Computers & Education 57 (2011) 2011–2024

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Computers & Education

journal homepage: www.elsevier .com/locate/compedu

Fostering positive attitude in probability learning using graphing calculator

Choo-Kim Tan*, Madhubala Bava Harji, Siong-Hoe LauFaculty of Information Science and Technology, Multimedia University, Jalan Ayer Keroh Lama, 75450 Bukit Beruang, Melaka, Malaysia

a r t i c l e i n f o

Article history:Received 23 November 2010Received in revised form8 May 2011Accepted 9 May 2011

Keywords:Cooperative/collaborative learningImproving classroom teachingInteractive learning environments

* Corresponding author. Tel.: þ60 6 2523427; fax:E-mail address: cktan@mmu.edu.my (C.-K. Tan).

0360-1315/$ – see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.compedu.2011.05.005

a b s t r a c t

Although a plethora of research evidence highlights positive and significant outcomes of the incorpo-ration of the Graphing Calculator (GC) in mathematics education, its use in the teaching and learningprocess appears to be limited. The obvious need to revisit the teaching and learning of Probability hasresulted in this study, i.e. to incorporate GC in the teaching and learning of Probability, specifically on theissue of attitudes towards learning probability. The objective is to examine the effectiveness of GCinteractive learning, particularly on students’ attitudes towards Probability. A total of 65 foundationstudents participated in this study; 32 students in the experimental group and 33 in the control group.The teaching approaches varied between the groups. While the experimental group experienced the GCapproach, the control group encountered the conventional teaching approach of chalk and talk. Students’attitude towards learning probability was assessed using the Probability Attitude Inventory (PAI), wasadministered prior to and after the study. The results show significantly difference in the improvedattitude towards Probability between the groups, particularly in terms of usefulness of Probability,interest in Probability and self-concept in Probability. This study provides evidence that learning Prob-ability with GCs benefits students.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Over the decades, two of themain concerns of themathematics educators around theworld, among others include Thailand (Prakitipong& Nakamura, 2006), Chile (Ramirez, 2005), United States (Fields, 2005) are the unsatisfactory performance in Mathematics and negativeattitudes towards Mathematics.

Numerous studies (Almeqdadi, 2005; Noraini, 2006; Utts, Sommer, Acredolo, Maher, &Matthews, 2003) have been conducted to addressthese issues and examine the difficulties students face in mathematics (Jones, 2000; Wisenbaker, Scott, & Fadia, 1999). Different teachingand learning methods have been experimented to motivate and increase students’ interest towards Mathematics. Among the methodsadopted include, such as adopting cooperative learning and peer interaction (Jones, 2000; Magel, 1998), learning via videos (Esteban,Gonzalez, & Tejero, 2000), Internet-based instruction (Utts et al., 2003), geometer’s sketchpad (Almeqdadi, 2005) and graphing calcu-lator (GC) (Ellington, 2003; Forster, 2004; Noraini, 2006).

Although research evidence highlights positive and significant outcomes of the use of the GC in education, there appears to be limitedstudies of its use in teaching and learning Mathematics. Most of the research is in the teaching and learning of Algebra, Graphs andFunctions, Straight Lines, Geometry, Trigonometry, Statistics and Calculus (Arnold, 2008; Horton, Storm, & Leonard, 2004; Jones, 1995;Thompson & Senk, 2001; Waits & Demana, 1999b). However, research on Probability learning with GC appears to be limited.

The obvious need to revisit the teaching and learning of Probability has resulted in this study, i.e. to address the issue of unfavourableattitude towards learning Probability. It is aimed at developing positive attitudes and behaviours towards learning Probability amongundergraduates. The two research questions of interest are:

(1) To what extend did the incorporation of GC change the students’ attitude towards Probability?(2) Are there significant differences in the students’ a) perceived usefulness of Probability, b) interest in Probability and c) self-concept of

Probability between the experimental and control groups?

þ60 62318840.

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2. Literature review

The conventional approach of talk-and-chalk obviously does not promote much interaction and students appear to be passive learners(Duatepe-Paksu & Ubuz, 2009; Neo, unpublished; Rosnaini, Mohd Arif, & Lim, 2009). The limited or hardly any interaction and discussionssessions make the learning process appear boring or mundane. There is no opportunity to discuss or actively explore concepts that studentsdo not understand. They may eventually not be able to understand the teaching materials and/or retain pertinent mathematical conceptsand hence, find the lessons boring or/and uninteresting (Honeycutt & Pierce, 2007).

