data follow a nor-mal distribution (Z1 = .990, p > .05; Z2 = 1.085, p > .05; Z3 = 1.048, p > .05). Table 2shows the descriptive statistics of the mathematics post-test scores for the threedifferent learning environments and across the four different learning styles.

The one-way ANOVA results demonstrate that (Table 3) for the learningenvironment that implemented lecture- and exercise-based instruction throughout thelessons, the assimilators and convergers scored significantly higher than did theaccommodators and divergers (F = 5.237, p = .002). There was no differencebetween the assimilators’ and convergers’ scores. In the other two learning environ-ments, where 50% (F = 1.755, p = .161) and 30% (F = 1.907, p = .134) of instruc-tional time was dedicated to manipulative use, the difference in mathematicsachievement levels across all of the learning styles disappeared.

Experiment 2

In experiment 2, a comparative group design was used to compare the students’mathematics achievement levels across different learning environments and learningstyles and to test whether learning environments and learning styles impose an inter-action effect on mathematics achievement. The independent variables were the three

Educational Psychology 285

different learning environments (100% Abstract, 70% Abstract-30% Concrete and50% Abstract-50% Concrete) and two different learning styles (northern and south-ern), and the dependent variable was mathematics achievement. The learning styleswere divided into two groups due to the restriction on the sample size. Accordingly,following the conventions used in the literature, assimilators and convergers wererelabelled as southerners, and accommodators and divergers were relabelled asnortherners (Kolb & Kolb, 2005).

Participants

In the second experiment, 78 seventh-grade students were randomly assigned tothree different learning environments. Learning environment A included 16 (59.3%)northerners and 11 (40.7%) southerners, learning environment B included 15(57.7%) northerners and 11 (42.3%) southerners, and learning environment Cincluded 15 (56.0%) northerners and 11 (44.0%) southerners. The distribution oflearning styles did not differ across the groups. All groups were taught by the sameteacher who taught for the first experiment. Of the 101 students who participated inthe first experiment, 78 also participated in the second experiment.

Design and treatment

Although the first experiment offered the advantage of a larger sample size, the factthat the three treatments for this experiment were provided in sequence may haveaffected the study results. The second experiment did not have this limitationbecause the treatments were carried out simultaneously in different groups, therebyincreasing the reliability of the study results.

As shown in Figure 3, the groups were given a mathematics achievement testprior to the treatment. The pre-test scores (F(2–75) = .041, p > .05) indicate no statisti-cally significant differences among the groups. A comparison between prep-test

Table 2. Descriptive statistics of mathematics achievement by learning style in differentlearning environments.

Learning environment Learning style N M SD

Abstract–concrete (50–50%) Diverging 33 11.03 3.81Assimilation 24 13.00 3.33Converging 21 11.52 3.14Accommodating 23 11.13 3.30Total 101 11.62 3.50

Abstract–concrete (70–30%) Diverging 33 12.69 2.66Assimilation 24 14.54 4.54Converging 21 12.81 3.44Accommodating 23 14.00 2.39Total 101 13.45 3.36

Abstract (100%) Diverging 33 9.66 3.41Assimilation 24 12.96 3.26Converging 21 12.19 4.61Accommodating 23 9.73 3.68Total 101 10.99 3.94

286 Z. Kablan

Table

3.One-w

ayANOVA

results

ofmathematicsachievem

entby

learning

styles

indifferentlearning

environm

ents.

Instructionalenvironm

ent

Sum

ofsquares

dfMean

square

Fp

Differences

effect

size

Abstract–concrete

(50–50%)

Between

62.886

320.962

1.755

.161

.05

With

in1158.81

9711.947

Total

1221.70

100

Abstract–concrete

(70–30%)

Between

62.883

320.961

1.907

.134

.06

With

in1066.16

9710.991

Total

1129.05

100

Abstract(100%)

Between

217.026

372.342

5.237

.002

*2>1,

2>4,

3>1,

3>4

.14

With

in1339.96

9713.814

Total

1556.99

100

*p<.05.

Educational Psychology 287

scores for the nationwide tests also revealed no statistically significant differencesamong the groups (F(2–75) = 1.067, p > .05). The lessons for the second experimentwere all taught by the same teacher. The four lessons spanned a total of 160 min.Students received the post-test after the treatment, and the groups were thencompared across learning environments and learning styles.

