Developing Mathematical Communication Skills of Engineering ... - Core

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Roselainy Abdul Rahman a *, Yudariah Mohammad Yusof b , Hamidreza ..... Kashefi, H., Zaleha Ismail., Yudariah Mohd Yusof., and Roselainy Abd. Rahman.
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Procedia - Social and Behavioral Sciences 46 (2012) 5541 – 5547

WCES 2012

Developing mathematical communication skills of engineering students Roselainy Abdul Rahman a *, Yudariah Mohammad Yusof b , Hamidreza Kashefi c, Sabariah Baharun d a b

Razak School of Engineering and Advanced Technology, UTM International Campus, Kuala Lumpur, Malaysia Department of Mathematics, Faculty of Science, Universiti Teknologi Malaysia (UTM), 81310 Johor, Malaysia c Regional Centre for Engineering Education, Universiti Teknologi Malaysia (UTM), 81310 Johor, Malaysia d Malaysia-Japan International Institue of Technology, UTM International Campus, Kuala Lumpur, Malaysia

Abstract In Malaysia and also elsewhere in the world the demands for graduates who have employability skills such as ability to think critically, solve problems and can communicate are highly sought in the workplace. In the early 2006, the development of such skills was recognized as integral goals of undergraduate education at Universiti Teknologi Malaysia. Since then rigorous efforts have been made to inculcate these skills amongst the undergraduates. In this paper, we will share some of our experiences in coping with the challenges of changing our teaching practices to accommodate this quest though focusing on communication. For mathematics learning to occur, we believed that students should participate actively in the knowledge construction and be abl e to take charge of their own learning. Taking these aspects into consideration, we had developed a framework of active learning and used it to guide our instruction in engineering mathematics at UTM. Here we will discuss the strategies that we had designed and employed in engaging students with the subject matter as well as to initiate and support students’ thinking and communication in the language of mathematics. Some students’ responses that gave indications of their struggle, progress and growth encountered in the research implementation will also be presented. © 2012 2012Published PublishedbybyElsevier Elsevier Ltd. © Ltd. Selection and/or peer review under responsibility of Prof. Dr. Hüseyin Uzunboylu

Open access under CC BY-NC-ND license. Keywords: Communication, Mathematical Thinking, Multivariable Calculus, Students’ Obstacles

1. Introduction Generic skills are becoming major and important requirements set by many stakeholders as due to business and industrial competitiveness, graduates who can think critically, solve problems and communicate, to name a few, are highly sought by employers (Akop et al., 2009; McCray, 2001). A special report from the Steering Committee of the National Engineering Education Colloquies (2006) has also recommended that there should be a study to investigate how the various elements of “innovation, critical thinking, systems thinking, biology, mathematics, physical sciences, engineering sciences, problem solving, design, analysis, judgment, and communication relate to each other to characterize the core of engineering as a profession.” Thus, in Malaysia, six out of eight domains of competencies

* Corresponding Author name. Tel.: +6-019-755-7580 E-mail address: [email protected]

1877-0428 © 2012 Published by Elsevier Ltd. Selection and/or peer review under responsibility of Prof. Dr. Hüseyin Uzunboylu Open access under CC BY-NC-ND license. doi:10.1016/j.sbspro.2012.06.472

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to be emphasized by higher education providers, refers to generic skills which includes communication and team working (Malaysian Qualification Framework, 2005). Since 2006, UTM have included the development of generic skills as goals of undergraduate engineering education but studies at UTM (Sabariah et al, 2007; Roselainy, 2009) had indicated that the goals have not been translated into implementation successfully. In addition, there has been a radical shift in the mathematical skills needed at the modern workplace which has not been fully recognized by the formal education system (Hoyles, 2007). Taking the above aspects into consideration, we had developed some innovative teaching approaches that support students in their mathematical learning, that empower students with more successful ways of thinking about mathematics, enhance their awareness of how to acquire efficient techniques in constructing knowledge and encourage them to participate and to take charge of their own learning, enhance mathematical communication skills, in written and verbal forms. 2. Our Approach Our model of teaching and learning (Roselainy, 2009; Roselainy et al, 2010) focused on three major aspects: the mathematical knowledge development, mathematical thinking processes as well as the appropriate generic or soft skills. The emphases were on student which include critical thinking and problem solving, knowledge construction, competency in procedures and techniques and the generic skills (communication, teamwork and self-directed learning). In developing the mathematical pedagogy for classroom practice, we adopted the theoretical foundations of Tall (1995) and Gray, et al (1999) and used frameworks from Mason, et al (1985) and Watson & Mason (1998) and the works of Meyers & Jones (1993) to support the important elements of effective active learning. preconceived preferences for procedural learning. Thus, the teaching acts implemented were aimed at shifting m rote learning towards understanding the procedures and recognising their mathematical powers, and to enhanc Roselainy, et al, 2007). We have since expanded effective use of technology (Kashefi, et al, 2011). 3. Active Learning We had adopted an active learning mode to integrate the mathematical learning and the enhancement of generic skills to promote a learning culture in which students could be active participants in their mathematical knowledge construction and development by using the following strategies; (a) Specially designed classroom tasks tasks that sses and structures that we wished them to learn. The tasks were categorized as Illustrations, Structured Examples and Reflection with prompts and questions; (b) Classroom activities these include working in pairs, small group (informal & formal), quick feedback (minute paper, muddy and writing; (c) Encouraging communication specially designed prompts and questions initiated both written and oral mathematical communication through discussion and sharing of ideas among the students and written assignments; (d) Supporting self-directed learning understanding of mathematical concepts and techniques, and (e) Using both summative and formative types of assessment. We found that encouraging students to talk, to listen, to read, to write and to reflect on their mathematical learning and problem solving, would enhance their awareness of their own thinking as well as their communication skills. Consequently, they are able to improve their understanding, gain new insights into the problem and communicate their ideas in a mathematical manner. In this way, students became more responsible for their learning and were able to think for themselves. All the mathematical tasks used in the classroom were compiled as a workbook. This helps students to focus on the tasks-at-hand rather than on writing and copying everything down. Furthermore students prefer having some