Preliminary findings provide evidence for changes in attitude, with the adoption of GCs. Positive attitudes and increased motivation tolearnmathematics with GC have been reported (Ellington, 2003; Ha, 2008; Milou, 1999; Schrupp, 2007; Stick, 1997). Students have reportedthat the learning is more interesting, exciting and enjoyable (Jones, 1995; Schrupp, 2007; Waits & Demana, 1994). They developed a positiveattitude towards using GC in learningMathematics, in general, and Statistics, in particular, which in turn affects their attitudes towards usingGC in learning Mathematics and Statistics and vice versa (Kor, 2008).

Studies found GC to be a user friendly, useful and effective tool to encourage students to enjoy learning mathematics (Noraini, 2006;Schrupp, 2007). They favoured the use of GC in learning Mathematics and found that it is easy to use (Ha, 2008; Hasan, Azizan, &Kassim, 2005; Mohd Ayub, Ahmad Tarmizi, Abu Bakar, & Mohd Yunus, 2008; Noraini, 2004; Seth & Willis, 2004; Waits & Demana, 1994).It also enhances their skills, knowledge, understanding of concepts (Kor, 2008; St. John,1998), andmathematical problems solving (Schrupp,2007). It provides excellent learning experiences and engages them in ‘real’ mathematics, i.e. a new experience to them (Waits & Demana,1994). In addition, it promotes appreciation for Mathematics, particularly in using real data/experiments and presenting real world problemswhich may not be solved easily with a paper-and-pencil approach (Kor, 2008; Schrupp, 2007). Consequently, students enthusiastically andwillingly work harder in learning mathematics when their interest towards Mathematics improves and lessons are transformed frommundane tasks to interesting and fun (Kor, 2008; Noraini, 2004;Waits & Demana, 1994). Students’ interest and enthusiasm is obvious whenthey are able to ‘see the whole picture’ of the topic (Abd Rahim, 2008).

There is also an obvious shift in the students’ perceptions towards solving Mathematical problem as they are freed from tediumcomputation (Dick, 1992; Stacey, 2004). Firstly, the multiple representation features of GC enables them to meaningfully resolve problems,taking multiple perspectives into consideration and solving problems in different ways (Rosihan & Kor, 2004). Students are more partici-pative and more willing to spent time in resolving Mathematical questions and display “a sense of self-confidence to work independently”(Noraini, 2004). Secondly, the ‘drawing and calculation’ functions of GCs enable students to ‘visualize’, i.e. graphically or numerically, therelationships between the Mathematics concepts, “make sense” of these concepts and interpreting the conclusions, which consequentlyraises their level of confidence in solving mathematical problems (Nasari, 2008; Scariano & Calzada, 1994; van der Kooij, 2001). The reducedlevel of anxiety, higher level of motivation and continued interested in learning mathematics encourages students to ‘talk’ about Mathe-matics, which eventually becomes a norm among students (Rosihan & Kor, 2004; Waits & Demana, 1994, 1998).

With a more enjoyable, exciting and interesting learning environment, students tend to place higher worth in mathematics (Waits &Demana, 1994). Waits and Demana (1999a) elaborate that the adoption of GC enables students “to see that mathematics has value.[and they]. findmathematics more interesting and exciting. [and it provided]. excitement and interest in mathematics” (p.5). Studentsperception towards the usefulness of Mathematics and its importance to their future and career tends to form as they experience using GCsin Mathematics lessons (Kor, 2008; Schrupp, 2007). They are able to ‘see’ the significance of Mathematics in their daily lives and tounderstand the reasons for learning Mathematics (Rosihan & Kor, 2004; Schrupp, 2007). They find Mathematics as a practical subject, i.e.a wholesome and community related subject (Rosihan & Kor, 2004).

3. Methodology

3.1. Participants

The target population of this study comprised foundation/pre-university students at a university in Malaysia. 65 students participated inthis study; 32 students in the experimental group (24 males and 8 females) and 33 students in the control group (29 males and 4 females);age ranged from 17 to 21 years old. An independent sample t-test on students’ performance of the previous trimester’s mathematics subjectwas conducted to verify similarity of the two samples. The Levine’s test for equality of variances (F ¼ .054, p > .05) was not significant,therefore it can be assumed equality of variances for both groups. A closer look shows that the mean and standard deviation of both groupsare almost similar, as displayed in Table 1. The mean scores of the experimental and control groups are 73.12 and 73.061 respectively; withstandard deviations of 19.874 and 19.733 respectively. Hence, it can be concluded that both groups are homogeneous as there is nosignificant difference between the groups, p ¼ .989 (> .05).

3.2. Instrument

The students’ attitude towards learning Probability was assessed, using the Probability Attitude Inventory (PAI) (see Appendix), adaptedfrom “Mathematics and Science Attitude Inventory” under the “Project EDGE” of Rochester Institute of Technology (Rochester Institute ofTechnology, 1999). The only adaptation made was a change in the term “mathematics” to “probability”. On a 5-point Likert scale, ranging

Table 1Mathematics background.

