Available online at www.sciencedirect.com
ScienceDirect Procedia - Social and Behavioral Sciences 102 (2013) 204 – 211
6th International Forum on Engineering Education (IFEE 2012)
Enhancing Students’ Understanding in Integral Calculus through the Integration of Maple in Learning Tuan Salwani Awang @ Salleh a,*, Effandi Zakaria b a
Mathematics Unit, Technical Foundation, Universiti Kuala Lumpur Malaysia France Institute, Bandar Baru Bangi, Selangor 43650, Malaysia b Faculty of Education, Universiti Kebangsaan Malaysia, Bangi, Selangor 43000, Malaysia
Abstract Students entering engineering and engineering technology courses at the university are expected to be competent in mathematics. This is due to the fact that mathematics, including calculus is a primary gateway to an engineering and engineering technology careers. However, the issue of under preparedness of students with diversity in mathematics background enrolling these two courses is at an alarming rate. One factor which can be improved to minimize this gap is to enhance the teaching technique of the subject. With the evolution of various forms of technology, a new approach of teaching integral calculus was developed in this study. Accordingly, this paper discusses the process of integrating Maple software in the teaching of the first year integral calculus topic. In addition, this paper also discusses how this approach can help to deepen the engineering technology students’ understanding in integral calculus topic at the university. Two groups of Technical Mathematics 2 with various mathematics backgrounds were randomly chosen to undergo the treatment. As an initial measurement, a pre-test on integral calculus was administered to these students to check their integral calculus background. The highest mark for the test was far below the passing mark, indicates that students involved in this study have a minimum understanding in secondary school integral calculus. These students were exposed to the treatment for five weeks. At the end of the treatment a post-test was administered to see the effect of the treatment on students’ understanding in integral calculus. Students in the experimental group were found to outperform their peers in the control group in both constructs investigated in this study. The effectiveness of the use of the technology in learning this topic may be enhanced further if its prerequisite topics were also exposed to the same strategy. ©2013 2013The The Authors. Published by Elsevier Ltd. © Authors. Published by Elsevier Ltd. Open access under CC BY-NC-ND license. Selectionand/or and/or peer-review under responsibility ZaidiOmar, Omar, Ruhizan Mohammad Yasin, Roszilah Selection peer-review under responsibility of Professorof Dr Mohd Mohd. Zaidi Associate Professor Dr Ruhizan Mohammad Yasin,Hamid, Norngainy Mohd.Dr Tawil, Kamaruzaman Yusoff, Mohamad Rasul Dr Roszilah Hamid, Norngainy Mohd. Tawil, Associate ProfessorSattar Dr Wan Kamal Mujani, Associate Professor Dr Effandi Zakaria. Keywords: Engineering technology; Integral calculus; Maple software; Mathematical understandings.
* Corresponding author. Tel.: +6-013-298-6868 E-mail address:
[email protected]
1877-0428 © 2013 The Authors. Published by Elsevier Ltd. Open access under CC BY-NC-ND license. Selection and/or peer-review under responsibility of Professor Dr Mohd. Zaidi Omar, Associate Professor Dr Ruhizan Mohammad Yasin, Dr Roszilah Hamid, Dr Norngainy Mohd. Tawil, Associate Professor Dr Wan Kamal Mujani, Associate Professor Dr Effandi Zakaria. doi:10.1016/j.sbspro.2013.10.734
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1. Introduction Industrial nations’ economic competitiveness and growth depend on the adequate number of qualified workers entering the science, technology, engineering and mathematics [1]. At the university level, the number of young adults entering these courses particularly in engineering majors acts as an indicator of the national standing including economic growth. At this level, engineering education develops a pathway that students take in order to prepare themselves for engineering career [2]. However, on the other hand, the declining in mathematical preparedness of students enrolling to its related courses including engineering majors is a challenge to universities [3]. Both engineering and engineering technology courses are highly related to secondary school mathematics. Mathematics including calculus is a primary doorway to engineering careers [4]. However the problem in this subject occurred way before students start their engineering journey. At the secondary school level, students’ ability to differentiate and integrate does not reflect the real understanding of these topics [5]. Integration is considered more difficult topic by some students as it is the reverse of differentiation [6]. The difficulties to understand these topics were handled by some teachers at school by introducing integration as a rule of reversing the derivatives, that is as anti-differentiation [5, 7]). In Malaysia, integration is introduced in this manner in its secondary school system where the rules of integration are related to the reverse of differentiation. This “informal calculus” approach is later enhanced to “formal calculus” in their tertiary education level. However, this type of calculus requires the prior understanding in the limit concept which is proven to be complicated to students [8]. The list of difficulties in calculus is extended by the tendency for mathematics lecturers to make a mistake when they design a sequence for calculus instruction at the tertiary level. They tend to simplify a complex mathematical topic by breaking it up to small components. These components are arranged in a logical sequence which offer a meaningful understanding if students are able to relate them