Effects of the application of graphing calculator on students’ probability achievement

Computers & Education 58 (2012) 1117–1126

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Effects of the application of graphing calculator on students’ probability achievement Choo-Kim Tan* Faculty 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

a b s t r a c t

Article history: Received 18 June 2011 Received in revised form 24 November 2011 Accepted 30 November 2011

A Graphing Calculator (GC) is one of the most portable and affordable technology in mathematics education. It quickens the mechanical procedure in solving mathematical problems and creates a highly interactive learning environment, which makes learning a seemingly difficult subject, easy. Since research on the use of GCs for the teaching and learning of probability in institutions of higher learning appears to be limited, a study was conducted to examine the effects of the use of GCs on students’ performance in probability. The study involved 65 pre-university students in a private institution of higher learning in Malaysia. Hypothesis testing showed that there is a significant difference in the mean score of the post achievement test between the experimental and control groups. Students in the experimental group displayed higher achievement than their peers in the control group in the Probability Achievement Test (PAT). Findings also showed that the use of GCs benefits students of all levels, that is, high, average and low mathematics achievers. Qualitative data provides a more lucid picture of how GCs aid in improving understanding and performance. Recommendations made will provide researchers, policy-makers and educators an alternative means of enhancing the quality of probability education. Ó 2011 Elsevier Ltd. All rights reserved.

Keywords: Improving classroom teaching Interactive learning environments

1. Introduction Research evidence that supports the use of handheld technologies, such as GCs to enhance performance in mathematics appears to be growing. A review of the literature provides testimony for the GC as a multifunctional educational tool to improve the performance in mathematics; a subject which appears to record higher failure rates as a result of the inability to grasp mathematical concepts and subsequently solve problems. Griffith (1998) notes that “it may well be the case that technology does not help or hinder students in the memorization of facts, but technology does help students to develop conceptual understanding and problem-solving abilities” (p.76). Noraini (2006) and Ron (2004) concur that, in comparison to the conventional approaches, GC instructional approach enables students to grasp and understand mathematical concepts better. The incorporation of GCs in mathematics classrooms has been found to yield positive outcomes, including gains in students’ learning outcomes. It encourages student-centred learning and develops higher order thinking as it allows students to simulate various mathematical situations for modelling and investigative purposes (Simonsen & Dick, 1997). In the same vein, Martinez-Cruz and Ratliff (1998) assert that GCs can be used to bring real-world data into the class and permit students to focus on mathematics concepts. Students have reported of broader and better understanding as well as appreciation of specific topics, such as calculus, linear algebra, differential equations and statistics (Kor & Lim, 2003; Waits, 1992). Ong (2004) argues that the increased number of examples, sequenced from easy to difficult that can be observed within a shorter time frame leads to better understanding. Consequently, marked improvement is evident in students’ performance (Acelajado, 2004a; Burill, 2004; Ekthaicharern, 2004; Nor’ain, Rohani, Wan Zah, & Mohd. Majid, 2008; Ruthven, 1990). Albeit, a plethora of research evidence on the use of GCs in the teaching and learning of algebra, graphs and functions, straight lines, geometry, trigonometry, statistics and calculus (Arnold, 2008; Chinnappan & Thomas, 2004; Dunham, 1995; Dunham & Dick, 1994; Graham & Thomas, 2000; Heid, 1997; Horton, Storm, & Leonard, 2004; Jones, 1995; Kor & Lim, 2003; Laughbaum, 1998; Marshall, 1996; Nasari, 2008;

* Tel.: þ606 2523427; fax: þ6062318840. E-mail address: [email protected]. 0360-1315/$ – see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compedu.2011.11.023