Group Mean SD

Experimental 73.129 19.874Control 73.061 19.733

SD ¼ Standard deviation.

C.-K. Tan et al. / Computers & Education 57 (2011) 2011–2024 2013

from Strongly Disagree (1), Disagree (2), Neither Agree nor Disagree (3), Agree (4), to Strongly Agree (5), students were required to indicatetheir perceptions and beliefs towards Probability. 31 statements were included in this inventory and the data obtained from these state-ments were analyzed in terms of three aspects: (i) Students’ attitude towards the usefulness of Probability, (ii) Students’ interest inProbability, and (iii) Students’ self-concept in Probability.

3.3. Graphing calculator (GC) instruction and conventional instruction

Students in the experimental group used the pocket-sized GC in learning four main topics of their Probability course: Random Variable,Binomial Distribution, Poisson Distribution and Normal Distribution. The instructor incorporated GC into the lectures to demonstrate keyconcepts of Probability and to provide visual examples. In order to produce cohesive and pedagogically sound GC learning materials, self-designed GC instructional worksheets, i.e. based on the interactive and scaffolding learning framework of this study were utilized:

C encourage active interaction, i.e. communication and involvement among students, interaction between student and GC as well asbetween student and instructor. Students as active rather than passive learners.

C prompt students to learn with each other by sharing information, ideas in a cooperative and collaborative manner.C provide a conducive, fun and enjoyable environment, and to relate real life experience to the activities.C prompt the scaffolding process, with GCs as the scaffolding tool.

To integrate the above elements in the GC instructional worksheet, tasks, guidelines and questions that require students to ‘explain’,‘discuss’, ‘compare’ were incorporated to create a more interactive classroom environment. Examples, “What are the values of X between 6and 10?”, “Explain how you solve for the probability that X assumes the values between 6 and 10.”, “Discuss with your friends how you solvefor the probability of at least 10 chips fail if the value of CDF (10) is known from part (i) above.”, “Compare and discuss the solution of usingthe graphing calculator and the solution of using the statistical tables. Solve it and write it in the symbol.” etc.

The students’ Probability textbook was used as a teaching and learning tool in the control group. The self-designed instructionalworksheets (without the use of GC) that based on the interactive and scaffolding learning framework of this study were also used by thestudents in the control group. The syllabus and teaching materials for both groups were similar.

3.4. Procedures

This study adopted the quasi-experimental intact group research design. It was carried out concurrently for both groups in a 14 weekstrimester. In order to ensure consistency, only one instructor taught both the groups, using the same contents. The only difference was thatdifferent teaching and learning tools were incorporated as mentioned earlier.

At the beginning of the trimester (week 1), the pre-PAI was administered to both groups. They were informed of the confidentiality oftheir responses.

In weeks 2 and 3, the experimental group underwent four sessions of GC workshops in order to familiarize themwith the buttons on GCso that they will be able to master the key features necessary for the topics identified for this study. Each session lasted for an hour.

The intervention period commenced after the GC workshops, i.e. from week 4 to week 12. A typical lesson for both groups generallybeganwith the teaching of theories (15 min), conducting GC activities for the experimental group/solving problem questions for the controlgroup (100 min), and conclusion (5 min).

The self-designed instructional activity sheets in both groups provided opportunities for discussions, interactions and communication ineach lesson. A transformation is seen in the instructor’s role, i.e. to a scaffolder who guided, facilitated and offered suggestions, if necessary.The intensity of the scaffolding gradually lessened as students gained competency in mastering the GCs (for the experimental group) andcomprehending the topics (for both groups). The control group, on the other hand, was taught using the conventional approach of chalk andtalk. However, they had equal opportunities for instructor guidance and facilitation, class interactions and discussions too. Both groups keptjournals to record their experiences.

At the end of the intervention period, the PAI was re-administered to both groups. Statistical analysis, i.e. descriptive statistics, t-test,ANCOVA and MANCOVA were conducted, using SPSS 11.0.

4. Data analysis and results

4.1. Instrument validation

All data of this study was analyzed to determine the reliability and validity of the measurement scales. A reliable instrument is one thatgives consistent results (Fraenkel & Wallen, 2010). The instrument is considered reliable if the reliability coefficient is greater or equal to .7(Fraenkel & Wallen, 2010). The Alpha coefficients as shown in Table 2, which ranging from .8648 to .9741, i.e. more than .7, implies that theinstruments exhibited acceptable reliability.

