2011 3rd International Congress on Engineering Education (ICEED)
Promoting Creative Problem Solving in Engineering Mathematics through Blended Learning
Hamidreza Kashefi#l, Zaleha Ismail#2, Yudariah Mohammad YuSO(3, Roselainy Abdul Rahman+4 # Department of Science and Mathematics Education, Faculty of Education,
*
Universiti Teknologi Malaysia (UTM), 81310 Johor, Malaysia I
[email protected] [email protected] Department of Mathematics, Faculty ofScienc;, Un �versiti Teknologi Malaysia (UTM), 81310 Johor, Malaysia .
[email protected] +Col!ege of Science and Technology, UTM City Campus, Kuala Lumpur, Malaysia
[email protected]
Abstract- Mathematics is one of the most difficult courses to study
for
engineering
students.
In
addition,
problem solving [3], [4] but also may cause engineering students to encounter difficulties in the learning of engineering subjects or even mathematics. Mathematics is a prime constituent of the infrastructure of the education of engineering students. The main goal of mathematics instruction for engineering students is to enhance the students' ability to apply a wide range of mathematical techniques and skills not only in their engineering classes but more importantly later in their professional work [5]. In most engineering curricula, many topics are taught using mathematics and mathematical models. Knowledge of the prerequisite background in mathematics is therefore necessary for students. Research findings indicate that for most engineering students, mathematics has always been one of the most difficult courses to study. Many students struggle as they encounter non-routine problems that are not solved by routine problem solving methods. Many of the mathematical deficiencies and learning difficulties of engineering undergraduates have been well documented. Studies on students' learning difficulties indicate that understanding basic calculus as a pre-requisite plays an important role in the understanding of Engineering Mathematics [6]-[9]. Various problem areas have been identified in basic and multivariable calculus for engineering students. These include difficulties in learning specific mathematical topics, difficulties in algebraic manipulation, in absorbing complex new ideas in a limited time, in the recall of factual knowledge, students' beliefs and learning styles, poor problem solving skills, and the inability to select and use appropriate mathematical representations [6]-[11]. Many methods have been applied to help support students to overcome their difficulties in mathematics. Creative Problem Solving (CPS) as a multi step process can fundamentally improve the way students learn mathematics and support their generic skills such as communication, teamwork, and problem solving [12]. In fact, CPS not only combines aspects of other approaches in solving problems of engineering, science, and mathematics but can also use the capabilities of computers [4]. However, the literature review indicates that very little research has been done that supports
effective
communication, teamwork, and problem solving are important skills for graduate engineers in their professional work. This study proposes Creative Problem Solving as a framework to improve the way engineering students learn mathematics and support their critical problem solving, communication, and team working skills. The main goal of this paper is to illustrate the importance of using creative problem solving skills through blended learning environment as seen from an Engineering Mathematics students' students
participated
perspective. First year undergraduate in
an
experimental
group
where
instruction was provided in a blended learning format using a creative problem solving approach. The lecturer then used face-to-face instruction on the same subj ect to a control group. Results indicate that from the perspective of the students in the experimental
group,
the
ability
to
utilize
creative
problem
solving skills is the most important skill set when learning Engineering Mathematics.
KeywordsProblem
Blended
Solving;
Learning;
Engineering
Communication; Mathematics;
Creative
Face-to-face
Instruction; Teamwork
1.
INTRODUCTION
Current tendencies in technology and the workplace require engineers to have a greater variety of capabilities, skills, and a wider understanding of engineering as a discipline if they want to succeed [1]. Educational and enterprise managers agree that too many engineering students graduate without having effective communication, teamwork, and problem solving skills [2]. Industries also complain that graduate engineers are technically competent; however, they lack skills in critical problem solving, communications, teamwork, and making judgments [3]. This implies that the rapid pace of technological change in society has not been matched by a corresponding change in engineering education, with the result that the same material is being basically taught with the same tools and methods that have been in use for many years. The limitations of traditional teaching and learning styles may not only be the reasons behind engineering students' weakness in generic skills such as communication, teamwork, and
978-1-4577-1259-3/11/$26.00 ©2011 IEEE
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students' generic skills and knowledge construction in Engineering Mathematics through CPS and computers. In this study, based on our preliminary study at Universiti Teknologi Malaysia (UTM) we propose blended learning as an appropriate environment to promote engineering students' CPS. Then, we identify the importance of employing CPS skills through blended learning in order to support students to overcome their difficulties in Engineering Mathematics trom the students' perspective. II.