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Ong, 2004; Penglase & Arnold, 1996; Ruthven, 1990; Thompson & Senk, 2001; Waits & Demana, 1999) is found, regrettably, research on probability in institutions of higher education appears to be limited. This study was thus conducted primarily to examine the effects of the use of GCs on pre-university students’ achievement in probability lessons. The two research questions of interest are: (1) To what extent does the incorporation of GCs in the teaching and learning of probability improve students’ achievement? (2) Are there significant differences in probability achievement for students of different performance levels between the experimental and control groups? 2. Literature review A review of studies provides evidence that the use of GCs significantly improves students’ mathematics achievement and performance (Brooks-Young, 2009; Burill et al., 2002). Students have been found to score higher on performance measures and perform better in examinations. Similar findings on students’ improvements are also seen in specific topics, such as straight lines, functions and graphs, algebra, geometry, trigonometry, pre-calculus, calculus, quadratic equations, statistics and problem solving (Arnold, 2008; Chinnappan & Thomas, 2004; Dunham, 1995; Dunham & Dick, 1994; Graham & Thomas, 2000; Heid, 1997; Horton et al., 2004; Jones, 1995; Kor & Lim, 2003; Laughbaum, 1998; Marshall, 1996; Nasari, 2008; Ong, 2004; Penglase & Arnold, 1996; Ruthven, 1990; Thompson & Senk, 2001; Waits & Demana, 1999). One of the attributes for the improved performance is . the graphing-approach curriculum can include examples and problems for modeling real-world situations with functions that would be either too time-consuming or impractical without a GC. [GC] affords [students] . the ability to create equations, tables, and graphs quickly and the facility to move among the representations rapidly . [Students] were more comfortable . when working with realworld data and situations. [They] performed significantly better . on interpreting and translating questions (Hollar & Norwood, 1999; p. 224) and . are able to solve quadratics by factoring, using tables, graphing, and using the quadratic formula more accurately (Schrupp, 2007; p. 59). Similarly, Shore’s (1999) study has also found increased gains in procedural skills and conceptual understanding in Elementary and Intermediate Algebra. Acelajado’s (2004a, 2004b) studies showed consistent findings, that is, improved achievement, particularly within each of the 3 ability groups. The obvious significant difference was in the mean scores between the pre- and post-tests in the achievements of the different ability groups. The results favoured the high ability group in Acelajado’s (2004a, 2004b) studies. Although both Harskamp, Suhre, and Van Streun (2000) and Van Streun, Harskamp and Suhre’s (2000) studies found improved achievements among the students, the studies recorded that below-average students benefited most from the use of GCs. They note that the improvement could be linked to the increased use of the graphical solution strategies. Evidence of positive outcomes is not only seen among low ability groups, but also among students with disabilities, that is, students who are diagnosed with learning disabilities, health impairment and emotional impairment (Bouck, 2009). Bouck’s (2009) study on the use of GCs by students with disabilities provides evidence for positive outcomes, in favour of the incorporation of GCs. Both groups of students with and without disabilities participated in Bouck’s (2009) study. His study confirms the positive impact of the use of GCs on performance of both groups of students. Bouck (2009) highlights that even though students without disabilities performed significantly better in the post-test, students with disabilities had also made gains from the pre-test to post-test. The obvious impact of the incorporation of GCs in classrooms is not limited to only increase in achievement, but also increase in classroom interaction and in active students’ participation. GCs appear to be a tool that scaffolds students’ interactions. Vygotsky (1978) highlights the three stages of the learning process, that is, cannot yet do, can do with help and can do alone. Therefore, during classroom interaction, students are in the can do with help stage, that is, the stage of Zone of Proximal Development (ZPD); a temporary stage towards being able to do something on one’s own, a can do alone stage. In the can do with help stage, the use of GCs have been found to provide opportunities for scaffolding and interaction. Students have been found to be actively interacting and constantly communicating among themselves and with their teachers (Galbraith, Renshaw, Goos, & Geiger, 1999; Nor’ain et al., 2008; Noraini et al., 2003; Waits & Demana, 1994). Moreno-Armella and Cinvestar (1999) note that the use of GCs creates a conducive environment for students to pose and explore mathematical questions that require them to display and contrast the potentials of diverse resources as well as mathematical strategies. The GC environment also allows for active students’ participation in activities such as the formulation of conjectures, the search of patterns and the exploration of related questions. Farrell (1996) and Slavit (1996) also add that the GC environment allows for active involvement in group activities, as they investigate and solve problems. Conventional classrooms were evidently transformed into GC laboratories, where everyone ‘experimented’ on GCs and worked collaboratively in small groups to investigate patterns, analyse and compare results as well as solve problems through active discussions, interactions and communication. As students gained confidence of the use of GCs, they moved towards the can do alone stage, which Waits and Demana (1994) reported that students were adventurous and ‘self-explored’ the GCs. Dick (1992) and Hopkins (1992) found that students were able to interpret the results generated from GCs. Thus, it is in the GC laboratory that the students consequently constructed their own mathematical understanding, which subsequently resulted in improved performance. 3. Methodology 3.1. Participants 65 pre-university students who were pursuing their foundation programme at a private university in Malaysia were involved in the experimental study. The composition was 32 students in the experimental group (24 males and 8 females) and 33 students in the control

