Mathematical modeling as a tool to improve learning of mathematics Anna Friesel Associate Professor, Electronics and IT Department Copenhagen University College of Engineering, DK-2750 Ballerup, Denmark [email protected]
George Anastasiou Nicolakis B.Sc. student, Electronics and IT Department Copenhagen University College of Engineering, Ballerup, Denmark [email protected]
Abstract - This paper describes the experience in teaching mathematics as part of a project involving circuit theory, mathematical modelling, analogue electronics, and filter theory at the department of Electronics and Information Technology, at Copenhagen University College of Engineering in Denmark. Modern engineering students are difficult to motivate in order to learn pure theoretical issues such as solving differential equations. Especially when we talk about undergraduate engineering students, who do not expect to continue their education higher than a Bachelor of Science in Engineering. At the same time, and for the same reason, the fail rate in mathematics is very high during the first two semesters. This was the reason we decided to change the structure of our education and incorporate the theory with practical projects. This paper describes the scale project, in which the students design the electronic scale. In order to explain the dynamical behaviour of the scale, they have to learn more mathematics. This is the motivating factor. The examination results show improved learning potential, when using this method of educating mathematics. The students’ evaluations show a very positive effect on their experience with this “practical” way of learning mathematics. The evaluation results have shown higher pass rates and higher grading, especially for the practical oriented students. Index Terms - Engineering mathematics, problem-based education, projects, teamwork. INTRODUCTION Engineering education enters new challenges along with the possibilities we have of integrating new technologies in didactic process ,,,,. These are technologies like simulation programs, graphics and animations, and of course the access to the Internet resources from all over the world. The simulation programs are used for visualisation of theory in mathematics and physics, at all levels of education.
In particular the engineering students benefit from these technologies, as they actively use both Internet and simulation programs. Teaching theoretical mathematics today is a very challenging task, even more so when we talk about undergraduate, practical oriented engineering students. The engineering students’ expectations for their study program are more practical and experimental activities, then the calculations of the theoretical problems with pen and paper. Many universities and engineering colleges have the opinion that pedagogical activities must actively involve the students in order to motivate them to learn basics of mathematics and physics. Problem-based learning and working with projects has already been implemented in engineering educations in many different universities ,,,. This paper describes one of the methods, which actively engage the students for learning mathematics. This paper presents how we implemented the course of differential equations in the curriculum for undergraduates in the Electronics Department. The mathematics has become a part of the third semester course KRA3 ,,,,, which includes theory, a practical project (electronic scale), and teamwork. Each part of this course like: circuit theory, filter theory and mathematical modeling, ends with practical design and experiments. The documentation of the practical design and experiments are described in the reports and give the basis for the evaluation of this course. This is an advanced process of learning, where the students have the responsibility for combining the theory and practical electronics design. The motivation to learn basics of mathematics is improved, compared with traditional classical educational methods, where the students learn all the theory in pure math-courses during the first semesters of their education. The paper concludes with analyzing the results of the course examination, and the students own evaluations of this course. MATHEMATICS IN ENGINEERING EDUCATION The primary objectives of the Electronics Department curriculum at the Copenhagen University College of
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July 23 – 28, 2006 9th International Conference on Engineering Education T1B-22
Session T1B Engineering are to prepare its graduates to practice electrical and computer engineering technology with skills consistent with the requirements of today's industrial companies. Danish companies are traditionally active in discussions about development of the engineering education. This is actually part of the examination process in Denmark, where all the examinations on the university level involve, by law, an external examiner certified by the Ministry of Education. For engineering departments the external examiner is very often a company manager. The role of external examiner, among other things, is to keep the engineering curriculum up to date. The external examiner has a great opportunity to discuss the contents of the engineering courses including: pedagogical methods, experimental work, and projects when he or she participates in the examination. The external examiner has to report his/her conclusions about the examination level, and level of education to the chairman of the external examiners, whom is approved by the Ministry of Education. This procedure gives the industry the influence on the engineering education in Denmark. The industry can easily have influence to change the engineering departments’ curriculum according to the needs of the industry. On the other hand the students meet their future managers, and get the knowledge of the requirements in industrial companies at the same time.
