Academic journal article
*The Journal of Developing Areas*

# Teaching Probability with Graphic Calculator Instructional Approach

## Article excerpt

ABSTRACT

Positive effects have been reported when graphic calculator (GC) is used in Mathematics classrooms as it influences the way Mathematics is taught and learned. This study aims to examine the effects of GC instructional approach on students' Probability performance, particularly for students with different levels of Mathematics performance. Two groups of students were involved in this study; Group A adopted the GC instructional approach while group B used the conventional teaching approach. Students from different Mathematics performance levels, particularly the low-performing students, gained benefits when the GC instructional approach was adopted in learning Probability.

JEL Classifications: I20, I21, I22

Keywords: Probability Learning; Graphic Calculator Instructional Approach; Mathematics Performance

INTRODUCTION

Most students view Mathematics as a subject with many rules to memorise for them to solve mathematical problems, and the subject requires tedious, complex and boring calculations (Waits & Demana, 1999). Educators and researchers alike have found that teaching and learning Mathematics is becoming an increasingly challenging task, and thus, they are searching and trying out new pedagogies to improve the effectiveness of their teaching, particularly in approaching students with different abilities and levels. Some studies have also found that the adoption of a teaching approach, particularly one that involves information and communication technology (ICT), does leave an impact to students' learning because ICT makes learning Mathematics a richer and more experimental experience as students learn new content and pay greater attention to the processes that are not possible to be shown/explored without ICT (Heugl, 2004; Mkomange, Chukwuekezie, Zergani, & Ajagbe, 2013). Educators have adopted/incorporated educational technologies such as LEGO, Foldit, Mathway and Mathcad into their classes with the aim to improve and enhance students' understanding of complex Mathematical concepts. Moreover, technology has helped to enhance interactions among the students and their teachers. It also helps students to be independent learners (Mkomange et al., 2013). The adoption and integration of technologies into Mathematics classrooms include the use of graphic calculators (GC). Researchers have found some positive effects that could be gained from the adoption and integration of GC into the Mathematics curriculum, namely students' improved performance in algebra, geometry, trigonometry and calculus (Arnold, 2008; Nor'ain, Rohani, Wan Zah, & Mohd. Majid, 2011; Ocak, 2008; Spinato, 2011). However, most of the studies that looked at the use of GCs were conducted in developed countries and focused mostly on the effectiveness of its adoption in general instead of examining its effects on students with different levels of achievement. Moreover, limited research has been done on its adoption and integration in developing countries such as Malaysia, particularly Probability in its institutes of higher learning. The adoption and integration of GCs into the Malaysian curriculum and classrooms have been found to be limited too. This may be due to the fact that GC is a relatively new educational technology in Malaysian Mathematics education and its adoption is at its infant stage (Nor'ain et al., 2011).

The objective of this research is to examine the effects of the adoption of GC instructional approach on students' Probability performance, particularly students of different achievement levels.

LITERATURE REVIEW

One of the main issues in Mathematics education in many different nations is the unsatisfactory achievements. It is evident in some countries; for example, in the United States, students' performance in Mathematics has been recorded to be below average (Fields, 2005); in Hong Kong, most students were found to be weak in plotting and reading graphs (Leong, 2006), and in Chile, the Mathematics achievement in TIMSS1999 was very low (Ramirez, 2005). Similarly, Malaysia is facing the same problem. Malaysia's ranking in Mathematics in TIMSS 2011 (TIMSS 2011 International Results in Mathematics, 2013) dropped from the 20th spot in 2007 to 26th spot in 2011, and the average score fell from 474 in 2007 to 440 in 2011. Students perceive Mathematics as the most difficult subject to learn, particularly Probability, and they lack the confidence to solve mathematical questions (Effandi & Normah, 2009). Okello (2010) notes that students have the mind set that Mathematics involves complex formulae which are difficult to grasp, and this causes them to perform poorly in the subject. In addition, many are unable to relate their Mathematics knowledge with real life scenario, resulting in students not being able to understand the practicality and importance of Mathematics in their lives. They are unable to "visualise" the abstract concepts and mathematical symbols that represent the real world, thus, they tend to just memorise the formulae and facts, and thereafter, encounter difficulties in applying the knowledge in solving unfamiliar questions.