As in the first experiment, three learning environments were designed with vary-ing lengths of instructional time spent on abstract and concrete teaching activitiesdescribed in the literature. In contrast to the first experiment, the groups that partici-pated in the second experiment were exposed to only one type of learning environ-ment throughout the treatment period. Thus, comparisons were made betweenlearning environments and learning styles. All of the students were taught the sametopic: the Circle and Its Properties. As can be observed in Table 4, Group Areceived abstract instruction only, Group B received 70% abstract and 30% concreteinstruction, and Group C received 50% abstract and 50% concrete instruction. Theprimary difference in the lessons across the groups was the distribution of abstractand concrete activities; all other factors were controlled across the groups. The dura-tion of a single activity was the same for each group. The quality and implementa-tion of the activities did not vary across the groups.

As demonstrated in Table 4, the concrete activities that lasted 50 min in GroupB were the same as those administered in Group C. Group C was provided 30 moreminutes of manipulative activity than was Group B. The content delivered over30 min of manipulative use in Group C was taught through abstract instruction inGroup B. The content delivered through both abstract and concrete instruction inGroups B and C was taught only through abstract instruction in Group A.

Fidelity of implementation

The lesson plans were developed by the same group of specialists who contributedto experiment 1. Abstract and concrete activities were designed based on the litera-ture. The similarities and differences among the three learning environments weredetermined. Different proportions of the abstract and concrete activities to be admin-istered across the groups were specified in the lesson plans. Accordingly, if the same

Group A

Abstract (100%) Pr

etes

t

Pos

ttest

Lea

rnin

g E

nvir

onm

ent

Com

parison based on Northern and

Southern learning styles Group

B

Abstract-Concrete

(70%-30%) Pret

est

Pos

ttest

Groups C

Abstract-Concrete

(50%-50%) Pret

est

Post

test

Figure 3. 3 × 2 Factorial experimental design.

288 Z. Kablan

activity was used in several learning environments, the specialists ensured that theimplementation of that activity was exactly the same for all three learning environ-ments. Conversely, if an activity was to be differentiated across the learning environ-ments, the specialists assessed the theoretical grounds for differentiation withoutchanging the content. Similar to the first experiment, videos were randomly selectedand observed by the university instructors for adherence to the lesson plans andtiming guidelines.

Research instrument

The student learning styles determined in the first experiment were used in the sec-ond experiment. To test the mathematics achievement levels, a multiple choice testthat measures understanding, applying and analysing skills was developed.

Validity and reliability

The reliability value for the mathematics achievement test was computed asKR-20 = .76 after pilot testing. The average difficulty value was .50, and the averagediscrimination value of the test items was .44.

Data analysis

Statistical analysis of this study was conducted using SPSS 18. A two-way ANOVAwas used. Learning environments and learning styles were the factors; the interac-tion effect of these two factors was also examined.

Results of experiment 2

Before testing for significant differences among the groups, the data were tested fornormality. The Kolmogorov–Smirnov test results showed that the data followed anormal distribution (Z = .808, p > .05). Table 5 displays the results of the descriptivestatistics on mathematics achievement across the three treatment groups and for thesoutherner and northerner learning styles.

According to the two-way ANOVA results (Table 6), there were no differencesin the mathematics scores among the students in the different treatment groups(F = .155, p > .05), and there were no differences in the mathematics scores betweenthe students with different learning styles (F = 2.256, p > .05). However, instruc-

Table 4. Duration of concrete ve abstract activities in each treatment group.

Lesson 1 Lesson 2 Lesson 3 Lesson 4 Total time

Group A Abstract 40′ 40′ 40′ 40′ 160′ (100%)Concrete – – – – –

Group B Abstract 15′ 40′ 40′ 15′ 110′ (70%)Concrete 25′ – – 25′ 50′ (30%)

Group C Abstract 15′ 10′ 40′ 15′ 80′ (50%)Concrete 25′ 30′ – 25′ 80′ (50%)

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tional treatment and learning style did have an interaction effect on the mathematicsachievement levels (F = 4.495, p < .05).