The 31 statements were analyzed for validity by conducting a factor analysis, with the extraction method of Principal ComponentAnalysis, the rotation method of Varimax, and the coefficient displayed by size. As mentioned earlier, the analysis was based on the threeaspects (17 statements on aspect of students’ self-concept in Probability (SC), 9 statements on students’ interest towards Probability (IP), and5 statements on students’ attitude towards the usefulness of Probability (UP)). The rotation solution, as shown in Table 3, yielded threeinterpretable factors, i.e. the SC, IP and UP aspects. The SC aspect accounted for 37.67% of the item variance, the IP aspect accounted for22.59% of the item variance, and the UP aspect accounted for 11.86% of the item variance. The results of the factor analysis shown in Table 4reveal high validity scores.

Table 2Reliability.

Instrument Aspects Cronbach’s Alpha

PAI Overall .9567UP .8648IP .9505SC .9741

Note. UP: Perceived usefulness of Probability, IP: students’ interest in Probability, SC: students’self-concept in Probability.

C.-K. Tan et al. / Computers & Education 57 (2011) 2011–20242014

4.2. Comparison of results between experimental and control groups

4.2.1. ANCOVATable 5 shows the means and standard deviations of pre and post-PAI for both groups. The means of pre and post-PAI for both groups are

depicted in Fig. 1. Fig. 2 depicts the graph of post-PAI with confidence interval for both groups.ANCOVAwas conducted. The independent variablewas the Groups, i.e. experimental and control groups. The dependent variablewas the

post-PAI scores and the covariate was the pre-PAI scores. A preliminary analysis evaluating the homogeneity-of-slopes assumption indicatesthat the relationship between the covariate (pre-PAI) and the dependent variable (post-PAI) did not differ significantly as a function of theindependent variable (group), F(1,61) ¼ .005, p ¼ .947 (>.05). Based on this finding, we proceeded with the ANCOVA analysis.

The ANCOVA is significant, i.e. F(1,62) ¼ 71.491, p < .05.The estimated marginal means are shown in Table 6 and depicted in Fig. 3. Theexperimental group had the largest adjusted mean (3.965) and the control group had the smallest adjusted mean (2.948). Follow-up test,Holm’s sequential Bonferroni was conducted to evaluate pairwise difference among the adjusted means. It was chosen to control for Type Ierror across the pairwise comparisons. It has strong control and great power and allows for use with any set of statistical tests. There isa significant difference between the two groups, p< .05. That is, the experimental group recorded significantly higher scores than that of thecontrol group at the end of the treatment period.

4.2.2. MANCOVATable 7 presents the means and standard deviations of pre-UP, pre-IP, pre-SC, post-UP, post-IP and post-SC for both groups. The means of

pre-UP, pre-IP, pre-SC, post-UP, post-IP and post-SC for both groups are depicted in Figs. 4 and 5. Fig. 6 shows the graph of post-UP, post-IPand post-SC with confidence interval for both groups.

MANCOVA was conducted. The independent variable was the Groups, i.e. experimental and control groups. The dependent variableswere the post-UP, post-IP and post-SC scores and the covariate were the pre-UP, pre-IP and pre-SC scores. A preliminary analysis evaluating

Table 3Total Variance Explained.

Component Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings

Total % of Variance Cumulative % Total % of Variance Cumulative % Total % of Variance Cumulative %

1 15.730 50.742 50.742 15.730 50.742 50.742 11.676 37.666 37.6662 4.281 13.809 64.552 4.281 13.809 64.552 7.001 22.585 60.2513 2.343 7.557 72.108 2.343 7.557 72.108 3.676 11.857 72.1084 .856 2.762 74.8705 .692 2.233 77.1036 .678 2.187 79.2907 .649 2.093 81.3838 .634 2.045 83.4289 .561 1.808 85.23610 .496 1.599 86.83511 .461 1.486 88.32112 .396 1.277 89.59913 .375 1.210 90.80814 .359 1.157 91.96515 .326 1.053 93.01816 .285 .919 93.93717 .232 .750 94.68718 .219 .707 95.39319 .201 .650 96.04320 .190 .614 96.65721 .167 .538 97.19522 .160 .517 97.71223 .141 .454 98.16624 .105 .339 98.50525 .102 .328 98.83326 .089 .287 99.12027 .081 .262 99.38228 7.444E-02 .240 99.62229 5.587E-02 .180 99.80230 3.794E-02 .122 99.92531 2.336E-02 7.535E-02 100.000

Table 4Validity.

Question Aspects

SC IP UP

Q8 .931 .124 .057Q16 .841 .097 .078Q20 .830 .093 .108Q7 .824 .142 .074Q18 .818 �.013 .133Q24 .812 .250 .155Q4 .810 .343 .091Q22 .804 .162 .249Q27 .803 .380 .074Q25 .802 .185 .233Q13 .799 .208 .171Q19 .781 .179 .275Q5 .778 .355 .248Q30 .752 .195 .181Q23 .747 .316 .252Q12 .745 .278 .073Q17 .732 .276 .222Q31 .130 .880 .039Q11 .224 .829 �008Q28 .093 .816 .150Q2 .249 .812 .145Q14 .206 .810 .104Q29 .324 .786 .286Q1 .309 .780 .247Q10 .104 .779 .190Q15 .314 .778 .202Q21 .204 .174 .843Q9 .133 .293 .760Q26 .386 .117 .739Q3 .191 .036 .739Q6 .097 .239 .709

Note. Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization.Bold: the value is greater or equal to .7.