to form a global idea. Lecturers perceived this approach as a practical exercise without realizing that students may not be able to link between these components [9]. The most advantageous environment for learning mathematics including calculus is based on the integration of mathematical software such as Maple and Matlab [10]. Previous researches have discussed about various ways that have been implemented in improving students’ calculus learning through the use of educational technology as a tool [11-14] A well-designed instructional approach in using technology for the teaching and learning can produce positive impacts on students’ mathematical understanding. It is crucial for the lecturers to ensure that their students are able to acquire the real understanding while completing any mathematical tasks with technology. One appropriate educational technology known as Computer Algebra System allows students to concentrate on formulation solutions [15]. A Computer Algebra System (CAS) is a mathematical tool that integrates knowledge of many branches of mathematics into one system. CASs are designed to make an easy accessibility of knowledge and an automation of computations. These features help students to offload their computation load to the system [16] without compromising the mathematical understanding. The existence of these systems allow students to engage in the meaningful learning experience as the tedious computations are performed by CASs [17]. Furthermore, students can also develop their understanding through sensori-motor activity and visual imagery available in the systems. Through these low-level cognitive actions, students can comprehend the high-level theoretical concepts [18]. This situation creates an environment in which students construct their own mathematical concepts [19]. Students who are actively involved in entering information into the system are categorized as a constructivist user of technology and able to reflect on what they are studying in new and meaningful ways [20]. Furthermore, CAS offers more opportunity in learning as students are encouraged to not be afraid of doing any mistakes [19]. These mistakes and errors can be advantages for real learning if students are able to figure out the reasons behind the errors [21]. The integration of CAS in the teaching and learning of calculus matches with the nature of engineering technology fields. Engineering technology programs are oriented towards applications which provide students with introductory courses on mathematics and science, and an introduction to engineering fundamentals. These programs aim to have experiential learning as their main educational principle [22]. This principle suggests the
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emphasis of the richness in laboratory practical courses. This facet applies to all subjects in engineering technology programs, including mathematics. Thus, to facilitate calculus learning in this field, this study introduced the engineering technology students with the application of mathematical software, known as Maple in the teaching and learning of the topic. Maple was chosen because it is suitable for a variety of uses including a very difficult integration questions. Furthermore it requires minimum programming as compared to other mathematical software. The objectives of this study are twofold. The first objective of this study is to investigate the effectiveness of the use of Maple software in enhancing students’ integral calculus conceptual understanding. The second objective is to investigate the improvement of students’ procedural understanding in integral calculus through the application of the same mathematical software. The objectives are considered achieved if students who were exposed to this software in the teaching and learning of integral calculus outperform students in the group without this technology.
2. Methodology The research was conducted in January – June 2012 session in a technical university. Two groups of Technical Mathematics 2 with various mathematics backgrounds consisted of 101 students were randomly chosen to be involved in this research. This research employed the quasi-experimental non-equivalent control group design. One group was randomly assigned to be the experimental group and the other group became the control group. Students in the experimental group used Maple in their learning of integral calculus. The intervention is known as Maple as a Learning Tool treatment. On the other hand, students in the control group experienced the Integral Calculus lessons as usual (Figure 1).
Fig. 1. Study design
The two groups were taught by the same lecturer for both lecture and tutorial slots. The materials for the lecture of the experimental group were prepared by the researcher with help from the lecturer involved. The materials include a set of PowerPoint slides presentation with the elements of Maple outputs. The materials for the tutorial slots are laboratory activities prepared by the researcher based on Dubinsky’s APOS theory [23]. For the control group, the materials for both lecture and tutorial were fully prepared by the lecturer involved. The approach used in the experimental group focused on the development of concepts in integral calculus through the use of Maple software. In this study, students were given activities which stimulate their mind to further investigate the concept behind any integral concepts taught in the lecture by using the mathematical software. They worked cooperatively in groups to solve the activities given in order to discover the concepts underlie each integral subtopic. On the other hand, students in the control group were guided with procedure-based instruction. Students were given a step by step guideline to solve any given integral calculus problem.