Previous studies that have incorporated GCs into Mathematics lessons yielded positive results, namely students developed better understanding (Michelle, 2013; Serhan, 2006; Tan, 2012) and appreciation of the concepts of functions and graphs, algebra, statistics, applied calculus, and calculus (Graham & Thomas, 2000; Nasari, 2008; Nor'ain et al., 2011; Ocak, 2008; Serhan, 2006). GCs provided students with the opportunities to communicate and be actively involved in the classrooms, and could foster better interactions and communication between the students and teachers (Nik Rafidah, Zarita, & Safian, 2008; Noraini, 2004; Tan, Madhubala, & Lau, 2011a; Tan, Tan, Siti Fatimah, & Lew, 2013). GC eases students' learning as students no longer need to memorise formulae, and it reduces computational time needed to perform and solve complex mathematical tasks. It removes complicated and manipulative procedures and is able to provide accurate answers within seconds (Daire, 2010; Mohd Ayub, Ahmad Tarmizi, Abu Bakar, & Mohd Yunus, 2008; Tan, Madhubala, & Lau, 2011b). The time saved allows students to have more time to tackle more problems which in turn helps them develop better understanding of mathematical concepts.

Ong (2004) and Brooks-Young (2009) supported the adoption of GC as its functions could be effectively used to explain mathematical concepts to students and help them to focus on the concepts which could not be effectively done using the traditional methods. It enables the students to employ more sophisticated strategies to solve problems, particularly problems that cannot be easily solved using normal algebraic methods or normal calculators (Sundram, 2008). The ability of GC to expose the students to the various levels of differentiated instructions to understand mathematical concepts allows them to analyse problems in a flexible way and increase their motivation in learning (Ha, 2008). Students' ability in decision making could also improve with the use of GC as students must decide on the information that has to be entered into the GC, choose the operations or functions to be used and interpret the answers generated from the GC (Nor'ain et al., 2011).

The GC instructional approach is suitable to be employed to help students of various levels of abilities. Acelajado's (2004a, 2004b) studies reported a significant difference in the mean between the pre and post-test in the achievements of the different ability groups, i.e. an improved achievement within the high, average and low ability groups, respectively. Studies by Harskamp, Suhre and Van Streun (2000) and Van Streun, Harskamp and Suhre (2000) found that low-performing students achieved higher achievements. Students with learning disabilities also gained benefits from using the GC in their learning. Similar trend is also evident from Bouck's (2009) study in which the GC was used by two groups of students with and without learning disabilities. While students without disabilities showed a significantly improved performance in their posttest results, positive outcomes were also seen among students with disabilities. In addition, it is evident in Nor'ain et al.'s (2011) study, the GC approach was able to increase both average and low ability students' performance and induce better levels of their meta-cognitive awareness with less mental effort invested during the learning and testing of Mathematics.

METHODOLOGY

Participants and Instructional Approach

The participants were students whose age ranged between 17 to 21 years old, and all of them were pursuing their Foundation programme in a private university in Malaysia. A quasi-experimental study with a non-equivalent control group design was adopted. Two groups were involved in this study, i.e. group A (32 students) and group B (33 students). Both groups were taught using the same Probability syllabus that covered the four chosen topics: Random Variable (RV), Binomial Distribution (BD), Poisson Distribution (PD) and Normal Distribution (ND). The categorisation of the students into different levels of Mathematics achievement was based on the students' Mathematics grades in the previous final examination. Students who obtained grades A+, A or A- were categorised as high- performing students (HP), while the average-performing students (AP) were those who had obtained B+, B or B-, and the low-performing students (LP) were those who had obtained C+, C and F.

The GC instructional approach was employed in the teaching and learning for group A. It is a teaching and learning approach where the GC is used as a teaching and learning tool and GC instructional worksheets as modular lessons. The GC instructional worksheets were designed for the four chosen topics, in line with the syllabus. In contrast, group B adopted the conventional teaching approach, with the use of a textbook as a teaching and learning tool. Students used scientific calculators and paper-and-pencil in solving problems. The same instructor was used throughout this study in order to ensure consistency.