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group (29 males and 4 females). To verify that the two groups were homogeneous in their mathematics achievement, an independent samples t-test at 5% significance level was conducted using the marks that they had attained for mathematics in the previous final examination. The t-test confirms that both groups were homogeneous as no significant difference was found between the groups, p ¼ .989 (>.05). The mean score of the experimental group (73.12) was not significantly different from that of the control group (73.061); the standard deviations were 19.874 and 19.733 respectively. Based on the students’ mathematics scores in the previous trimester’s final examination, they were categorised into high, average and low achievers in each group. Students who obtained grades Aþ, A or A- (or 75 to 100 marks) were categorised as high achievers, Bþ, B or B- (or 60 to 74 marks) as average achievers and Cþ, C and F (or below 60 marks) as low achievers. 3.2. Instrument and instructional approaches A formal paper-and-pencil test, that is, Probability Achievement Test (PAT) was administered to measure the students’ performance. The test comprised four problem solving questions on Random Variable, Poisson Distribution, Binomial Distribution and Normal Distribution, as per the probability syllabus adopted at the private university. The maximum score for PAT is 80. The PAT was administered to both the experimental and control groups at the beginning and at the end of the study. To establish the validity of the PAT, it was reviewed and validated for its relevance and concordance with the syllabus by five experts in the field of probability and statistics with 12 years of experience teaching probability and statistics at the university. It was also assessed for reliability, using Cronbach’s alpha. A coefficient of .9259, with an acceptable reliability was recorded. The GC instructional approach was employed in the teaching and learning of probability in the experimental group. GC instructional worksheets were designed as modular lessons to be used during the intervention period. The conventional approach, using a textbook as a teaching and learning tool, was adopted in the control group. For consistency, one instructor taught both groups, using the same syllabus. 3.3. Procedure The study was conducted in a 14 week trimester and was designed in three stages, that is, in correspondence to the beginning of study, intervention period, and end of study. 3.3.1. Stage 1 – beginning of study Stage one was conducted at the beginning of the trimester (week 1). The pre-PAT was conducted as a group administration to all the 65 students involved in this study. In weeks 2 and 3, students in the experimental group attended the four 1-h sessions of GC workshops. The workshops were aimed at giving students the opportunity to explore the buttons and to be familiar with the pertinent key features of the GCs that are essential for the specific 4 topics on probability in this study. 3.3.2. Stage 2 – intervention period Weeks 4–12 were the intervention period. As mentioned earlier, both groups were taught the same subject (probability) and content (4 topics) by one instructor, with the exception of the instructional approach; the GC approach to the experimental group and conventional approach to the control group. During the course of this period, in each lesson, instructional worksheets were distributed to the experimental group. A general lesson in both groups began with the teaching of theory (approximately 5 min), followed by carrying out GC activities by the experimental group and solving problems by the control group (approximately 100 min), and finally concluding the lesson (approximately 5 min). Students in both groups were encouraged to interact and communicate with each other. As the lessons in the experimental group progressed, the role of the instructor was transformed to one who scaffold, guided, and facilitated. The intensity of the scaffolding gradually lessened as students gained mastery of the GC and better understanding of the topics. Both groups kept journals to record their experiences. 3.3.3. Stage 3 – end of study In week 13, the PAT was re-administered to both the experimental and control groups. Similar procedures of the administration of the pre-PAT as in Stage 1were adopted. Statistical analysis of descriptive statistics and t-test using SPSS 14.0 were conducted to analyse the results. The students’ comments recorded in their journals were analysed by a three-member panel who have 12 years of experience teaching probability and statistics at the university. The panel had categorised the comments based on the emerging themes. 4. Results The quantitative data are presented as descriptive statistics, that is, mean scores and standard deviations. The data are also presented graphically to highlight the similarities and differences in the results. The qualitative data, which complements the quantitative data provides a more lucid picture of the students’ perception of the use of GCs in their learning of probability. Table 1 shows the mean scores and standard deviations of the pre- and post-PAT for both groups. Fig. 1 displays the mean scores of the pre- and post-PAT and the mean difference of the post-PAT for both groups. As seen in Table 1, although the mean scores of the pre-test show that the control group performed better than the experimental group, there is no statistical significant difference between the two groups. The experimental group scored a mean of 1.99, with a standard deviation of 1.954, while the control group recorded a mean score of 2.95, with a standard deviation of 2.630. The mean difference between these two groups is .96, with a t-value of 1.678 and a p-value of .099 (>.05), which indicates that there is no significant difference in the mean scores in the pre-test. The results confirmed our assumption that the two groups are homogenous as mentioned earlier. It can be clearly seen in Fig. 1 that both groups appear to be at the same starting point before the study.