C.-K. Tan et al. / Computers & Education 57 (2011) 2011–2024 2015

the homogeneity-of-slopes assumption indicates that the relationship between the covariate and the dependent variables did not differsignificantly as a function of the independent variable, i.e. F(2,58)¼ .266, p¼ .768, for pre-UP and post-UP, F(2,58)¼ .074, p¼ .929, for pre-IPand post-IP, and F(2,58) ¼ 1.934, p ¼ .154, for pre-SC and post-SC. Based on this finding, we can proceed with the MANCOVA analysis.

The Pillai’s Trace of .607 is significant, F(3,58) ¼ 29.857, p < .05, indicating that we can reject the hypothesis that the populations meanson the dependent variables are the same for the two instructional methods.

The MANCOVA is significant, i.e. F(1,60) ¼ 11.337, p¼ .001, for post-UP; F(1,60) ¼ 71.215, p< .05, for post-IP, and F(1,60) ¼ 31.677, p< .05,for post-SC, controlling for pre-UP, pre-IP and pre-SC.

Univariate test results are significant for post-UP, post-IP and post-SC, with F(1,60) ¼ 11.337, 71.215, and 31.677, respectively, and p < .05.The estimated marginal means are shown in Table 8 and Figs. 7, 8 and 9. Holm’s sequential Bonferroni was conducted to evaluate

pairwise difference among the adjusted means. There are significant differences between two groups for post-UP, post-IP and post-SC,p < .05. That is, the experimental group recorded significantly higher scores than those of the control group after the study in all thethree aspects.

4.2.3. t-test resultsThe t-test results reveal that there is a significant difference in the overall means attitude before and after the study for both groups. The

control group had significantly lower overall means attitude after the study (M ¼ 2.95, SD ¼ .614) than before the study (M ¼ 3.31,SD ¼ .386), t(32) ¼ 2.944, p < .05, whereas the experimental group registered a significant higher overall means attitude after the study(M ¼ 3.96, SD ¼ .284) than before (M ¼ 3.24, SD ¼ .392), t(31) ¼ �9.114, p < .05.

In addition, there is no significant difference in pre-PAI between the two groups (t(63)¼ .762, p¼ .449), but there is significant differencein post-PAI between them (t(45.370) ¼ 8.561, p < .05). Before the study, the average attitude score of the experimental group (M ¼ 3.24,SD ¼ .392) is not significantly different from that of the control group (M ¼ 3.31, SD ¼ .386). However, the experimental group (M ¼ 3.93,SD ¼ .284) recorded significantly higher scores than that of the control group (M ¼ 2.95, SD ¼ .614) after the study.

Table 5Means and Standard Deviations of pre and post-PAI for two groups.

Group PAI Mean SD

Control Pre 3.31 .386Post 2.95 .614

Experimental Pre 3.24 .392Post 3.96 .284

2.5

2.7

2.9

3.1

3.3

3.5

3.7

3.9

4.1

Pre Post

PAI

Scor

es

experimental group

control group

Fig. 1. PAI scores before and after intervention.

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4.3. Qualitative data

This study evidently concurs with the literature on the improved attitude when GC is used. Closer examination also found that theexperimental classroom was transformed into a laboratory of active learners, exploring, experimenting, analyzing, discussing, comparingand having funwhile learning collaboratively, a seemingly, difficult subject. They developed confidence in solving problems andwere able to‘see’ the useful of Probability. An active ‘lab’ emerged, with students interactingwith each other, with the instructor andwith GC. The entriesinto the students’ journals verify these findings:

control groupexperimental group

Group

4.2

3.9

3.6

3.3

3.0

2.7

95%

C

I P

ost-test attitu

de

Fig. 2. Error bar chart for post-PAI.

Table 6Estimated Marginal Mean.

Group Mean Std Error 95% Confidence Interval

Lower Bound Upper Bound

Experimental 3.965 .086 3.794 4.136Control 2.948 .084 2.779 3.116

Note. Covariates appearing in the model are evaluated at the following values: Pre-PAI ¼ 3.28.

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C Perceived usefulness of probability

. We can solve the uncertainty in numerical quantity. For instance, the win or loss in each play of a game. We can use probability todetermine the outcome of the situation or experiment. . Interesting and can be applied in our daily life.. quite interesting and it may be useful in my future life.. I think this topic is more useful than other topics that we have learnt. Interesting and can be applied in daily life,. will help me in thefuture.. helps me to understand interrelation between values.