Research Alternative Hypotheses

There are three alternative hypotheses:

H1: There is significant difference between the LP in group A and group B in the students' overall Probability learning performance.

H2: There is significant difference between the AP in group A and group B in the students' overall Probability learning performance.

H3: There is significant difference between the HP in group A and group B in the students' overall Probability learning performance.

Instrument

To measure students' Probability achievement before and after the study, a self-designed test called the Probability Achievement Test (PAT) was designed. PAT contains four problem solving questions on the four chosen topics. Its objective was to measure students' knowledge and problem solving ability on RV, BD, PD and ND . Twenty marks were allocated for each question. A panel comprising five experts with 15 years of experience teaching Probability at the university validated PAT for its relevance and concordance with the syllabus. The acceptable reliability of Cronbach's coefficients of.72 was produced from the filed study. In order to attain the comparative data, a test-retest approach was employed in this study.

Procedures

The research was conducted during a 14-week trimester. In Week 1, the pre-PAT was administered to both groups A and B while GCs workshops were conducted only on group A students with the aim to provide them the opportunity to learn and master the important functions and commands of the GCs which would be essential for the four chosen topics for this study. Altogether four GCs workshops were conducted in weeks 2 and 3. The duration for each workshop was for an hour. The GC intervention period was from week 4 to week 12. During this period, the students in group A were taught with the GC instructional approach, while students in group B were taught with the conventional teaching approach. Each lesson for both groups began with the discussion of the theories and examples for about 15 minutes, followed by GC-aided instructional activities to group A and paper-and-pencil problems solving activities for group B for about 100 minutes, and finally, the lesson ended with a five- minute conclusion. The instructor, as a facilitator, guided and facilitated students' learning throughout all the lessons particularly during the problem solving stage. In addition, when necessary, the instructor encouraged the students in both groups to interact, discuss and exchange ideas actively during the activity. All the students in both groups were required to keep a journal to record their experience. After the GC intervention period, both groups A and B were again given the PAT, which served as the post-PAT. By using SPSS, the data were analysed for descriptive statistics and a t-test at 5% significance level was conducted. Qualitative data collected from students' journals were analysed with the help of a group of panelists of three experts who have 15 years of experience teaching Probability at the university.

FINDINGS

In this section, an overall picture for pre-PAT and post-PAT is presented first, followed by the results for the research hypotheses. At the beginning of the study, the students were unfamiliar with the subject, and thus they were unable to answer the questions asked in PAT, particularly BD, PD and ND. Therefore, the pre-PAT's t-value could only be generated for RV since all the values of average and standard deviation for BD, PD and ND were zeros for both groups A and B. The t-value and p-value for RV were -1.678 and .099, respectively; indicating that there was no significant difference in achievement between the two groups. In other words, group B's mean score of 2.95 ( SD = 2.63) was not significantly higher than that of group A (M = 1.99, SD = 1.95).

Figure 1 shows the overall results for all the topics in post-PAT for the students in both groups. The mean scores of all topics for group A were higher than that of group B. Group A recorded the highest mean score in PD, followed by ND, BD and RV, while group B had the highest mean score in RV, followed by PD, BD and ND. The t-test results confirmed that group A performed better than group B in all topics, i.e. group A's mean values in RV (SD = 1.94), PD (SD = 1.90), BD (SD = 1.24) and ND (SD = 1.62) were significantly higher than that of group B (SD for RV, PD, BD and ND were 3.74, 7.57, 6.93 and 8.11, respectively), p < .001.

The following paragraphs discuss the findings for the research hypotheses. However, the categorisation of students into various Mathematics performance levels that has been analysed will be presented first and the frequencies of each category in groups A and B are shown in Table 1. Both groups have equal number of LP, but group A has one more AP than group B and two fewer HP than group B.