Figure 4 provides a visual representation of the interaction effect of learningenvironments and learning styles on mathematics achievement. The analyses of theBonferroni pairwise comparisons reveal that the southerners (M = 13.00) outper-formed the northerners (M = 8.38) in the 100% abstract learning environment(p = .004). No other significant differences were found between students of differentlearning styles within the same learning environment. Pairwise comparisons alsoshow that the northerners (M = 12.20) in the 70–30% abstract-concrete learningenvironment outperformed the northerners (M = 8.38) in the 100% abstract learningenvironment (p = .009). No other significant differences were found among studentsof the same learning style across the different learning environments.

Discussion and conclusion

The findings of the first experiment confirmed those of previous descriptive studiesthat assimilators and convergers exhibit higher levels of academic performance thando accommodators and divergers (Biçer, 2010; Davies et al., 1997; JilardiDamavandiet al., 2011; Kablan & Kaya, 2013; Kurbal, 2011; Özkan et al., 2004). However, thecurrent experimental study found that such learning differences did emerge in envi-ronments where only traditional methods were used. In other environments whereconcrete experiences were used in combination with traditional methods, the differ-ences across learning styles were not statistically significant. This finding indicatesthat combining learning methods that are conducive to students who prefer northernlearning styles with traditional methods would help close the achievement gapbetween students with different learning styles in mathematics education settings.Similar results were reported for the second experiment. While teachers who use theo-ries, ideas and abstract learning activities in their lessons can meet the needs of

Table 5. Descriptive statistics of mathematics achievement by learning style and instruc-tional treatment.

Northern Southern Total

Treatment N M SD N M SD N M SD

Abstract (100%) 16 8.38 4.21 11 13.00 4.47 27 10.25 4.82Abstract–concrete (70–30%) 15 12.20 4.28 11 10.18 3.81 26 11.35 4.14Abstract–concrete (50–50%) 14 10.50 2.82 11 12.00 4.07 25 11.16 3.43Total 45 10.31 4.10 33 11.72 4.17 78 10.91 4.16

Table 6. Two-way ANOVA results of mathematics achievement by treatment group andlearning style.

Source Sum of squares df Mean square F p Effect size

Treatment 4.904 2 2.452 .155 .856 .004Learning style 35.633 1 35.633 2.256 .137 .030Treatment × Learning style 142.006 2 71.003 4.495 .014* .111Error 1137.286 72 15.796Total 10619.000 78

*p < .05.

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abstract learners, they may lose the attention of those students who require concreteexamples and hands-on experimentation to understand concepts (Healey et al., 2005).

A finding from the second experiment also revealed that students with northernlearning styles exhibited higher mathematics performance levels in the environmentthat provided less time for manipulative use (30%) than did their counterparts in theenvironment where only abstract activities were used. These findings support theobservations of previous literature that students with different learning styles exhibithigher performance in certain learning environments but not in others (Davies, et al.,1997; Kvan & Jia, 2005; Tulbure, 2011). Although AC is crucial to mathematicseducation, abstract learning activities should not be considered the only strategy thatcan produce successful learning outcomes in mathematics. As demonstrated in thisstudy, concrete experiences must be combined with traditional mathematics methodsfor students who prefer these experiences.

It was shown that the advantage held by the northerners in the environment inwhich a reduced amount of manipulatives (30%) was used disappeared when thelength of time dedicated to manipulative use increased. This finding may reflect thearguments presented by mathematics educators that abstract thinking is critical formathematics learning and that concrete experiences can be used as tools for achiev-ing abstract thinking (Barmeyer, 2004; Kolb & Wolfe, 1981). The findings of thecurrent study indicate a potential threshold effect of manipulative use on mathemat-ics achievement. In other words, the effectiveness of manipulative use in mathemat-ics education is comparable to abstract learning to some degree. However, themethod may not offer additional benefits to students beyond this threshold becauseincreasing the time spent on manipulative activities detracts from the time reservedfor abstract learning activities. While educators support the use of concrete elementsin mathematics classrooms, they also warn of their limitations. Educators suggestincorporating more abstraction into mathematics curricula offered in elementaryschools (Levenson, Tirosh, & Tsamir, 2006; Wu, 1999).

Figure 4. The interaction effect of learning environment and learning style on mathematicsachievement.