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The major difference between these two approaches is the technique used during the tutorial classes. Students in the experimental group completed their integral calculus exercises using Maple software, whereas students in the control group used pen and paper approach. Prior to the integral lessons, a pre-test on integral calculus was administered to these students to check their integral calculus background and also to control the differences in the dependent variables. Then, these students were exposed to the treatment for five weeks, which is equivalent to 40 hours (eight hours per week). At the end of the treatment, a post-test was administered to see the effect of the treatment on students’ understanding in this topic. The instrument involved to measure students’ understanding in integral calculus topic is a set of Integral Calculus Test. This instrument was developed by the researcher and has been carefully piloted to ensure its reliability and validity. By using Rasch model, the reliability of the instrument was proven high with item reliability for both constructs (conceptual and procedural understandings) are 0.95 and 0.96 respectively. The person reliability for both constructs was also proven good, which are 0.77 for conceptual understanding and 0.86 for procedural understanding. The separation indices for item and person were also acceptable with the minimum value of 2.0 separations [24]. 3. Findings and Discussion 3.1 Pre-test results The data of both pre-test and post-test were analyzed using a statistical package PASW Statistics 18. The highest mark for the test was far below the passing mark which indicates that students involved in this study have a minimum understanding in secondary school integral calculus (Table 1). This fact is supported by a very high percentage (more than 60%) of the students in both groups scored 0% in both parts. The pre-test results also showed that students in both groups have a better conceptual understanding than procedural understanding. Students in the experimental group scored a slightly higher mean value in conceptual understanding compared to students in the control group. Nevertheless, students in the control group were a little better in procedural understanding. Table 1. Pre-test results for both groups Group
Experimental Group (n=26)
Control Group (n=25)
Maximum
Conceptual Understanding (CU) (Full Marks = 50) 17.50 (35%)
Procedural Understanding (PU) (Full Marks = 50) 9.26 (18.52%)
Minimum
0.00
0.00
Mean
2.79
1.28
Standard Deviation
4.76
2.14
Maximum
15.00 (30%)
7.41(14.82%)
Minimum
0.00
0.00
Mean
1.70
1.56
Interpretation
The marks for both constructs are below the passing mark at this university which is 40%. There are students who have never learnt integral calculus in their previous education stage. The mean is very low because there are 16 students (61.5%) scored 0% in both CU and PU. The spread of marks of the CU part is more that the PU part because the maximum mark in the first part is more than the maximum score in the latter. The marks for both constructs are below the passing mark at this university which is 40%. There are students who have never learnt integral calculus in their previous education stage. The mean is very low because there are 16 students (64%) scored
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Standard Deviation
3.66
2.37
0% in CU and 18 students (72%) scored 0% in PU. The spread of marks of the Conceptual Understanding part is more that the Procedural Understanding part because the maximum mark in the first part is higher than the maximum score in the latter.