Findings for H1

Table 2 shows that the mean values of all topics in post-PAT for the LP in group A were higher than the mean values for the LP in group B. Group A recorded the highest mean score in PD, followed by BD, ND and RV, while group B had the highest mean score in RV, followed by PD, BD and ND. The LP of group A scored significantly higher mean values in all topics in post-PAT (SD for RV, PD, BD and ND were 2.66, 1.03, 1.94 and 2.14, respectively) compared to those LP in group B (SD for RV, PD, BD and ND were 5.18, 5.42, 5.94 and 1.44, respectively). The greatest mean difference between the LP of two groups was recorded for ND (16.31), followed by BD (13.90), PD (12.21) and RV (6.43). In other words, among the four topics, LP performed significantly better than their counterparts particularly in ND, followed by BD, PD and RV, when the GC instructional approach was employed.

Findings for H2

Table 2 shows that the mean values of all topics in post-PAT for the AP in group A were higher than the mean values for the AP in group B. The highest mean score recorded by the AP in group A was found in ND, followed by BD, PD and RV. On the other hand, group B had the highest mean score in RV, followed by PD, ND and BD. Group A (SD = 1.86, 3.05, .95 and 1.25 for RV, PD, BD and ND, respectively) performed significantly better than group B (SD = 3.01, 8.95, 7.24 and 7.75, for RV, PD, BD and ND, respectively) in all the four topics after the study, p < .05. Furthermore, similarly with the LP, the greatest mean difference between the AP of two groups was found in ND (13.55), followed by BD (13.28), PD (10.22) and RV (7.11). That is, among the four topics, the AP performed significantly better than their counterparts, particularly in ND, followed by BD, PD and RV when the GC instructional approach was employed.

Findings for H3

Table 2 also displays the mean values of RV, PD, BD and ND for the HP in both groups A and B after the study. The HP in group A achieved almost maximum marks for the mean values in all topics except BD, with the highest mean value recorded in PD ( SD = .49), followed by ND ( SD = .48), RV (SD = .90) and BD ( SD = .90). On the contrary, group B had the highest mean score in PD (SD = 5.43), followed by RV (SD = 2.35), ND (SD = 6.23) and BD (SD = 4.65). The HP in group A recorded higher mean values than group B, in which the t-test results confirmed that they performed significantly better than group B in all the four topics, p < .05. The mean differences between the HP in group A and group B in all the four topics were not as large as those for the LP and AP. They were in the range of 4.13 to 5.88. BD (5.88) had the greatest mean difference, followed by ND (5.76), RV (4.69) and PD (4.13).

In summary, for all the research hypotheses, the t-test results showed that there were significant differences between the LP (AP and HP) in group A and group B in the students' overall performance. All three levels, i.e. LP, AP and HP, recorded larger mean differences in BD and ND, i.e. greater than 5 for both topics for the HP, greater than 13 for both topics for the AP, and greater than 13 and 16, respectively for the LP. Also, it was noticed that the mean difference got smaller by the levels, i.e. from LP to HP.

Qualitative Findings

In order to provide a clearer picture on the students' views about the GC instructional approach and the conventional approach, group A and group B students' comments in their journals were analysed and are discussed below. Pseudonyms are used to cite the students' comments.

Group A's Journal Entries

GC instructional approach has facilitated the students' learning particularly in topics of distributions that need the use of statistical tables. As such, it helped to increase students' understanding because students could use the built-in functions of GC such as binomcdf and binompdf to solve distributions problems without referring to the statistical tables. Students in group A used the saved time to carry out many tasks. For instance, Mutu cited that "... I use many functions of GC to solve problems. My understanding in Probability improves as it helps me to understand the concept..." and Siti commented that "... I found the GC functions that I used to solve Poisson distribution problems same as using statistical tables to solve them. I am able to solve the problems with GC and statistical tables. Now, I understand the concept and know how to solve the problems using the statistical tables." In addition, the students could tabulate the data and draw the graphs in the GC with its 'LIST' function and graph function. The graphical representation allowed students to observe and visualise the pattern of distributions particularly for the normal distribution. As such, students' understanding improved. Guang noted that "... GC shows me the normal curve and the shaded area as the probability. It made me understand better on ND..."