all analytical, creative, and critical thinking and that it can be used to strengthen the quality of teamwork, thinking and communication skills of students in whole brain during of its stages [4]. The five distinct steps in CPS correspond with different thinking modes [4]. In fact, CPS is a framework which employs many different thinking skills and thinking tools. The analytical aspects of quadrant A thinking are needed to analyze clues and data about real problems and this quadrant is also involved at the highest level in evaluation and critical judgment of ideas. The organizational aspects of quadrant B thinking are needed during idea implementation for planning, execution, follow-up, and the final evaluation process. The interpersonal aspects of quadrant C thinking are needed for teamwork during problem definition, during idea generation, and during implementation. The imaginative, wishful aspects of quadrant D thinking are needed during problem definition, during idea generation, and during implementation and judgment [4], [19]. CPS also allows the use of computers to enhance thinking, learning, and communication [4]. Computers can be used to focus on the fundamentals, to investigate and defme the problem and its broader context, to model the problem, and to visualize the problem and results graphically. Tn fact, computers are the best analogy of the human brain; however, unlike the brain, computers cannot synthesize infonnation and think creatively. Tn the context of teaching, learning, and thinking at least four distinctly different ways of using computers and their relations with four quadrants of brain are: (i) database and data processor (calculator), (ii) teaching machine, (iii) communication tool, and (iv) simulator and visualize (graphics) [4]. By using the potential of whole brain, CPS can support students' communication, teamwork, and critical problem solving skills. Computers and the way they are used in the CPS process play an important role to support these skills. Blended learning benefits from these powerful tools creating a cooperative learning environment such as online forums during Engineering Mathematics class to support students' critical thinking [20]. Using CPS not only can help students in Engineering Mathematics learning but also can encourage students' generic skills such as communication, teamwork, and problem solving [2], [3], [21].
CREATIVE PROBLEM SOLVING
CPS is a descriptive model of the process that can be used to understand and define problems, generate ideas, and refine solution for implementation [13]. The CPS model has changed and expanded in many ways since its roots were planted six decades ago. There have been a dozen different models of CPS developed that can be considered and these different versions are like different updates of various software packages [14]. The roots of CPS are found in Osborn's works [15] which have been followed by many researchers and developers who have presented a variety of different models, or approaches to, CPS, in varied settings specifically in colleges and universities [16], [17]. The most important goal in using CPS within an educational setting is to enable students to improve their ability to deal successfully and creatively with real problems and challenges [16]. Some researchers used CPS framework in engineering, science and even mathematics courses [2], [4]. However, in the case of mathematics, there have not been many studies done at the undergraduate level. Lumsdaine and Lumsdaine [4] suggested a CPS framework in teaching and learning of mathematics for engineering students. The framework from Lumsdaine and Lumsdaine is based on Herrmann model. According to the Herrmann model [18], the brain can be visualized as a four quadrant metaphorical model. The upper left quadrant is labeled A (mathematical, analytical, critical thinking), followed by B (sequential, controlled, routine thinking), C (interpersonal, empathetic, symbolic thinking), and D (imaginative, visual, conceptual thinking) in a counter clockwise direction. Each quadrant is characterized by distinct ways of thinking, knowing, and processing information (see [4]). Engineering curriculum itself on average is extremely skewed toward a strong preference in quadrant A, with lesser preferences in quadrant B and quadrant D thinking, and the least preference in quadrant C thinking [19]. Therefore in general, many engineering students, because of the traditional teaching styles, are predominantly left-brain thinkers [12] who then suffer from a lack of effective generic skills. Research has confirmed that quadrant C and D activities must be part of the engineering curriculum [3]. Lumsdaine and Lumsdaine [4] stated that CPS has five distinct steps: (i) Problem Definition (ii) Idea Generation (iii) Creative Idea Evaluation (iv) Judgment, (v) Solution Implementation. The authors show the relations between these stages and the four-quadrant thinking of brain in Herrmann Model [18]. They believed that the process of CPS involves
TTT. BLENDED LEARNING ENVIRONMENT There are many definitions of blended learning in the literature review; however, the tenn is still vague [22]. The three most common definitions of blended learning are: the integrated combination of instructional delivery media, the combination of various pedagogical approaches, and the combination of face-to-face and online instruction [22], [23]. In this study, we defme blended learning as the integration of traditional learning activities with some technological aids which is familiar with the third one [24]. The definition of blended learning as the combination of face-to-face and e learning instructions identifies an environment to support
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•
students' mathematical knowledge construction and generIC skills. The three aspects of learning focused upon in this study were: the development of mathematical knowledge construction, mathematical thinking as a type of CPS different thinking, and generic skills (see Fig. I). The theoretical foundation on the development of the strategies for mathematical knowledge construction and the enhancement of students' mathematical thinking were based on the works of Gray and Tall [25] and Tall [26]. The frameworks from Lumsdaine and Lumsdaine [4] and Watson and Mason [27] were used to design classroom activities and tasks based on CPS approach.