C Students’ interest in probability

. It was amusing and also fun (learning with graphing calculator). I am happy and satisfied. I don’t have anymore problem .I have exciting experiences with the graphing calculator.. quite interesting.

C Students’ confidence in probability

. it enables me to calculate the answers faster and more efficiently.. less room to make careless mistakes.. helps me to double-check myanswers. . to detect where I have gone wrong in the calculation and improves my understanding.. it is more efficient while learning something . letting me double check my calculations/method.. I have identified the method to obtain a solution. . I am confident because I understand the method of solution.. I am now able to interpret the questions and answer that I got. I think with graphing calculator, my problem solving skills have improved.. I can answer the questions confidently and can’t wait or the next class .

C Collaborative learning

. We brain-storm and discuss with each others, also ask help from lecturer because we are confident .

. we interact more now.lecturer and friends helped in conducting tasks, this made me to be more confident to solve problems now.I donot scare to solve more problems.

Control groupExperimental group

4

3.8

3.6

3.4

3.2

3

2.8

Est

imat

ed M

argi

nal M

eans

Fig. 3. Estimated marginal means of Post-PAI.

Table 7Means and standard deviations of pre and post-PAI for three aspects.

Group PAI Mean SD

ControlUP Pre 3.61 .549

Post 3.31 .794IP Pre 2.93 .445

Post 2.44 .749SC Pre 3.40 .470

Post 3.12 .905

ExperimentalUP Pre 3.43 .590

Post 3.88 .512IP Pre 2.87 .507

Post 3.76 .426SC Pre 3.42 .518

Post 4.10 .286

C.-K. Tan et al. / Computers & Education 57 (2011) 2011–20242018

. we discuss the questions thoroughly and try to figure out the concepts behind the questions .

. we interact by discussing every question and help each other when we find difficulties .

In contrast, entries in the control group’s journals provide evidence that, as compared to the GC approach, the conventional approach didnot provide much room for interaction among them. Students’ confidence level in problem solving was low and they were not able to ‘see’the useful of Probability:

. It is not as what I expect. I thought it is useful before I learnt this subject, but now I can’t see .

. abstract subject, can’t apply it. I don’t bother, I also don’t understand.

. I don’t want to discuss with my friends. I would like the answers to be given straight away. Hate to calculate. Boring subject. I don’t know howto do my exercises, I don’t understand ... not as what I expected. I am good in mathematics but hate long calculations. I know how to solve the question coz after the classes, I dolots of exercises and study myself. I don’t know how to solve, everyone do their own work . I don’t know them . no choice but ask lecturer.. I am not sure whether my solution is correct .. Probability is difficult, very abstract . don’t know how to start with the workings. . too many formula and difficult calculation.. bored with this class, don’t “see” the concept of probability, my friends know a little only even though discuss with them.. The class is boring. Just sitting and doing our own work. Lecturer asked me to discuss with friends, but I don’t know what to discuss & theyalso don’t want to discuss. Never mind, just sit & listen to lecturer.

Of the 33 students in the control group, only 5 students had perceived the usefulness of Probability.

5. Discussions and implications

As a conclusive result, both groups recorded significant differences before and after the study, and between groups in their attitudestowards Probability after the study. The experimental group showed significant improvement in attitudes towards Probability, andsignificantly better attitude than the control group at the end of the study. On the other hand, the control group had significantly less

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

Pre Post

PAI S

core

s

UP

IP

SC

Fig. 4. PAI scores before and after the study for control group based on three aspects.

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

4

4.2

Pre Post

PAI

Scor

es UP

IP

SC

Fig. 5. PAI scores before and after the study for experimental group based on three aspects.

C.-K. Tan et al. / Computers & Education 57 (2011) 2011–2024 2019

positive attitude and interest towards Probability after the study than before the study. The results are consistent with previous research(e.g. Ellington, 2003; Ha, 2008; Kor, 2008; Milou, 1999; Munawir & Raihan, 2008; Noraini, Tay, Ding et al., 2003; Schrupp, 2007; Stick, 1997;Wilson & Krapfl, 1994) confirming that GC approach fosters positive attitudes and increased motivation to learn mathematics. Theexperimental group showed both increased attitudes and better attitude than the control group in all three aspects after the study.

The significantly higher attitude recorded by the experimental group might be due to the learning environment which was moreinteractive and supportive. Students looked forward to learning new things and having fun with GC. Students actively participated in

control groupexperimental group

Group

4.5

4.0

3.5

3.0

2.5

2.0

95%

C

I

Post-students' self-concept inprobability

Post-students'interest towardsprobability

Post-students'attitude towards theuse of probability

Fig. 6. Error bar chart for three aspects.