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Table 1 Pre-PAT and Post-PAT between the two groups. PAT

Group

Mean

SD

MD

Pre

Experimental Control Experimental Control

1.99 2.95 75.71 42.19

1.954 2.630 5.037 23.162

95% confidence interval of the difference Lower

Post

Upper

.96

2.110

.185

33.52

25.139

41.902

SD ¼ standard deviation, MD ¼ mean difference.

However, the difference between the groups after the intervention is obvious. The experimental group recorded a mean score of 75.71, with a standard deviation of 5.037, whilst the control group recorded a mean score of 42.19, with a standard deviation of 23.162 on the posttest. Hypothesis testing shows that there is a significant difference at 5% significance level (t-value ¼ 8.118 and p-value < .05) in the mean score. The mean score of the experimental group is significantly higher than the control group in the post-test. This evidently shows that the experimental group outperformed the control group. We can see that a mean difference of 33.52 is found between the groups in the postPAT (Fig. 1). Figs. 2 and 3 depict the mean scores of the pre- and post-PAT, with confidence intervals for both groups. The width for the confidence interval is less than 2 in the pre-PAT for both groups, that is, 1.14 and 1.87 for the experimental and control groups respectively (Fig. 2), which are approximately equal. However, in the post-PAT, the width for the confidence interval for the control group (16.43) is wider than that of the experimental group (3.63) as seen in Fig. 3. This is due to a higher standard deviation and a lower mean score recorded by the control group. Based on the levels of achievements in mathematics as mentioned earlier, an analysis was also conducted to examine the students’ probability achievement. Table 2 shows the total number of students in the different levels of achievements in mathematics for both groups. There is no obvious difference as the number of high, average and low achievers are almost similar in both groups. Table 3 shows the results of the different levels of achievers in both groups for the pre- and post-PAT. As we can see, there are no significant differences in the students’ achievement in probability between the two groups for all levels before the study, p > .05. The mean scores of the pre-PAT of the high achievers (M ¼ 2.59, SD ¼ 2.163), average achievers (M ¼ 1.14, SD ¼ .876) and low achievers (M ¼ 2.14, SD ¼ 2.470) in the experimental group are not significantly different from the control group (M ¼ 3.75, SD ¼ 2.993; M ¼ 2.25, SD ¼ 2.024 and M ¼ 2.14, SD ¼ 2.249, respectively). However, a marked difference is seen in the post-test, that is, there are significant differences in mean scores between the groups for all levels, p < .05. The results show that the mean scores of the high achievers (M ¼ 77.94, SD ¼ 1.589), average achievers (M ¼ 75.45, SD ¼ 5.395) and low achievers (M ¼ 71.64, SD ¼ 6.785) in the experimental group are significantly higher than the control group (M ¼ 57.48, SD ¼ 13.358; M ¼ 31.29, SD ¼ 24.921 and M ¼ 22.80, SD ¼ 14.933, respectively). The highest mean difference between the groups is recorded by the low achievers (48.84), followed by the average achievers (44.17) and high achievers (20.43). Figures 4, 5 and 6 display the graphical representation of the mean scores, with confidence intervals for the high, average and low achievers in both groups for the pre- and post-PAT. Similar to the pre-PAT, the high, average and low achievers for both groups recorded a small size of confidence intervals and are approximately equal. The width of the confidence interval for the high achievers in the experimental group is 2.5 and the control group is 3.18 (Fig. 4). The width of the confidence intervals for the average achievers in the experimental and control groups are 1.17 and 2.9 respectively (Fig. 5). For the low achievers, the width of the confidence interval is 4.29 for the experimental group and 4.16 for the control group (Fig. 6). In the post-PAT, the confidence intervals for all 3 levels of achievers in the control group are wider than their counterparts in the experimental group. The width of the confidence intervals for the high achievers in the experimental and control groups are 1.83 and 14.24 respectively (Fig. 4). For the average achievers in the control group, the width of the confidence interval is the widest among all the levels of achievers in the control group, that is 35.65, and is also wider than the average achievers in the experimental group, 7.25 (Fig. 5). The width of the confidence interval for the low achievers in the experimental group of 12.55 is narrower than that of the control group, 27.62 (Fig. 6). This implies that the standard deviations are higher and the mean scores are lower for the control group than the experimental group for all the 3 levels.