•
Strategies- synchronous and asynchronous web communication facilities such as chat, email, and discussion board used to support students' oral and written communication were included. Other strategies such as working in pairs, small group (informal and formal), critical thinking and problem solving, students' own examples, doing assignments, reading and writing in the face-to-face and web environment were also used. Assessments- incorporate both the summative and formative types such as quizzes and tests, quick classroom feedback and written assignments in face to-face and online formats. Fig. 2 shows the model of blended learning which is used as a guide to classroom instruction. Tasks Computer and Web Aided
Assessments
Stmtcgies
�
Communication
Communication
Teamwork
Teamwork
Problem Sol\'ing
Fig. 2 Model of blended learning
Problem Solving
IV. METHOD
Fig. I Focus of mathematical leaming
This study is part of a project concerned with using blended learning to promote CPS in Engineering Mathematics at Islamic Azad University of Kermanshah (lAUKSH) in Iran. Two classes enrolled in Engineering Mathematics courses participated in this study in the fall semester of 2011. One class with 59 enrolled students was selected for CPS experiment and the other class with 57 enrolled students served as a control group where teaching was conducted in the face-to-face instruction mode. The profile of students in both groups was similar in terms of size, gender, ethnicity, a n d previous educational level achieved. The first-named author with more than 8 years teaching experience of teaching in Engineering Mathematics courses taught both groups. The Engineering Mathematics offered by TAUKSH is a three credit undergraduate course and covers functions of several variables, partial derivatives, multiple integrals, vector functions and vector calculus. In order to cover set learning outcomes, we decided to use the same book resources for both control and experimental groups. The book entitled "Engineering Mathematics for Independent Learners" [31] was translated to Persian and used as a main resource. For the experimental group the instructional design of the textbook was changed based on researchers' methodology to inculcate CPS. Thus, we changed and designed prompts and questions based on CPS to increase the students' understanding, teamwork, and communication by organizing the contents in the specified manner. With special permission, the interactive web-based tools from the e-book "Calculus: Early Transcendentals" [32] were adopted. These books together
The blended learning environment would give students the opportunities to benefit from both the face-to-face and the e learning environments [28]. The following aspects were given due consideration in the development and implementation of blended learning in the Engineering Mathematics course. • Classroom tasks- in designing the tasks, existing tasks were modified to focus students' attention on mathematical knowledge construction, thinking processes and generic skills. The mathematical tasks that were used in the classroom were compiled on a website that was divided in different sections as html and word formats of the textbook. The word format tasks were categorized as Illustrations (using examples with complete solution and explanation) with prompts and questions, Structured Examples (using typical examples and then generic examples to lead students towards a generality), Reflection (asking important ideas and concepts), Review exercise, and Further Exercises. The prompts and questions were intended to direct and guide students' awareness of the fundamentals of doing a problem. • Computer and web aide- by using the Modular Object-Oriented Dynamic Learning Environment (MoodIe) as a course management system, students could access lecture notes, interactive web-based tools, animations, videos, forums module, chat module, journal module, assignments, assessments, survey, and feedback (http://mathed.utm.my/math) [29], [30].