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In the second experiment, while manipulative use increased the mathematics per-formance of northerners, the performance of southerners neither increased nordecreased compared with the condition in which manipulatives were not used. Thereare two possible explanations for this outcome. First, while manipulatives were usedin the second and third learning environments, a considerable amount of time wasspent on AC activities. Southerners may still have taken advantage of the abstractlearning activities though the manipulative activities did not appeal to them. Kolb andcolleagues (2001) highlight that to meet the needs of different students in a classroom,all learning modes must be considered. When manipulatives and abstract learningactivities begin to be used together during mathematics instruction, the needs of dif-ferent learning styles can be addressed. Second, southerners have the ability to solveproblems, understand a broad range of information that they are able to consolidateinto logical form (Kolb et al., 2001), and demonstrate superior intellective skills(Bostrom, Olfman, & Sein, 1988). These attributes allow southerners to be moresuccessful in mathematics learning activities. Thus, though the proportion of theirpreferred learning activities was reduced, they were not affected negatively.

The finding that southerners did not benefit from the manipulative activities canbe explained by Kolb’s learning theory. Boyatzis and Mainemelis (2011) emphasisedthat students who are accustomed to or prefer traditional learning activities may havedifficulty adapting to new learning activities. It is important to note that the teacherswho participated in this study and Turkish mathematics teachers in general tend toprefer traditional, whole-class methods. This situation may have provided an unfairadvantage to the students participating in the study who preferred southerner learn-ing styles. Conversely, Mainemelis, Boyatzis, and Kolb (1999) stated that studentswith concrete learning styles are more flexible than students with abstract learningstyles. Because northerners prefer manipulative activities and are better at adaptingto new learning situations, they might have benefited from the concrete experiencesto a greater degree than did the southerners.

Recommendations

It can be concluded that using manipulatives in mathematics teaching settings bene-fits certain learning styles to some degree. First, given the needs of concrete learners,teachers are advised to incorporate manipulatives into their mathematics teachingmethods. It is argued that using manipulatives in mathematics classrooms is morebeneficial for students who are transitioning from a concrete to abstract understand-ing of mathematical ideas (White, 2012). Manipulatives can ease the transition fromconcrete to abstract thinking; however, an excessive use of manipulatives may nothelp students more than a moderate use.

The findings of this study indicate that the development of flexible learningstyles that allow students to adapt to different learning environments can help stu-dents increase their achievement levels in mathematics. Consequently, many learningsituations in classrooms require using more than one learning style, and studentswho have not developed adaptive flexibility in learning may face learning difficultiesin these situations (Gilakjani, 2012). Adaptive flexibility in learning is defined as‘the degree to which one changes learning style to respond to different learning situ-ations in their life’ (Kolb et al., 2001, p. 25) or the ability to alternate between con-crete experience and AC modes to adapt to concrete and abstract learning situations(Mainemelis, Boyatzis, & Kolb, 2002).

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While instructional approaches offered should be conducive to student learningstyles, students should also make efforts to adapt their learning styles to newlearning situations (Gilakjani, 2012; Hansen, 1997; Loo, 1997; Rush & Moore,1991). Some researchers have argued that developing flexible learning styles instudents that will assist them in different learning activities is more beneficial thanproviding learning style-based instruction (Hayes & Allinson, 1996; Healey et al.,2005). In fact, the ELT emphasises the importance of using all four learning modes:concrete, reflective, abstract and active (Kolb et al., 2001). The results indicatingthat manipulative use does not guarantee meaningful learning (Burns & Hamm,2011; Clements, 1999) might also be dependent on whether students are able toadapt to such learning activities. For the purpose of developing adaptability inlearning styles, educators are advised to inform students of the different learningstyles, and determine students’ strengths and weaknesses based on these preferences.By specifying the learning activities that are less preferred for each learning style,teachers can help students become more competent in these activities and developflexibility in their learning styles (Coffield, Moseley, Hall, & Ecclestone, 2004;Isemonger & Sheppard, 2003; Moallem, 2007). Previous research conducted on thisissue has shown that students can adapt their learning styles depending on thelearning task and subject area with which they are presented (Hayes & Allinson,1996; Jones et al., 2003). Future research may focus on ways to develop flexiblelearning skills in students that will help them adapt to different learning situationsand, consequently, achieve better learning outcomes.

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