80 75.71

70

MD:

PAT score

60 33.52

50

control group

40 42.19

30 20 10 0

experimental group

Control: 2.95

Pre

Expected results: 1. Pre: no significant different 2. Post: significant different

Experimental: 1.99

Post

Fig. 1. PAT scores before and after intervention.

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4.0

95% CI Pre-PAT

3.5 3.0 2.5 2.0 1.5 1.0 experimental group

control group

Group Fig. 2. Error bar chart for pre-PAT.

This study evidently concurs with previous studies conducted by other researchers including Brooks-Young (2009) and Nasari (2008) on the increased performance. As mentioned earlier, an analysis of the students’ comments made in their journals were categorised based on emerging themes. Their comments, cited verbatim will provide a more lucid picture of their experiences. Pseudonyms are used to cite the students’ comments. Additional comments cited can be seen in Appendix 1. 4.1. User-friendly It is not surprising that majority of students had found the GC easy to operate. The ‘friendly’ interface with no complicating commands allowed students to key in the data and explore functions of the GC with ease. For instance, Jaya noted in his journal that “the GC has a friendly graphical interface that allows us to key in data easily and retrieve answer precisely, which increase my understanding” and Lew recorded that “the GC makes me understand probability. No complicated commands involved when I enter the data into the GC.” 4.2. Crunching machine The GC is perceived as a crunching machine as it saved students the tedious calculations, sped up as well as made the calculations easier. The crunching machine helped minimize careless mistakes students tend to make during the computation process. This is evident in David’s comment: “the GC saves a lot of time and less room to make careless mistakes . it enables me to calculate better. I can re-confirm my answer with the GC and my friends.” In addition, students found that they were able to use the ‘saved’ time to crunch more problems, thus giving them the opportunity to do more practices. With the additional practices, students found that it helped in better understanding of the subject. This is evident in Shuba’s comment: “I can solve the problems faster. My understanding has increased because I am now having more time to solve more exercises. The GC saved many manual calculation steps and time.” 4.3. Active participation The incorporation of GCs had provided opportunities for active participation in class activities. Students constantly interacted with each other, with the GC as well as with their instructor. They discussed the questions and solutions, and compared as well as confirmed their answers with their peers and with the GC. For instance, Lily noted that “we discuss the questions thoroughly and try to figure out the concepts behind the questions. We interacted with GC and friends to get the answers” while Fahim noted that “[the] GC made me talked a lot. I talked with my friends the GC method to solve problems. We discuss and compare results .” The more capable ones were also found

95% CI Post-PAT

80 70 60 50 40 30 experimental group

control group

Group Fig. 3. Error bar chart for post-PAT.

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Table 2 Frequency and Percentage for different level of mathematics background. Group