10
with some web-based resources formed the spatial website for the course (see Fig. 3).
tasks, and to post messages and questions on the discussion board. In the control group, students did not use any technology and computer tools and did not have access to online tasks or course content from the course home page. The topic was also taught in 3 meeting hours per week including lecture and tutorial sessions over a period of 14 weeks. The tutorial session was combined as part of the lectures, thus each week the meeting consisted of two 1 hour and 30 minutes class with a mix of lectures and activities. Data for both studies was collected through structured questionnaires (distributed at the end of the course) about students' difficulties encountered in Engineering Mathematics course and the ways of overcoming them based on their perspective. The structured questionnaire included two open ended questions (Question 1 and 3) and a ranking question (Question 2) as follows. 1. What are the greatest difficulties that you have faced so far in Engineering Mathematics? 2. How would you rank the following methods (Table 1) to help your learning in Engineering Mathematics? (1 for the most important . . . 13 for the least important)
.. _ ...... _l_l_tI"O C -:- III ...... __
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oof' �
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.2:!:::1::' -1
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k
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... . r:;J .
... ,(I
..... ,,;L.....,..).I........,. ..... ......... ,!o'''..u.'i .. ''',.,....�jl ....... .J,oII"'' ... �,...,. .. . ... =-'",)............... ........, ....... ...;
-
-kkbr
.
Fig. 3 The e-Iearning section of blended learning [30]
One of the web resources used was an interactive web based tool that was used to help students in the learning of different topics that students had difficulties with based on our preliminary study at UTM [8], [9]. Fig. 4 represents an interactive tool that was used to teach the concept of two variable functions and its graph. Tn addition, a course discussion board was used to foster student-student and student-instructor communication during the course providing a collaborative and shared space for a global community. The textbooks for the control group were the same as with the experimental group; however, these textbooks were used without the corresponding use of any CPS strategies and computer tools in the class. .------[)-- Y�-o-x
show function III show 9rids I!:I
(�
TABLE I THE METHODS OF SUPPORTING STUDENTS' LEARNING Method
Ranking Choosing problem related to the real world or your major Simplified concept Peer teaching (teaching at your level)
Lecturer encouragement Individual homework and assignment Group project
z
•___.�
•
lx.
Average
Choosing relevant topics
Function} assigns to
",h
p,int
Quiz/test/assessment or exam
(x. y) in the
Group work (collaboration) in the class
domam. a reel number z=}('.
fj \
Y)'
Classroom discussion with the peers and lecturer
Peer and lecturer onlineloffline suppons in the outside of class (cliat, email, discussion board.,,)
Y))
Using computer facilities (software, animation, calculator ".) Using online facilities (website, web learning modules, online assessment ".)
(x, y)
E
3. With the exception of the above noted methods, do you have any suggestion to help your learning in Engineering Mathematics? For data analysis, we adopted the qualitative analysis method of Miles and Huberman [33] as the main framework in analysing the data obtained from students' responses. Three stages of this method that were used to analyse the students' responses to the open ended questions are: data reduction, data display, and conclusion drawing used.
Domain of f
Fig. 4 A typical interactive web-based tool [32]
In the experimental group, topics were taught over a period of 14 weeks with 3 meeting hours per week consisting of 2 hours face-to-face and 1 hour laboratory session. Tn t h e lecture session, the mathematical concepts were introduced to the whole class. After the students had established a general idea of the concept, they then proceeded to the laboratory session. Tn the laboratory session, online activities directed students to perform interactive mathematics
V.
RESULTS
Analysing students' responses in the control group revealed that imaging and sketching in 3- dimensions posed the greatest difficulties for the majority of students in the learning of Engineering Mathematics. Most students mentioned that too many concepts/facts/theorems/formulas, memorizing,
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forgetting methods and formulas, complex calculations, and recalling prior knowledge were the reasons of their learning difficulties. Table 2 represents how students ranked the methods that can help them in the learning of Engineering Mathematics. Simplified concept and peer teaching were the highest rated methods and using computer facilities (offline and online) were the least important methods. Furthermore, some methods such as online and offline communication, group work, group project, and even classroom discussion that are important skills in CPS were not given high ranking based on the students' perspectives.