physics. On the other hand, the industry of today requires engineers with practical skills in hardware and software together with other abilities like: the ability to work in teams, communication and language abilities. In order to fulfill all these requirements, and at the same time to motivate the engineering’s students to learn mathematics, we decided to combine the practical analog circuit and filter theory with differential equations and mathematical modeling. The theoretical part of this third semester course (KRA3) includes: • Circuit theory • Filter theory • Differential equations • Laplace transformation • Mathematical Modeling • • •
The practical part of KRA3 includes: Specification, analysis and design of electronic scale. DCproject (figure 1). Design of the metal detector, AC-project and filter design (figure 2). Mathematical Modeling of the electronic scale, comparison between measurements and simulations (figure 3).
At the department of Electronics and Computer Science a part of this curriculum is the basics of mathematics and
FIGURE 1 BLOCK-DIAGRAM OF ELECTRONIC SCALE, DC-PROJECT.
Low pass filter
FIGURE 2 BLOCK- DIAGRAM OF THE METAL DETECTOR, AC-PROJECT.
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July 23 – 28, 2006 9th International Conference on Engineering Education T1B-23
FIGURE 3 MEASUREMENT AND SIMULATION OF THE ELECTRONIC SCALE.
FIGURE 4 KRA3 TIME SCHEDULE .
Each of the practical parts of this course should be described in a report, and is the basis for the oral examination. The course is composed of approximately 60% of the scheduled lectures, unequally divided during the semester, there is about 75% at the beginning, 50% in the middle, and the last two weeks there are no lectures. All the time is totally allocated for the project teamwork at the end of the semester.
The example of the semester time schedule is shown in figure 4. During the course the educational process is supported with PC-based exercises. The simulation programs like SPICE and MATLAB are used to support the comprehension of the learning process. The simulation programs are also used to verify the theoretical calculations of the practical
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July 23 – 28, 2006 9th International Conference on Engineering Education T1B-24
Session T1B measurements. Tests, simulations, and measurements are very important parts of this course. Another important issue of this course is to make students work with the project as a team. Our students actually do this from the very beginning, from the first semester, but the projects they work with become more and more sophisticated. This course provides the students with following abilities: • To identify, formulate and solve technical problem from a given specification. • To apply the knowledge of mathematics, science, and electronics. • To participate in the design and implementation process of an electrical system. • To conduct experiments, as well as to analyze and interpret the data. • To communicate effectively in written, oral, and in electronic media. • To work in a team.
mathematics in this practical approach to the theory, especially for the students with previous practical experience. The examination results are highly improved and the overall dropout due to theoretical mathematics is much lower. ACKNOWLEDGEMENT I would like to thank my colleagues, the professors Lars Maack and Ib Lemvig Christoffersen at the Copenhagen University College of Engineering, for our good discussions and close cooperation during this course. Thanks to all the students for valuable discussions on future development of this course, and special thanks to all the students who contribute for the evaluation of this course. Thanks to I.Stauning and N.Storm for technical support in connection to the practical part of the project. REFERENCES 
EVALUATIONS The examination is a group examination, but the marking is individual. The evaluation is based on a general impression of the level achieved by the student relative to the objective of the course. The examiners, teachers, and the external examiner have read the students three reports in advance. The students present the results of their work, both as the oral presentation and as a practical demonstration of the hardware. The questioning, which is individual, begins after their presentations. After questioning, the students leave the examination room and the examiners discuss their presentations. The marks are given individually, and the students are invited back into the examination room for the explanation of their marking. After the examination the students can evaluate the course anonymously on the Campusnet. The students’ evaluation shows, that 78% of the students are either very satisfied or satisfied with this form of mathematics education mixed together with theory and practical electronic projects. Evaluations also show that 82% of our students were very highly or highly motivated to learn mathematical methods covered in this course. We work at the moment on more detailed statistics of the students’ evaluations of this project/course. CONCLUSIONS After two semesters of completed KRA3-course and the evaluation of this course, we can make the conclusion, that the main objectives have been achieved. The students have got a better understanding of the mathematical tools in engineering. We have also observed much better understanding of more advanced engineering problems during the fourth semester course in system dynamics and control theory. The students own evaluations show an increased motivation to learn
   
      
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