High achievers

Average achievers

Low achievers

Total

Experimental Control Total

14 (43.8%) 16 (48.5%) 30

11 (34.4%) 10 (30.3%) 21

7 (21.8%) 7 (21.2%) 14

32 33 65

to scaffold their peers within the ZPD. The scaffolding of each others’ learning process had helped students grasp the concepts, solve probability problems and gradually solve the problems independently. Some of them noted that they were able to solve the problems independently eventually after interacting with or being scaffold by their peers, instructor and GC. John succinctly wrote in his journal that “I have more opportunities to discuss when carrying out GC activities. We help each others to understand the concepts and questions as well as to solve the problem. Sometimes we seek help from the lecturer. Finally, I find that I am able to solve the problems independently.” 4.4. Thinking tool Students noted that they were able to interpret and analyse probability problems using different methods with the GC. They were more critical and able to provide explanations on how they made decisions on the different method of solving probability problems as well as how they derived the solutions. The saved time gave students the opportunity to explore the GC, which consequently enabled them to think of the best possible method of solving probability problems. This is evident in Peter’s and Mariam’s entries respectively: “I am more critical now after carrying out the GC activities through communication with my friends” and “since the GC made the calculation faster, I have more time to try different methods using GC to solve the probability questions.” 4.5. ‘Visualizer’ and ‘graphic artist’ Majority of the students found that the graphical representation available in the GC enables them to ‘visualize’ the probability. They were able to “see the probability” as well as “draw graphs and tables”. This is evident in Halim’s comment that “the GC shows the shape and shaded areas of the graph” and Tan’s comment that “the graphing calculator shows us the normal curve and probability. I see the ‘probability.” The graphical representations of the data enabled them to understand the relationships between the value of a random variable and its probability. The GC had evidently substituted the statistical tables for the probability distributions as noted by Chin, “the GC is an alternative way to get the answers rather than using the statistical tables. 1 now prefer to use the GC.” Entries by the control group (Appendix 1), however, presented a picture of tedious computation, less interesting and passive participation, despite constant encouragement to interact was made: 4.6. Tedious computation The students in the control group found that the manual calculation, using the statistical tables involved many formulas and laborious steps before they could solve the probability problems, hence the possibility of making careless mistakes was greater. Some of them highlighted that they had to go through the process of ‘try and error’ when solving the problems and yet not able to obtain the solutions to the problems. This is recorded by Steve that “working with mathematics takes a lot of time because we need to do it by try and error until we get the correct answer with the long steps and workings. I tend to make careless mistakes. So I have to check and check many times. . Take up a lot of time.” Chris too noted in his journal that “sometimes I am not sure which page of the statistical Table 1 should refer to. It is very tedious . Also, there are many tedious formulas and steps involved in solving the problem. I spent a lot of time to solve the problems, but sometime I still can’t get the correct answer.” Hence, it is not surprising that most of the students found the subject difficult, as seen in Susan’s comment: “Probability is difficult, very abstract . I don’t know how to start with the workings . too many formula and difficult calculations.” Table 3 Pre-PAT and Post-PAT between the groups based on the mathematics background. Mathematics performance level

Group

Mean

SD

MD

95% confidence interval of the difference Lower

Pre-PAT High Average Low Post-PAT High Average Low

Experimental Control Experimental Control Experimental Control

2.59 3.75 1.14 2.25 2.14 2.14

2.163 2.993 .876 2.024 2.470 2.249

Experimental Control Experimental Control Experimental Control

77.94 57.48 75.45 31.29 71.64 22.80

1.589 13.358 5.395 24.921 6.785 14.933

SD ¼ standard deviation, MD ¼ mean difference.

Upper

1.16

3.140

.819

1.11

2.622

.395

.00

2.751

2.751

20.43

13.305

27.616

44.17

26.178

62.153

48.84

34.657

63.027

C.-K. Tan / Computers & Education 58 (2012) 1117–1126

80

1123

Pre-PAT Post-PAT

95% CI

60

40

20

0 control group

experimental group

Group Fig. 4. Error bar chart for PAT of the high achievers.

4.7. Less interesting and passive participation Majority of the students found the learning of probability as less interesting. This could be due to the fact that majority of the students were passive in class. Despite constant coaxing and encouragement to discuss with peers, students barely interacted with each other or even with their instructor. Comments highlighting that there was less interaction in class and that most of them preferred to solve the problems on their own rather than discuss or communicate with their peers were found in many of the students’ journal, including Minah and Seow: “I don’t like learning probability because it is not interesting. Most of the students in the class prefer to solve the problems themselves and they do not want to discuss. Boring, the class is quiet and students are passive even though my lecturer keep asking us to interact” (Minah) and “we seldom communicate and discuss the problems. My lecturer always asks us to discuss among ourselves to solve the problems, but most of us try to solve the problems ourselves without having the discussion. We do not communicate with each other” (Seow). 5. Discussion Findings of this study provide evidence that the use of GCs improved students’ performance in probability. Similar to the previous research findings including Harskamp et al. (2000) and Van Streun, Harskamp, and Suhre (2000), this study concludes that the use of GCs benefits all levels of students (high, average and low achievers). This is evidently seen in the significant performance of the experimental group. The greater mean difference between the control and experimental groups clearly indicates that low and average achievers benefited the most. Results of this study also indicate that the GC instruction approach is an alternative to the conventional approach, especially in probability that involves understanding of concepts and tedious calculations. The outcomes of this study are consistent with the studies by Acelajado (2004a), Burill (2004), Ekthaicharern (2004) and Nor’ain et al. (2008) that highlight the GC as a useful and user-friendly tool in improving students’ probability performance. The students noted that the GC enabled them to perform several types of calculations that cannot be performed by other calculators. It enabled them to go beyond the seemingly impossible calculations and was a ‘crunching machine’, ‘interaction tool’, ‘thinking tool’, as well as ‘visualizer and graphic artist’. The experimental classroom was transformed into

80

Pre-PAT Post-PAT

95% CI

60

40

20

0

experimental group

control group

Group Fig. 5. Error bar chart for PAT of the average achievers.

