integrating computer algebra systems (cas) into ...

INTERNATIONAL JOURNAL Of ACADEMIC RESEARCH

Vol. 3. No. 3. May, 2011, II Part

INTEGRATING COMPUTER ALGEBRA SYSTEMS (CAS) INTO INTEGRAL CALCULUS TEACHING AND LEARNING AT THE UNIVERSITY 1

2

Tuan Salwani Awang Salleh , Effandi Zakaria 1

Mathematics Unit, Universiti Kuala Lumpur Malaysia France Institute, 43650 Bandar Baru Bangi, Selangor, 2 Faculty of Education, Universiti Kebangsaan Malaysia, 43000 Bangi, Selangor (MALAYSIA) E-mails: [email protected], [email protected] ABSTRACT

This paper discusses the need to integrate technology into the teaching and learning of integral calculus at the university level. This issue was raised due to the decline in students’ integral calculus performance in the recent final examination at the university involved. The existing mathematics knowledge gap was identified as the major contributor to this problem. Decline of mathematics performance at the secondary school level and mismatch between students’ learning styles with teaching methods were identified as the factors for the existence of gap in mathematics knowledge at the university level. The literature review has shown that the use of computer algebra systems (CAS) in the teaching and learning of mathematics can act as a catalyst in understanding the abstract concepts in mathematics. Therefore, an innovative change in the teaching and learning of mathematics, particularly in integral calculus need to be implemented. Key words: Integrating technology, Computer Algebra Systems (CAS), Integral calculus 1. INTRODUCTION Mathematics is one of the core subjects in any engineering fields, including engineering technology and science fields. It is also known as the backbone of the success in these fields. In both fields, “mathematical models” are everywhere, “modeling” is a central activity (Kent & Noss, 2000). However, students entering the higher education, particularly in these two fields were found to have insufficient basic mathematics skills and knowledge. This scenario is not only becoming an issue in Malaysia (Ahmad Fauzi Mohd Ayub, Mohad Zin Mokhtar, Wong Su Luan, & Rohani Ahmad Tarmizi, 2010), but it is also becoming the worldwide phenomenon in adding up burden to mathematics educators. One of the biggest challenges in teaching mathematics skills based at the university level is the under preparedness of students enrolling in the mathematics related fields (Varsavsky, 2010). What will happen next are very predictable, low levels of success and engagement with university level mathematics. The increasingly weaker mathematics background of university students and its consequences have been reported around the world (Varsavsky, 2010). Apart from under preparedness and lack of strong basic mathematics skills and knowledge, another reason why the high rate of declining in students’ mathematics performance in Malaysia is that, they consider mathematics as a difficult and tedious subject (Effandi Zakaria, Lu Chung Chin, & Md. Yusoff Daud, 2010). Furthermore, a common method of delivering information to all students is normally applied in the university. Whereas, researches on students’ learning style have shown that every student has a different preference of learning style (Felder, 1988). At the university involved in this study, mathematics is not a specialization offered as a major course. However, mathematics is one of the compulsory subjects for graduation and it is used as a tool for the engineering technology courses. Since this subject is not a major specialization, the time allocated for this subject is very limited. Therefore, students have a very little time to study mathematics because they have to concentrate on the core technical subjects. Traditional method of teaching mathematics was found to be ineffective for all topics due to the poor performance in certain topics of mathematics, such as calculus integral. Thus, an innovation in teaching and learning of mathematics should be considered to bridge the gap of knowledge in mathematics to ensure the st quality of future engineers and scientists in the 21 century. 2. ENGINEERING TECHNOLOGY According to Kamsiah and Marlia (2008), with regards to institutional output of engineering graduates, Accreditation Board for Engineering and Technology (ABET) has highlighted that graduates of Bachelor of Engineering Technology should be able to:  Demonstrate an appropriate mastery of the knowledge, techniques, skills, and modern tools of their disciplines.  Apply current knowledge and adapt to emerging applications of mathematics, science, engineering and technology.  Conduct, analyze and interpret experiments and apply experimental results to improve processes.  Apply creativity in the design systems, components and processes appropriate to program objectives.  Function effectively in teams.  Identify, analyze and solve technical problems.