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documentation with them when they are working in the class. The prompts and questions that we have developed were placed at the relevant problems so as to guide students in their thinking and to make explicit the processes and structures that they were learning. and communication. 4. The Observation used to informed us on issues such as what were the concepts the students were attending to, were any of the prompts and questions misleading or do we have to develop other possible means of imparting the mathematical (Sabariah, et al., 2007). We then modified, improved and introduced changes where appropriate to the workbook, our delivery and the assessments and implemented them during the next academic session. We worked as a team and each played the role of teacher as researcher. A questionnaire requesting comments on our approach was administered to the students at the beginning and again at the end of the semester of the academic year 2010/11. However, most of the data were collected mainly through self5.

Responses

early in the course and some that were observed later in the Here, we will share some of the student course. We encouraged students to work in groups or in pairs so as to facilitate discussion amongst them. Most students enjoyed the opportunities to participate in their own learning. We observed and noted various difficulties and misconceptions displayed by the students, particularly during the first half of the semester. Many faced difficulties to communicate mathematically. By using structured questions with prompts and questions, students usually can give some response to the questions asked although not necessarily correct. See excerpt below of a student work in the task under Illustrations (Figure 1). The responses indicated that he was not sure of what is .

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Student A claimed that he understood the concepts but could not articulate their knowledge very well to their friends and in written form. Some students also struggled in their attempts in writing out mathematical ideas in English as indicated in the following excerpt (Figure 2). The course was conducted in English and so were most of the other courses taken by the students. In the Making Sense section, students were requested to explain in writing how explanations was written in short form rather than full explanations.

Figure 2. A

writing out mathematical ideas in English

During an interview conducted at the end of the course, two students (Tikah and Fizah) were asked to solve the following questions (Table 1): Table 1. Two typical questions Question 2 Let R be the region enclosed by (a)

Express

y

x ,y

0 and x

y dA

as an iterated integral in rectangular coordinates.

y dA

in terms of polar coordinates.

R

(b)

Express R

1.

Prompts/Questions Can you identify and sketch the region of integration? How do you write the iterated integral? o What are the limits of integration in the rectangular coordinates? In polar coordinates?

Roselainy Abdul Rahman et al. / Procedia - Social and Behavioral Sciences 46 (2012) 5541 – 5547 Question 3 Use polar coordinates to evaluate the double integral.

y dA ; R is the disk x2

(a)

y2

R

4.

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Prompts/Questions How do you change Cartesian to polar form? o Integrand o Differential of integration (dA) o Domain of integration How do you find the limits of integration? What basic integration techniques do you need?

Below is a copy of their work (Table 2). As they worked, they were discussing with each other; about what dA means in Cartesian coordinates and which order to take. However, this was not clearly indicated in their working. Looking at the limits and their sketch, it was clear that they had chosen the order dx dy . There was more discussion on what dA was in polar coordinates and how the limits should be read. They had worked out the conversion and knew the technique of how to read the limits but made a mistake in identifying the lower limit of r. Table 2. Question 2

during interview Question 3

While working on Question 3, they got zero as an answer and were quite concerned as they felt that it was wrong. decided to submit what they had written. In themselves, these are not significant pieces of work but looking at the work with the added element of the on-going discussion, verbalisation of mathematical ideas and techniques, showed a remarkable shift in mathematical behaviour for the students. Although they were just as concerned about checking answers at the back of the book, they did not request for an work when they finished with it. Many students appreciated the opportunities provided and were willing to participate in the class activities. They showed some progress in their ability to construct ideas on their own, to discuss with their friends and lecturers as well as communicate mathematically in class. See example below of an ex written work,

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indicating his efforts in making explicit his reasoning and response to questions asked (Figure 3). His emphasis was no longer only on computation.