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80

Pre-PAT Post-PAT

95% CI

60 40 20 0

experimental group

control group

Group Fig. 6. Error bar chart for PAT of the low achievers.

a laboratory of active learning, exploring, experimenting, analysing, discussing, comparing and having fun while learning, a seemingly, difficult subject. As a ‘crunching machine’, it enabled students to solve probability problems faster, lessened the tedious calculations and hastened the process of obtaining answers. It had eventually reduced the need to write out the workings and steps, a feature of the conventional approach, and lessened the possibility of making careless mistakes in the calculations. It also enabled students to draw normal curves within seconds, which consequently enabled them to be more focused and more observant. Thus, the saved time allowed students to solve more probability problems. This had obviously resulted in better understanding of the subject, and consequently improved performance. Students also felt that the lessons with the use of GCs were more interesting and enjoyable as they were more involved and more interactive. They constantly communicating, discussing, comparing and exchanging views on the solutions that they obtained with the use of GCs. Students were obviously interacting with their peers as they helped each other in solving the problems.The active participation and interaction created a student-centred learning environment. The instructor, GC and competent peers scaffold the social and physical interactions in the probability classroom. Through the guidance of the instructor, the collaboration with peers and the use of GCs, students in the experimental group recorded better performance than the control group. Through the scaffolding process, students were gradually able to solve probability problems independently, a scenario where Vygotsky (1978) expounds, what students are able to do in collaboration with more competent peers or teacher now (can do with help), they will be able to do independently in the future (can do alone). The social interaction (student–student; student-instructor) and physical interaction (student-GC) had mediated the development of students’ higher mental functions in probability, especially in the probability problem solving. This had resulted in improved performance in probability. Consistent with Griffith (1998) and Ron (2004), the GC formed a ‘thinking tool’ that enabled students to develop conceptual understanding and mathematical problem-solving abilities and in this study, particularly in the topics of Random Variable, Binomial Distribution, Poisson Distribution and Normal Distribution in probability. Similar to Dick (1992) and Hopkins (1992) findings, it enabled students to calculate the probability of events, which consequently enables them to focus on understanding, setting up and interpreting results. The exploration of problem solving with GCs improved the students’ higher order thinking, which is also consistent with Simonsen and Dick’s (1997) finding. As a ‘visualizer and graphic artist’, especially in the topic of Normal Distribution, it enabled students to visualize the graphs. Students were able to ‘see’ the Normal curves, which consequently enabled them to ‘see’ the ‘probability’ of a Normal variable through the ‘area under the normal curve’, which aided in problem solving skills. This helped students understand the probability concepts better. The positive outcomes of this study provide support for GCs to be used extensively in mathematics, particularly in probability and even in sciences education. As Nor’ain et al. (2008) assert that “it is timely to make widespread the use of GCs” in Malaysia where we “should take advantage of the vast potential of the GC for all students’ beneficial.” With consistent findings, ministries of education ought to consider exploring the possibility of incorporating GCs at all levels of education. Interesting and meaningful GC instructional activities ought to be developed to enhance classroom interactions and to increase interest to learn probability. Educators, especially probability instructors ought to be more creative in designing instructional activities and strategies in the teaching and learning of probability by incorporating GCs into the lessons. The possibility of granting the use of GCs in examinations ought to be considered too especially in countries that have not considered its use during examinations. The use of GCs during examination eliminate the tedious process in calculations and enable higher thinking skills’ questions, such as questions on formulations of mathematical models and interpretation of answers to be tested. 6. Conclusion Given the advancement of handheld technology and the increasing use of GCs, better understanding and implementation of effective GC approach would certainly enhance its use and educational value of such educational technology in the mathematics world. With the obvious increase performance and meaningful learning found in this study, it clearly displays the fact that the GC is worth considering as a teaching and learning tool. More extensive research ought to be conducted, especially in probability subject at the institutions of higher learning to draw a more conclusive result. It can be extended to include other subject domains and in other educational settings to see its wider effects. Acknowledgements We would like to thank StatWorks (M) Sdn Bhd for the GC loan and the Foundation.Center for the permission of conducting this study to foundation students.