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 Communicate effectively.  Recognize the needs for and posses the ability to pursue lifelong learning.  Understand professional, ethical and social responsibilities.  Recognize contemporary professional, societal and global issues and aware of and respect diversity.  Have a commitment to quality, timeless and continuous improvement. (Kamsiah M. Ismail & Marlia Puteh, 2008) The points listed above have clearly indicated that engineering technology graduates should be able to apply their knowledge in engineering and technology fields effectively. Furthermore, they must also be able to adapt their knowledge in emerging applications of mathematics and science fields. This view has been supported by the following definition given by Cheshier, as follows: “Engineering technology is the profession in which a knowledge of mathematics and natural sciences, gained by higher education, experience, and practice, is devoted primarily to the implementation and extension of existing technology for the benefit of humanity. Engineering technology education focuses primarily on the applied aspects of science and engineering, aimed at preparing graduates for practice in that portion of the technological spectrum closest to product improvement, industrial practices, and engineering operational functions” (Cheshier, 2006) Engineering technology courses are oriented towards application to support the engineers. They provide students with introductory courses on mathematics and science, and qualitative introduction to engineering fundamentals. Therefore mathematics is one of the importance subjects in this field. In Malaysia, the ideal ratio for engineer technologists to engineers is 2:1, but the current ratio is 1:3. Therefore the universities involved in producing engineering technology graduates are responsible to overcome the shortage of engineer technologists to realize the vision 2020; Malaysia as an industrial nation. 3. INTEGRAL CALCULUS Calculus is one of the main subjects in mathematics for engineering technology courses. It comprises of two main topics, namely differentiation and integration. Students face difficulties in these two topics in calculus, particularly integration, due to the following reasons:  Differentiation is a forward process; hence the difficulties faced by students in this topic are not as severe as those in the backward process in integration.  The nature of integration: it is the reverse of differentiation process; this explains why it is also known as anti-derivative. Integration is a tool for calculation area and volume which requires a good skill in sketching graph. (Shafia Abdul Rahman, 2005). Tall (1992) has listed some difficulties that students face in learning calculus. They are:  Restricted mental images of functions.  Difficulties in distinguishing the meaning of “dx” in dy/dx (in differentiation) and “dx” in integration.  Algebraic manipulation or lack of it.  Preference for procedural methods rather than conceptual understanding.  Difficulties in translating real-world problems into calculus formulation.  Difficulties in absorbing complex new ideas in a limited time. (Tall, 1992). The analysis of students’ performance, particularly in integral calculus topic, for July – December 2010 session in the university involved in this study, has been done thoroughly. The results for two mathematics subjects for both levels; diploma and bachelor degree programs offered at the university were investigated. The subjects are WQD 10202 (Technical Mathematics 2) for diploma degree program and FKB 13102/ FKB14102 (Engineering Mathematics 1) for bachelor degree program. They are compulsory subjects for graduation in Diploma in Engineering Technology and Bachelor in Engineering Technology degrees at this university respectively. They are also pre-requisite for the higher mathematics subjects, namely FKD 22302 (Mathematics for Technologists 3) at diploma degree level, and FKB 13202/ FKB 14202 (Engineering Mathematics 2) at bachelor degree level. The topics covered in WQD 10202 are complex numbers, trigonometry, function, differentiation and integration. Whereas the topics covered in FKB13102/14102 are matrices, complex numbers, polynomials, differentiation and integration. In July – December 2010 session, there were 143 students enrolled for the first time in WQD 10202. However, there were only 132 students sat for the final examination held in November 2010, while the remaining 11 students were absent for a various reasons. The percentage of students passing the subject is 79.7%. This percentage is considered high as it is just 0.3% below the targeted percentage set by the university, which is 80%. However, the percentage of students achieved a pass in integral calculus topic in this examination is extremely low. Due to this reason, the students’ examination scripts were examined. The final examination paper consists of three parts, Part A, B and C. The analysis was done to investigate students’ performance in Part B and C only. Part A was not analyzed because it consists of multiple choice questions, whereas both Part B and C consist of short answer and long answer subjective questions respectively. Students were asked to answer all questions in Part B, while in Part C, they have to answer only four out of five questions.