Figure 3.

In general, the findings from such responses and observations indicated that students made conscious efforts to use appropriate terminologies in their explanation and solutions to the problems. They were no longer afraid to speak up, able to think through their responses before articulating and are also better at expressing what they had discussed in their own words. 6. Conclusion We had adapted and modified questions and structure them in a manner to build concepts and ideas, to draw students to misconceptions and to introduce some applications. We had provided and promoted an active learning environment to engage students in their learning and to participate actively through the diverse activities created. The mathematical tasks were designed so that students could experience for themselves the mathematical processes such as the process of identifying the general class of problems they were working on. Throughout, the students were actively supported in discussing, verbalising and writing out their understanding of the mathematical ideas and concepts. The diverse activities had generated the students as well as providing them with opportunities to take charge of their learning. At the beginning, students were uncomfortable with the activities as they were different from their usual learning experiences. However, after a few sessions, they gradually adapted to the new environment showing particular enthusiasm working in groups, sharing of ideas and working out the mathematics for themselves. Thus the environment created had facilitated both thinking and communication skills among the students, making for a much livelier class. Acknowledgements he authors acknowledged the Ministry of Higher Education of Malaysia and Universiti Teknologi Malaysia for the financial support via UTM Razak School Research University Grant (No: 4B014) given in making this study possible. References Akop, M. Z., Mohd Rosli, M. A., Mansor, M. R., and Alkahari, M. R. (2009). Soft skills development of Engineering undergraduate students through Formula Varsity Engineering Education (ICEED), International Conference on Engineering Education (ICEED 209), 7-8 Dec, Kuala Lumpur, Malaysia. Gray, E., Pinto, M., Pitta, D., and Tall, D. O., (1999) Knowledge Construction and Diverging Thinking in Elementary and Advanced Mathematics, Educational Studies in Mathematics, 38(1-3), pp 111-133.

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Kashefi, H., Zaleha Ismail., Yudariah Mohd Yusof., and Roselainy Abd. Rahman. (20 1). Promoting Creative Problem Solving in Engineering Mathematics through Blended Learning, The 3rd International Congress On Engineering Education (ICEED 2011/IEEE), pp.14-18, 7-8 December, Malaysia. Malaysian Quality Agency. (2005). Malaysian Qualification Framework: MQA Guide book. Mason, J., Burton, L., and Stacey, K. (1985). Thinking Mathematically. Addison-Wesley Publishing Company, Inc., Wokingham, England. McCray, P. D. (2001). Business View on Math in 2010 C.E. CUPM Discussion Papers About Mathematics and the Mathematical Sciences in 2010: What should students know? The Mathematical Association of America. Meyers, C., and Jones, T. B. (1993). Promoting Active Learning: Strategies or the College Classroom. San Francisco: Jossey-Bass Publishers. McCray, P. D. (2001). Business View on Math in 2010 C.E. CUPM Discussion Papers About Mathematics and the Mathematical Sciences in 2010: What should students know? The Mathematical Association of America. Roselainy Abd. Rahman. (2009). Changing My Own and My Students Attitudes Towards Calculus Through Working on Mathematical Thinking , Unpublished PhD Thesis, Open University, UK. Roselainy Abd. Rahman., Yudariah Mohd Yusof., and Sabariah Baharun. (2007). Enhancing Thinking through Active Learning in Engineering Mathematics, In CD Proceedings of Fourth Regional Conf. on Engineering Educ, Johor Bahru, 3 5 Dec. Roselainy Abd. Rahman., J. H. Mason., and Yudariah Mohd Yusof. (2010). Factors Affecting Students' Change of Learning Behaviour, The 3rd Regional Conference on Engineering Education & Research in Higher Education, Kucing, Sarawak, 7-9 June. ical Competency through Mathematical Thinking. In CD Proceedings of Third Intern. Conference on Research and Education in Mathematics, The Legend Hotel, Kuala Lumpur, 10-12 April. Special Report, The National Engineering Education Colloquies. (2006). Journal of Engineering education, p 257-261, Oct . Tall, D. O. (1995). Mathematical Growth in Elementary and Advanced Mathematical Thinking. In L. Meira & D. Carraher, (Eds.), Proceedings of PME 19, Recife, Brazil, Vol. I, pp 61-75. Watson, A., and Mason, J. (1998). Questions and Prompts for Mathematical Thinking. ATM, Derby.