Moreover, students found that GCs simplified the tedious and complex calculations needed during problem solving. Chin responded that "... solving probability problems with the GC involves entering only a few data into it and it does all the calculations for me. Therefore, I could practise more exercises. It gets rid of the complicated calculations that I am scared of." Cindy cited that "I can get the solution in few seconds, it is quite interesting. Now, I am confident to solve difficult questions." Students also found that the GC increased their confidence level especially in solving difficult questions as it was easier to understand the concepts, and reduced the manual calculations and time.

The joy of learning Probability with the GC instructional approach was also recorded in the students' journals. Students actively interacted with GCs, peers and lecturer. They actively participated and discussed the methods of using GC's multifunctions, ideas and steps of getting answers; hence, their understanding improved. Low-performing students found that after learning with their peers, they have not only improved in their understanding, but have also become more confident to solve the problems independently and teach others as well. Swee stated that "Learning probability with GCs is fun and I enjoy it very much", Henry cited that "... after discussing with my friends and lecturer and using GCs, I found that I like this subject. I am also happy to know more friends by interacting with them" and Minah, who was a low-performing student, responded that "I could grasp the Probability concepts and solve the problems independently. I am glad to use the GC as I am now able to teach my friend when she could not get the solution."

Group B's Journal Entries

On the other hand, comments by group B showed that the conventional approach was uninteresting, and the manual calculations were complex and tedious. The majority of the students, particularly the LP, were less interested towards learning Probability. Keng noted that "I didn't like to solve the problem due to long calculations. It is very boring too." As such, they adopted the 'wait and see' approach, i.e. to wait for the answers from others as they admitted that the complicated calculations caused them to get the wrong answers most of the time. Students also felt that they spent more time in solving problems using the paper-and-pencil method. Therefore, they could not practise more exercises in a day to attempt more difficult questions. This could be seen in Joe's journal that "Too much time was used to solve the questions. Moreover, the subject involves many formulas which are difficult to memorise, sometimes I am lazy to solve the problems till I give up to solve them" and Jojo recorded that "We spent a lot of time to write the formula and substitute the figures in the formula. A lot of time was also spent on reading the probability values from the statistical tables. Thus, we cannot solve many problems." There was less interaction among students especially the LP. Susan cited that "We have less discussion and interaction because I could not solve the problems and I did not discuss with my friends who also could not solve the questions completely. Most of the time, I wait for my lecturers to provide the solutions and answers."

DISCUSSIONS

The findings from this study are consistent with previous research's findings such as Ha (2008), Michelle (2013) and Serhan (2006) that the GC helped students to improve mathematical understanding and hence results in better performance. Furthermore, previous studies have proven that the GC helped students' learning in algebra, statistic, applied calculus, calculus and pre-calculus (Graham & Thomas, 2000; Harskamp et al., 2000; Nasari, 2008; Nor'ain et al., 2011; Ocak, 2008) . However, the results of this study provided extra support that the GC approach can help students' Probability learning in the topics of RV, PD, BD and ND. The LP, AP and HP showed improved performance in Probability which is in harmony with previous studies (Acelajado, 2004a, 2004b; Bouck, 2009; Nor'ain et al., 2011) which ascertained that students of different Mathematics ability levels gain benefits from the GC instructional approach. The findings are consistent with Harskamp, Suhre and Van Streun (2000) and Van Streun, Harskamp and Suhre's (2000) studies, which is the fact that the LP greatly benefited from the GC instructional approach as a greater mean difference between the LP in groups A and B was recorded, particularly for BD and ND. Their minimum scores in PD and ND were greater than the maximum scores for the LP in group B.