Table 8Estimated Marginal Means of three aspects.

Dependent Variable Group Mean Std. Error 95% Confidence Interval

Lower Bound Upper Bound

Post-UP Experimental 3.879 .121 3.638 4.120Control 3.305 .119 3.068 3.542

Post-IP Experimental 3.761 .111 3.539 3.983Control 2.437 .109 2.218 2.655

Post-SC Experimental 4.090 .122 3.847 4.333Control 3.121 .120 2.882 3.361

Note. Covariates appearing in the model are evaluated at the following values: Pre-UP ¼ 3.52, Pre-IP ¼ 2.90, Pre-SC ¼ 3.41.

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knowledge and skills acquisition, during the GC activities. Some students sought help frommore capable peers and their instructors, if theyencountered difficulties in solving the problems and/or the functions of using GCs. In addition, students explored the GC activities throughthe discovery approach and were seen sharing answers and information during the discovery process. They appeared to enjoy the sharingand interactive process. It saved their time in solving problems by speeding-up the tedious calculation process. As highlights by Stacey(2004), GC has been used functionally and pedagogically in which students appreciated getting the answers. Consistent with Kor (2008)and Schrupp (2007) GC made interesting changes in Probability classrooms and enhanced the students’ appreciation and enjoyment oflearning mathematics.

The results of this study are also consistent with other studies (Nasari, 2008; Scariano & Calzada, 1994; van der Kooij, 2001), particular interms of helping students to “visualize” the concepts, graphically or numerically, understanding the relationships between concepts, and“make sense” of probability concepts. This in turn increases their confidence in applying mathematics and interpreting the conclusions. Theresults are also consistent with previous studies that support the use of GC in boosting students’ self-confidence in learning mathematics(Abd Rahim, 2008; Acelajado, 2005; Anis, 2008; Graham, Headlam, Sharp, & Watson, 2008; Jones, 1995; Khairiree, 2003; Noraini, 2003;Noraini, Tay, Goh, et al., 2003; Rosihan & Kor, 2004).

Students began to find Probability relevant to the daily lives, i.e. how the situations in the world apply the Normal distribution. Theyfound the lessons enabled them to understand the probability of an incident to occur and that they can apply what they learnt into real lifesituations and solve daily life’s problems. The knowledge gainwould also be useful in their future, as they would be able to apply probabilityto determine the outcome of a certain situation in their daily life as well as in their workplace. They also displayed better understanding ofthe usage of the formula in the Probability course. These findings are also consistent with previous research that found that GC helpedstudents see the usefulness of mathematics and highly regard its importance to their future and career (Kor, 2008; Schrupp, 2007; Seth &Willis, 2004; Waits & Demana, 1994). The quantitative findings of significant differences in students’ attitude towards learning Probabilityafter using GC support this evidence.

The mean score in the control group, which, on the other hand, had declined after the intervention showed slightly better attitudetowards Probability before the intervention than after the intervention (mean difference ¼ .36). However, after the intervention, the

Control groupExperimental group

3.9

3.8

3.7

3.6

3.5

3.4

3.3

Est

imat

ed M

argi

nal M

eans

Fig. 7. Estimated marginal means of post-UP.

Control groupExperimental group

3.75

3.5

3.25

3

2.75

2.5

Est

imat

ed M

argi

nal M

eans

Fig. 8. Estimated marginal means of post-IP.

C.-K. Tan et al. / Computers & Education 57 (2011) 2011–2024 2021

students displayed the same attitude in the aspects of the usefulness of Probability and self-concept in Probability as before, except for theaspect of interest towards Probability. This could be because they had expected that they would develop some level of interest towardsProbability (mean ¼ 2.93) before the study. However, they found the lessons of the conventional approach were not to their expectations,especially in the aspect of interest towards Probability. It is clear that although students underwent a full course on Probability, most ofthe students, who did not use GC, appear to be unable to ‘see’ the relevancy of Probability in their daily lives and appear to display lower

Control groupExperimental group

4.2

4

3.8

3.6

3.4

3.2

3

Est

imat

ed M

argi

nal M

eans

Fig. 9. Estimated marginal means of post-SC.