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The analysis for Part B shows that only 5.3% scored more than 75% (Excellence performance), 0,8% scored between 60% - 74% (Good performance) and only 1.5% scored 40% - 59% (Pass). The remaining 94.7% did not manage to score more than 40% (Fail) in this topic. A further detail analysis on the answer scripts of those who have failed in this topic was conducted. The result indicates that 23.5% of them did not attempt the questions related to integral calculus, whereas 52.3% of these students attempted to answer the integral calculus questions, but did not manage to score any marks. In Part C, the analysis reveals that only 12.9% of the candidates attempted the questions related to integral calculus, out of which, 70.6% have failed to achieve more than 40%. However, 5.9% have scored an excellence performance, and 17.6% have managed to score a good performance, which indicating that the questions asked were not difficult. Based on the analysis done on both parts, it implies that the high passing rate achieved in this final examination result was obviously not contributed by the marks obtained in integral calculus topic. At the bachelor degree level, the mathematics examination format is different. The questions were not segregated into three parts as in diploma’s final examination paper. Instead, the examination paper consists of five subjective questions, where each question is represented by different topic. The candidates were required to answer only four questions. The total passing rate is 77%. This percentage is considered as satisfying, as it is only 3% below the targeted percentage. However, the scenario at the diploma level is even worse. There was only 6.8% chose to attempt the questions involving integral calculus. Nobody has scored an excellence or good performance and there was only one student managed to get 43.3%, which is only a pass. Again, it can be concluded that integral calculus was not the contributing factor to the high passing rate achieved. This problem cannot be kept silence as calculus including integral calculus is the foundation for higher mathematics subjects, as well as in technical subjects. Therefore, an immediate action needs to be figured out as this problem will affect the number of students graduated on time. 4. INTEGRATING TECHNOLOGY The traditional teaching approach can be enhanced by integrating technology in teaching and learning of integral calculus topic. The studies conducted by mathematics instructors throughout the world related to this approach have given the positive impacts on students’ understanding (Berry, Lapp, & Nyman, 2008; Cook, 2006; Kocsis, 2007; Noinang, Wiwatanapataphee, & Wu, 2008; Wiwatanapataphee, Noinang, Wu, & Nuntadilok, 2010). Computer Algebra Systems (CAS) Computer algebra systems (CAS) have the capabilities in manipulating algebraic symbolic form of an equation. Therefore, they have been widely used to support teaching and learning of mathematics at the university (Noinang, et al., 2008; Robinson & Burns, 2009; Wiwatanapataphee, et al., 2010). Over the past two centuries, many symbolic packages have been developed using Mathematica and Maple softwares. These symbolic packages enable students to understand the concepts taught better with the availability of their visual supports (Noinang, Wiwatanapataphee & Yong 2008). According to Robinson and Burns (2009), CAS like Maple and Mathematica have been used as a complement to the traditional teaching approach. There are several reasons why CAS is integrated into the curriculum as a teaching aid. The first reason being, CAS is able to perform any complex calculations. This feature will help students to focus on the high-level mathematical calculations and hence, they manage to grasp the new concepts introduced more effectively. CAS will also benefit the mathematics educators especially for those who are lacking of artistic skills, as most mathematics topics, particularly 3-dimensional calculus involves a lot of complicated graphs (Cook, 2006). In addition, mathematics educators have a chance to incorporate the real applications in their teaching. This element is important in motivating students and therefore will improve their performance. In terms of mathematics curriculum, Heid (2001) stated that CAS serves as a reorganizer in mathematics teaching and learning as the adjustment moderator. In this case it is able to facilitate the balance, the sequence, and the priorities of conceptual and procedural knowledge in the mathematics curriculum (Heid, 2001). CAS is not only important in improving students’ understanding of the mathematical concepts per se, but its applications will be useful in preparing students for their future career needs in the industrial world. 5. RELATED THEORIES IN TEACHING AND LEARNING WITH TECHNOLOGIES In this study, three theories will be proposed as a basis for the theoretical and conceptual framework. The theories are cognitive load theory, distributed cognition theory and cognitive theories for multimedia learning. Cognitive Load Theory Cognitive load theory (CLT) is mainly related to the study of complex cognitive tasks to process several elements of information simultaneously before effective learning can begin (Paas, Gog, & Sweller, 2010; Paas, Renkl, & Sweller, 2004). This theory emphasizes on the mechanical design of teaching methods to reduce the burden on students’ cognitive system. This will allow the use of their limited cognitive resources effectively (Beckmann, 2010). The theory which focuses on the role of working memory emphasizes the control of high cognitive burden in the development of any suitable teaching techniques (Nor’ain Mohd Tajuddin, Rohani Ahmad Tarmizi, Mohd Majid Konting, & Wan Zah Wan Ali, 2009; Paas, et al., 2010). This approach can also be