as shown in Table 3. Using computer facilities (offline and online) and peer and lecturer online/offline supports outside of class had the highest ranking and individual homework and assignment and lecturer encouragement were ranked the lowest. Unlike the students' opinion in the control group, CPS skills had the highest ranking in supporting students' learning. TABLE III RANKING METHODS FROM EXPERIMENTAL GROUP STUDENTS Method
Method
Average
Ranking
Simplified concept
4.45
Peer teaching (teaching at your level)
4.63
Individual homework and assignment
4.71
Classroom discussion with the pee rs and lecturer
5.10
Lecturer encouragement
5.24
Choo s ing problem related to the real world or your major
6.10
Choosing relevant topics
6.57
Qui2ftestlassessment or exam
6.85
Group project
8 . 05
Peer and lecturer online/omine supports in the outside of class (chat,
9. 26
3.62
Peer and lecturer onlinelomine supports in the outside of class (chat, email, discussion board . .�
3.89
Group work (collaboration) in the class
4.65
Classroom discussion with the pe ers and lecturer
4.83
Group proj ect
5.16
Peer teaching (teaching at your level)
5.93
Simplified concept
6. 27
Choosing relevant topics
Choos ing p roblem related to the real world or your major
7.47
7.79
Quiz/testlassessment or exam
8.40
Lecturer encouragement
9.79
Individual homework and assignment
10.50
The majority of students suggested that using computer tools is the best way to overcome students' difficulties in visualization. Some students suggested that solving more exercises and examples can help students to overcome their difficulties in the learning of Engineering Mathematics.
email, discussion board ... )
Using online facilities (website, web learning modules, online assessment . .. )
2.16
Using online facilities (website, web learning modules, online
7.49
Group work ( co llaboration) in the class
Using computer facilities (software. animation� calculator. ,. )
Using computer faciliti e s (software, animation, calculator ... ) assessment ...)
TABLE II RANKING METHODS FROM CONTROL GROUP STUDENTS
Average ranking
10.49 11.08
VI. CONCLUSION
Many students suggested that more and different examples, exercises, assignments, and tutorials can help them to overcome their difficulties (with the exception of the above noted methods). Some of them mentioned supporting their learning by solving some problems that are the same as the final exam problems. Many students believed that the imaging and sketching in 3-dimensions represented their greatest difficulties in the learning of Engineering Mathematics. However, a few of them mentioned that online and offline computer facilities can help them to overcome these difficulties. The rest of students did not suggest any alternative methods to support imaging and sketching in 3dimensions. On the other hand, as shown in Table 2, using offline and online computer facilities as the most important ways to support visual thinking scored the lowest ranking among all methods from the perspective of the control group of students. Students' responses in the experimental group indicated that for the majority of them, the lack of basic skills and a solid mathematical background were the greatest difficulties students encountered in the learning of Engineering Mathematics. A few students had difficulties in the learning of specific mathematical topics and these difficulties may also be related to poor a mathematical background. Students in the experimental group ranked the methods that helped them in the learning of Engineering Mathematics
This study investigated the importance of using CPS skills through blended learning environment on students' learning in Engineering Mathematics. The comparison made between two contexts verified that communication, teamwork, and computer tools were the most important methods that support students' learning. Many engineering students in the control group noted that they cannot understand Engineering Mathematics because of too many concepts/ facts/ theorems/ formulas, memorizing, forgetting methods and formulas, complex calculations, and recalling prior knowledge. For these students, imaging and sketching in 3-dimensions were the greatest difficulties that they encountered when doing non-routine problems in Engineering Mathematics. Most students suggested that using computer facilities (offline and online) can support their visualization. However, they ranked the computer facilities (offline and online) as the lowest important methods to support their learning by visual thinking (see Table 2). According to students' responses in the experimental group, most students believed that the lack of basic skills and prior knowledge were the reasons for their difficulties in the learning of Engineering Mathematics. For most students in this group CPS skills and online and offline computer tools were the most important methods to help them in their
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[13] S. G. Isaksen, G. 1. Puccio, and D. 1. Treffinger, "An ecological approach to creativity research: Profiling for creative problem solving," The Journal of Creative Behavior, 27,pp. 149-170,1993.
difficulties. Some students suggested that doing a lot of exercises is also a relevant way to help students. The analysis of experimental students' responses revealed that different thinking skills and tools from CPS such as communication, teamwork, and visualization supported by computer tools were the highest ranking methods to help students in the learning of Engineering Mathematics. These results confirmed that promoting CPS skills through blended learning not only can help students in the learning of Engineering Mathematics but also can support their generic skills such as communication and teamwork.
[15] A. F. Osborn, Applied Imagination: Principles and Procedures of Creative Thinking, New York: Charles Scribner's Sons,1953.
ACKNOWLEDGMENT
[18] N. Herrmann, The Creative Brain, Brain Books, Lake Lure, North California,1988.
The authors acknowledged the Ministry of Higher Education of Malaysia and Universiti Teknologi Malaysia for the financial support via Research University Grant (No: Q.J 130000.7126.03J07) given in making this study possible.
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