The multivariate test was also conducted to determine whether the level of the conceptual and procedural understandings between the two groups is indifferent. The Hotelling’s T2 obtained is 0.023 (F = 0.556, p = 0.577), indicated that there was no significant mean different between both constructs in both groups. The independent t tests were also found to be insignificant (tCU = 0.913, p = 0.366 > 0.05, tPU = -0.432, p = 0.667 > 0.05). The Levene’s Test of equality of error variances was also performed in order to determine whether the two groups are homogeneous in both parts prior to the integral calculus lessons. The results indicated that the assumption was met where both groups were homogeneous in both parts (FCU(1,49) = 1.641, p = 0.206 > 0.05 and FPU(1,49) = 1.237, p = 0.271 > 0.05). Therefore, if there is a significant different in students’ mean performance score after the treatment, the effect can be claimed as solely because of the treatment itself. 3.2 Post-test results The mean values for the experimental group for both constructs were higher than the mean values of the control group (Table 2). Even though students in the control group outperformed students in the experimental group in terms of procedural understanding of integral calculus in the pre-test, but after the treatment, the outcome was reversed. Students in the experimental group were found to be better than students in the control group. However, the differences were analyzed further to confirm whether they are significant or not. Thus multivariate analysis was performed to determine the differences. Table 2. Post-test results for both groups Group Experimental Group (n=26)
Control Group (n=25)
Maximum Minimum Mean Standard Deviation Maximum
Conceptual Understanding (CU) (Full Marks = 50) 47.92 2.08 19.87 11.68 41.67
Procedural Understanding (PU) (Full Marks = 50) 41.94 1.61 15.32 12.73 35.48
Minimum Mean Standard Deviation
0.00 7.08 10.27
0.00 7.23 10.53
Hotelling’s T2 multivariate test was employed to determine if there are any differences in students’ mean performance score in both conceptual and procedural understanding parts after the Maple as a Learning Tool treatment. Table 3 shows the outcomes of the test and its corresponding t tests. Table 3. Multivariate and univariate tests results Variables Maple as a learning tool treatment Conceptual Understanding Procedural Understanding
Hotelling’s T2 0.494
t test
4.144 2.470
Probability, p 0.000 0.000 0.000
Effect size, η2 0.331 0.259 0.111
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The Hotelling’s T2 was found to be significant, showing that there was a difference in students’ mean performance score between those using Maple as a learning tool with those without this technology in their learning. Hence, further statistical analyses were performed to assess which variables were responsible for significant main effects. The analyses done were the corresponding univariate analysis for Hotelling’s T2 which are independent t tests. The independent t test for the conceptual understanding was found to be significant. This implies that there was a difference in students’ performance between the groups of students who used Maple with the students who did not use Maple. Similarly, the independent t test for the procedural understanding also gave a significant mean difference between the two groups. However, statistical significance is not sufficiently useful to be claimed as a sole noteworthiness significance [25]. Therefore, in this case the effect size values of the treatment were also investigated. Table 3 also shows the effect size values for the effectiveness of the strategy in terms of the whole treatment as a total and also in terms of its individual constructs, which are conceptual understanding and procedural understanding. In PAWS Statistics 18 package, these effect size values were measured in terms of eta squared values (η2). The values obtained are 0.331, 0.259 and 0.111 for the whole treatment, conceptual understanding and procedural understanding respectively. The effect size value more than 25% is considered as a large effect on the effectiveness of any teaching and learning strategy as it is equivalent to Cohen’s d = 0.8 [26]. In this study, the strategy as a whole contributed 33.1% to the effectiveness of learning integral calculus. Even though both constructs in integral calculus were significantly improved more in the experimental group than in the control group, but their effects are unequal. By integrating Maple software in the teaching and learning of integral calculus, students’ conceptual understanding in this topic was found to improve significantly higher than their procedural understand. In this study, students’ conceptual understanding in integral calculus was enhanced by 25.9% and their procedural understanding in this topic was increased by 11.1%. 4. Conclusion The integration of Maple activities in integral calculus tutorial classes was found to give positive effects on enhancing engineering technology students’ understanding in this topic. The activities which were designed based on Dubinsky’s APOS theory were found to be able to engage students into a more meaningful learning. The indicators that were used to measure the effectiveness of this strategy are integral calculus conceptual and procedural understandings. The performance of students in the experimental group was found to be significantly better in both understandings than their peers who were not exposed to the mathematical software. In this study, Maple as a learning tool strategy has raised students’ integral calculus performance by 33.1%. Students’ conceptual understanding in integral calculus has improved more than their procedural understanding, where the former has increased by 25.9% and the latter has improved by 11.1%. These values show that the integration of Maple software in the learning of integral calculus has made a significant contribution to the effectiveness of calculus teaching for engineering technological field. However the effect of this strategy on procedural understanding was found to be significant with only a medium effect. One of the contributing factors to this result is possibly because of the short period of the intervention duration. This is due to the fact that this study is only limited to the integration of Maple software in the teaching and learning of integral calculus, which is equivalent to only one third of the total duration in Technical Mathematics 2 teaching and learning hours. Therefore, the improvement of this type of understanding may be enhanced further by implementing this strategy in a wider scope to include not only integral calculus topic but also to include its other prerequisite topics such as function, limit and differential calculus. By implementing this strategy in other topics in calculus, the intervention duration automatically will be lengthened. With the longer intervention time frame, it is hoped that the effects of integrating Maple software in the learning of integral calculus on students’ understanding is larger.
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