implemented to ensure the achievement of a meaningful learning (Paas, Renkl & Sweller 2004). In this sense, CLT uses the information about human cognitive design in developing a suitable teaching method. During learning, there are three types of cognitive processing that can contribute to cognitive load. They are intrinsic processing, extraneous processing and germane processing. Intrinsic processing is very essential in comprehending lesson, and it depends on the number of interacting elements in students’ mind at any one time. Our working memory is not only occupied with the interacting of useful elements, but it is also interrupted by the weaknesses of procedures in teaching designs. This kind of load is known as extraneous processing which needs to be minimized in optimizing the learning outcomes. Robinson and Burns (2009) believed that the use of CAS will reduce this kind of load. The third load is also resulted from the teaching design, but in a positive way. A good teaching design will reduce the extraneous processing load and will increase germane processing load, in which students engage in deep cognitive processing. What will happen is that, students will involve in organizing the materials mentally and also manage to relate them to their prior knowledge, and hence will improve their understanding. Distributed Cognition Theory Distributed cognition term refers to the processes that have two criteria. Firstly, they are cognitive in nature in the sense that they involve formation of representation of certain things. Secondly, they are formed between individuals and non-human technical devices (List, n.d.). Unlike other cognition theories, this theory emphasizes interaction between individuals with the resources in the environment. However, like other cognition theories, distributed cognition try to understand the organization in human cognitive system (Hollan, Hutchin, Kirsch, 2000), which consists of cognitive activities involved in memory, making decisions, inferential, reasoning and learning (Hollan & Hutchins, 2009). To design effective human-computer interactions, at least the following three distributions must be considered:  Cognitive processes are distributed in a social group.  Cognitive processes are the result of the interactions between internal and external structure.  Processes are distributed along a time line, where the later events depend on the earlier events (Hollan, Hutchins, & Kirsh, 2000). Cognitive Theory for Multimedia Learning Mayer and Moreno have derived three assumptions on how our mind works in a multimedia learning environment. First, assumption is dual channel, where all humans possess two channels in digesting verbal and visual information separately. Second, assumption is limited capacity, which implies that information processing capacity in both channels is limited. Last assumption is active processing that characterizes learning as an active cognitive process that occurs in two different channels. The relation between all assumptions is summarized in Figure 1.

Fig.1. Cognitive Theory of Multimedia Learning (Mayer & Moreno, 2003) 6. CONCLUSION The literature review has shown that CAS is a catalyst in improving students’ understanding of difficult abstract mathematical concepts. Therefore, this paper would like to propose a transformation in teaching and learning of mathematics at the university level using CAS. The focus is to integrate CAS as a medium to improve the understanding of this subject particularly in integral calculus topic. This idea will hopefully give benefits to students from engineering technology courses because it is more hands-on in nature. This kind of approach is believed to be suitable in enhancing their understanding as its nature matches their style of learning in technical subjects.

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