The GC instructional learning environment provides students with the opportunities for communication and active involvement in the classrooms, and fosters better interactions and communication between the students and teachers (Nik Rafidah et al., 2008; Noraini, 2004; Tan et al., 2011a; Tan et al., 2013). During the GC hands-on activity sessions, students had active communication and interactions with their peers, and they were willing to help each other to deliberate the steps on how to solve the Probability problems using the GC approach and the paper-and-pencil approach. They also exchanged ideas on the solution and compared the answers between the GCs and the paper-and-pencil approach, which further helped them to increase their understanding. Therefore, students showed significant improvement in Probability. In addition, students showed more interest in using the GC in classrooms as they felt that the Probability learning was less stressful, and they found it easy to relate the theories with their real life situation. They enjoyed learning Probability with the GC instructional approach. This was evident from their journals that learning Probability with the GC was fun, and they, particularly the LP, were more confident in solving Probability problems. Furthermore, these LP students were found to be independent learners after using the GC. Not only were they able to solve the problems without the help of the competent peers and lecturers, but they were also able to help others. In contrast, students who did not use the GCs in learning were passive and had little interaction despite given encouragements. To these students, learning was boring.

Moreover, the results of this study provided evidence that the GC instructional approach eases students' learning. It reduces the tediousness of memorising formulae, computational time, and complicated and manipulative procedures, which have also been supported by the findings of previous studies (Daire, 2010; Mohd Ayub et al., 2008; Tan et al., 2011b). This allows students to have more time to explore more problems which in turn enhances students' understanding of mathematical concepts. The multi-functions of GCs provide students the opportunity to explore various strategies for problem-solving and various levels of differentiated instructions in studying mathematical concepts. The GC is able to help students confirm the mathematical working steps and answers that they would obtain with the paper-and-pencil approach, which eases students' learning and increases their confidence level. The functions such as binompdf and binomcdf allow students to solve binomial problems without referring to the statistical tables or writing the formulae with the paper-and-pencil method. Hence, students are able to go beyond obtaining an answer that cannot be solved by using a normal calculator. The GC is particularly useful for solving PD, BD and ND problems that involve more parameters in the formulae. It is a useful and user-friendly educational technology device that enables students, particularly the LP, to visualise the graph, especially the Normal curves, and this has improved their problem solving skills. This is in line with previous studies (Brooks-Young, 2009; Ha, 2008) that the GC enabled students to understand Mathematics problems in various ways and hence, improve their performance.

IMPLICATIONS AND CONCLUSIONS

The GC instructional approach benefits all students regardless of their previous mathematics achievement levels. This study found that the LP in the conventional classrooms did not perform significantly better in the Probability topics particularly in BD and ND if the conventional approach was used, while the findings provided evidence that the LP who used the GC instructional approach performed significantly better in these two topics than their counterparts. Evidently, the GC instructional approach enabled the LP to achieve higher performance in these two topics. Thus, this approach could be an alternative innovative instructional approach to improve Probability performance in higher learning institutions, and the GC should be widely adopted particularly in Mathematics education, in Malaysian higher learning institutions and in teaching Probability. It is also recommended to lecturers who are teaching Binomial Distribution and Normal Distribution especially to the LP.

The findings of this study show that the GC is a powerful and influential device in solving complicated mathematical problems. The complex procedures and tasks in Mathematics are simplified with the adoption of GC instructional approach; hence this has changed the students' perception to "learning Mathematics easy and fun now". The Ministry of Education should encourage the use of GC in all higher learning institutions. All Mathematics and Science lecturers must be encouraged to provide an interactive learning environment by adopting the GC instructional approach. Probability lecturers should be more creative and innovative in designing lesson plans, activities and strategies in the teaching and learning of Probability with the use of the GC. Moreover, the setting of the examination questions could be enhanced by incorporating the GC. Therefore, during the examination, students should be tested in their ability to create mathematical models using the GC and to interpret the answers from the GC.

In conclusion, it is clearly seen that students showed improvement in their performance, displayed increased confidence level and enjoyed learning as the GC crunched numbers and helping them to have more time to explore more problems. It also helped them think about probability spatially by visualising density curves and provided them more opportunities for interaction, discussion and collaboration. It is therefore recommended that the GC instructional approach is adopted in the teaching-learning process in order to create an interactive and meaningful learning environment and experience.

**[Reference]**

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**[Author Affiliation]**

Choo-Kim Tan

Multimedia University, Malaysia

Choo-Peng Tan

Multimedia University, Malaysia

Corresponding Author's Email: cktan@mmu.edu.my

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