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Appendix 1

Experimental group Themes

Students’ comments

User-friendly

 “The graphing calculator is user-friendly. We can just key in the data in the distribution table and get the answer through formula. Therefore, I understand the lesson.”  “It has a nice interface where we can see the probability clearly. No difficult commands when entering the data. Easy.”  “It has provided a user-friendly interface with interactive design.”

Crunching machine

 “Learning becomes easy with graphing calculator, will take a lot of time using a normal one”  “It makes the calculations easier and I get to know binomial probability distribution function better.”  “I understand the lesson because it makes the calculation simpler, saves a lot of time.”  “It fastens the calculation process and has many functions and methods to solve the problems . the normal calculator cannot do.”  “It lessens the manual calculation . also minimizes the chances of making careless mistakes.”  “We didn’t need to calculate manually. The GC saves our time . It makes things easier and improves our understanding of the topic. I checked my answer with my friends’ answer.”

Active participation

 “I enjoy the lesson and interact with my group members.”  “GC allowed us to have more discussions. We discuss together to make sure everyone understand the activity and get the correct answer.”  “After the discussion for the GC activities, we are able to solve the activity on our own as we have already understood the concept and are able to use the graphing calculator without guidance.”  “Understanding of the subject has improved. I understand the topic slightly better after discussing with friends and interacting with GC.”  “We discuss together on how to use the mode of calculator to solve the problem.”  “We brainstorm and discuss with each others. . also asked help from lecturer, confirmed answers with GC.”  “Yes (he was able to solve the problem independently), I have identified the method to obtain a solution. . I am confident because I understand the method of solution. This is because I have more chances to discuss and interact with friends when I use GC.”

Thinking tool

 “I feel my performance in probability is better than other subjects. I am now able to interpret the questions and answer that I got. I think with GC, my problem solving skills had improved.”  “During the activities, we exchange views and discuss on various methods to solve the problem. After listening to everyone, we choose the best method to solve the problem.”  “It (GC) helps because it is more practical as we carry out the experiment we understand the concept more. This is because to use the GC to calculate, we need to understand the concept thoroughly. I have to understand the concept before using GC or while using the GC.”  “I was able to explain and justify the solutions to my friends. We also exchanged ideas.”  “I explored various functions of the GC and also various methods to solve the problems.”  “We write down the explanation on the worksheet on how to solve the problems. We also interpret the probability values after obtaining them from the GC.”

‘Visualizer’ and ‘graphic artist’  “It shows me the pattern of binomial distribution.”  “The graphing calculator shows the basic concept of normal distribution.”  “We can create the table in the graphing calculator by just key in the data in the distribution table and get the answer through formula.”  “It shows me the area of the normal curve. Now I understand that it is the probability.”  “We have chance to draw the graphs using graphing calculator. The convenience of drawing graph helps me to understand better.”  “I can see the area under the normal curve. I understand that it is the probability for normal distribution.”  “I use the list function of the GC to get a table in order to solve the problems for random variables.” Control group Themes Tedious computation

Students’ comments  “The calculation is very long . got mistakes if I do fast . I can understand better only after do lots of revision.”  “. not as what I expect . I am good in mathematics but hate long calculations, it is time-consuming. I know to solve the question because I do lots of exercise and study myself after the classes .”  “For Binomial, Poisson and Normal Distributions, we use the statistical tables most of the time. It is very time consuming.”  “How to change the x-value to z-value for Normal Distribution??? . headache if ask me to memorize the formula for it .”  “Ah, I don’t like to solve the problems because I hate the long calculations and many steps to go through. Need to spend more time on computations. Sometimes, I get the wrong answers.”

Less interesting and passive participation

 “I feel bored solving this .”  “I feel that learning probability is very “sian” (boring). friends are also quiet . they do their own work. “sian”(boring). However, I can solve the questions. I study on my own after the class.”  “I don’t know how to solve, everyone does their own work . I don’t know them also . I have no choice, but to ask my lecturer.”  “. bored with this class, don’t “see” the concept of probability, they (peers) know a little bit even though, I discuss with them.”  “. boring, . a lot of formula, cannot remember, . cannot get answer after calculating for so long. sometimes give up . don’t know if my answer is correct or not .”  “. my friends also don’t know how to solve, no need to ask them .”  “. we have less discussion and interaction .”  “I don’t want to discuss with my friends. I don’t know to do my exercises, I don’t understand, but I do not discuss.”  “I don’t like long calculations, so no need to discuss. I prefer to wait for the answers.

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