C.-K. Tan et al. / Computers & Education 57 (2011) 2011–20242022

self-concept in Probability. This is evident from students’ journal, i.e. of the 33 students, only 5 students had perceived the usefulness ofProbability and most of themwere unsure of the solutions and answers they presented. Obviously, students appeared to be disinterested inProbability, which could be due to the delivering mode that they found to be boring. The significant lower attitude displayed by the controlgroup may also be due to the limited interactions among students. Despite the instructor’s encouragement to discuss, they were mostlypassive and did not seek assistance when in difficulties. The majority of students expressed that Mathematics is difficult, even beforeundertaking the Probability course. This perception and attitude prevailed, and in fact ‘worsen’ after the Probability course. They appear tobe bored of listening to the lecture and ‘bored’ in solving problems, using the pencil-and-paper approach and the one-way, non-activelearning environment. Consequently, the students found it difficult to understand, which could also be because they were not able to ‘see’the usefulness of Probability in lives and future careers. The findings are consistent with Duatepe-Paksu and Ubuz (2009), Neo(unpublished), Rosnaini et al. (2009), i.e. in the conventional approach, students are mostly passive and rarely volunteer to participate,and solve the questions displayed on the whiteboard. Thus it is not surprising that the level of confidence in Probability declined. Studentswere clearly not motivated in the Probability classrooms as they found the lesson uninteresting, which is consistent with Honeycutt andPierce’s (2007) study, i.e. the students were less interested and unable to understand, and retain important concepts.

This study provides significant implications in the adoption of GC. As highlighted in the Literature Review section, there is evidence inchanges in attitude when GCs are adopted in the classrooms, i.e. among others, increased motivation, enjoyment, appreciation in mathe-matics. This study shows empirical evidence that GC instruction can be an alternative innovative teaching approach to develop positiveattitude in mathematics learning, especially in Probability lessons.

The interesting results yielded in this study would certainly encourage educators to adopt GCs in their classrooms, particularly inProbability courses. Educators could design effective GC instructions which provide hands-on experiences, particularly since mathematicsinvolves complex concepts and computations. GCs would facilitate students’ learning, especially in solving more complex mathematicsproblems which is challenging. The GC instructional worksheets of this study could be adopted or adapted for this purpose. Alternativelyinstructors could design their own instructional worksheets to facilitate effective use of GCs, adopting the interactive and scaffoldingframework of this study as an option.

The interactive, cooperative and supportive learning environment, such as scaffolding frommore competent peers and adults (teacher),which the GC instructions generated, had helped overcome learning barriers as well as anxieties and elevated students to the higher level ofdevelopment as they gained conceptual and procedural knowledge. It is important for educators to apprehend their students’ attitudetowards the subject and an effective tool in order to undertake appropriate measures/actions to continuously enhance the learning ofProbability.

6. Conclusion

In concluding, learning Probability with GC has clearly benefited students in this study. It has proven to be a powerful learning tool insolving mathematical problems and has transformed students’ attitude towards learning Probability. It provides great opportunities tofoster positive attitudes towards mathematics, in general, and Probability, in particular. This is a clear indication that students who use GC intheir learning positively, perceive the usefulness of Probability in lives, develop an increase in interest in Probability learning and greatersense of confidence in Probability. With these positive findings, GC is seen as a valuable educational tool that ought to be adopted inmathematics classrooms as well as further extended to other domains and scopes. Future research could examine the effects of GC in otheraspects and in different educational settings. A better understanding and implementation of effective GC instructions will enhance theadoption and educational value of such educational technology.

Acknowledgements

We would like to thank StatWorks (M) Sdn Bhd for the GC loan, the Foundation Center for the permission of conducting this study tofoundation students, Rochester Institute of Technology, and all the respondents.

Appendix. Questions in the Probability Attitude Inventory (strongly disagree. strongly agree):

1. Probability is something which I enjoy very much.2. Solving Probability problems is fun.3. There is little need for Probability in most jobs.4. When I hear the word Probability, I have a feeling of dislike.5. I would like to spend less time in school doing Probability.6. Probability is helpful in understanding today’s world.7. No matter how hard I try, I cannot understand Probability.8. I often think, "I can’t do it," when a Probability problem seems hard.9. It is important to know Probability in order to get a good job.

10. I enjoy talking to other people about Probability.11. Sometimes I do more Probability problems than are given in class.12. I remember most of the things I learn in Probability.13. I would rather be given the right answer to a Probability problem than to work it out myself.14. It is important to me to understand the work I do in Probability.15. I have a real desire to learn Probability.16. It scares me to have to take Probability.17. I have a good feeling towards Probability.18. If I don’t see how to do a Probability problem right away, I never get it.

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19. I usually understand what we are talking about in Probability.20. I feel uneasy when someone talks to me about Probability.21. Probability is of great importance to a country’s development.22. I would like a job which doesn’t use any Probability.23. I am good at doing Probability problems.24. I can get along perfectly well in everyday life without Probability.25. It makes me nervous to even think about doing Probability.26. Probability is useful for the problems of everyday life.27. I don’t do very well in Probability.28. I would like to do some outside reading in Probability.29. Probability is easy for me.30. Most people should study some Probability.31. Sometimes I read ahead in my Probability book.

The Probability Attitude Inventory was adapted from Mathematics and Science Attitude Inventory, Project EDGE, Rochester Institute ofTechnology, 1999.

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