Available online at www.sciencedirect.com
Procedia - Social and Behavioral Sciences 56 (2012) 117 – 125
International Conference on Teaching and Learning in Higher Education (ICTLHE 2012) in
conjunction with RCEE & RHED 2012
Supporting Engineering Students’ Thinking and Creative Problem
Solving through Blended Learning
Hamidreza Kashefia, Zaleha Ismaila, Yudariah Mohammad Yusofb *
a
Department of Science and Mathematics Education, Faculty of Education, Universiti Teknologi Malaysia, UTM Skudai, 81310 Johor, Malaysia
b
Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, UTM Skudai, 81310 Johor, Malaysia
_____________________________________________________________________________________________________
Abstract
Creative Problem Solving as a framework to encourage whole-brain thinking which employs different thinking skills and tools is not
sufficiently emphasized in universities. Research findings indicate that for most engineering students, mathematics has always been one of
the most difficult courses to study. Previous researches tried to overcome students’ difficulties in the engineering mathematics by using
some methods based on supporting mathematical thinking. In this paper, we shall discuss and propose a learning environment for supporting
students’ thinking and creative problem solving in engineering mathematics. Blended learning is suggested as an environment to support
students’ thinking powers through creative problem solving.
©
PublishedbybyElsevier
Elsevier
Ltd.
Selection
and/or
peer-review
responsibility
of Centre
of Engineering
Education,
© 2012
2012 Published
Ltd.
Selection
and/or
peer-review
underunder
responsibility
of Centre
of Engineering
Education,
Universiti
Universiti
Teknologi Malaysia
Teknologi Malaysia
Keywords: Blended learning; Creative Problem Solving; Engineering Mathematics; mathematical thinking; mathematical powers
_____________________________________________________________________________________________________________________________
1. Introduction
Current trends in technology and our increasingly complex society and the workplace require engineers have a greater
variety of capabilities, skills, and a wider understanding of engineering as a discipline, if they want to succeed (Pappas, 2002).
Educational and enterprise managers agree that too many engineering students are graduated without having effective
communication and teamwork skills (León de la Barra et al., 1997). According to Lumsdaine & Voitle (1993a), industries also
complain that graduate engineers are technically in competent, they lack critical problem solving skills, communications, team
working, and how to set up criteria to make sound judgment. Unfortunately, the rapid change of technology in the society
does not produce a corresponding change in the training and education of engineers (Lumsdaine & Voitle, 1993a). According
to Lumsdaine & Voitle (1993a), the same material basically is taught with the same tools and methods that have been used
fifty years ago. In other words, the traditional approach with a strong preference in analytical thinking (left-brain) worked well
in the past but does not produce the type of engineering graduates for the future human capital (Lumsdaine & Voitle, 1993a).
The limitations of traditional teaching styles due to the lack of employing of whole brain cause engineering students encounter
many problems in the learning of mathematics which play important role in engineering (Lumsdaine & Voitle, 1993a;
Lumsdaine & Lumsdaine, 1995b).
Mathematics is a prime constituent and infrastructure of the education of engineering students. The main goal of
mathematics learning for engineering students is the ability of applying a wide range of mathematical techniques and skills in
their engineering classes and later in their professional work (Croft & Ward, 2001). Many topics in most engineering curricula
* Corresponding author. Tel.: +6019-7609600; fax: +607-5566162.
E-mail address: yudariah@utm.my
1877-0428 © 2012 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Centre of Engineering Education, Universiti
Teknologi Malaysia
doi:10.1016/j.sbspro.2012.09.638
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Hamidreza Kashefi et al. / Procedia - Social and Behavioral Sciences 56 (2012) 117 – 125
are taught using mathematics and mathematical models. Knowledge of the prerequisite background in mathematics is
therefore necessary for students to learn many areas of study. Research findings indicate that for most engineering students,
mathematics has always been one of the most difficult courses to study. Many students will struggle as they encounter the
non-routine problems that are not solved by routine methods of problem solving.
Creative Problem Solving (CPS) as a framework that encourages whole-brain which employs different thinking skills and
tools can fundamentally improve the way students learn mathematics and support their generic skills such as team work and
communication. CPS can be used to strengthen the productivity, quality of teamwork, thinking and communication skills of
students in whole brain. Some researchers promote CPS in engineering, science, and even mathematics courses (Lumsdaine &
Voitle, 1993a; Wood, 2006; León de la Barra et al., 1997; Lumsdaine & Lumsdaine, 1995b); however, the literature review
indicates that there are very little researches that support students solving Engineering Mathematics problems and
mathematical knowledge construction in a creative manner through CPS. In these studies, researchers try to encourage
whole-brain which employs different thinking skills and tools to support students’ abilities in problem solving by promoting
CPS. In the case of mathematics there are researches that support students’ thinking powers in the learning of mathematics by
promoting mathematical thinking, but very few employ CPS (Mason, Burton & Stacey, 1982; Dubinsky, 1991; Shoenfeld,
1992; Yudariah & Tall, 1999; Gray & Tall, 2001; Tall, 2004; Roselainy, Sabariah & Yudariah, 2007).
In this paper, the theoretical framework for promoting mathematical thinking by using computer is discussed and
theoretical reasons for selecting blended learning to support mathematical thinking in mathematics through CPS are put
forward. A theoretical framework that supports blended learning by integration of the benefits of both face-to-face (F2F) and
computer environment has a rich structure to overcome students’ difficulties in mathematics.
2. Creative Problem Solving
According to Lumsdaine & Lumsdaine (1995b), based on the Herrmann model (1988, 2001) the brain can be visualized as
a four quadrants metaphorical model that are labeled A (mathematical, analytical, critical thinking), B (sequential, controlled,
routine thinking), C (interpersonal, empathetic, symbolic thinking), D (imaginative, visual, conceptual thinking) and each
quadrant is characterized by distinct ways of thinking, knowing, and processing information (see Fig 1). Engineering
education on the average by skewing toward a strong preference in quadrant C thinking has caused many engineering students
and even professors be predominantly left-brain thinkers (Lumsdaine & Lumsdaine, 1995a). This causes when engineering
students are graduated they will encounter many problems in their work place that require different thinking abilities
(Lumsdaine & Lumsdaine, 1995b). So the researches confirm that quadrants C and D activities must be part of the
engineering curriculum (Lumsdaine & Lumsdaine, 1995a).
Fig. 1. The four-quadrant brain model of thinking preferences developed by Herrmann
CPS that employs whole brain of students can play an important role to provide new generation of engineers for human
capital. The roots of CPS are found in Osborn's works (1953, 1963) and it followed by many researchers like Parnes (1967),
Isaksen & Treffinger (1985), Isaksen, Treffinger & Dorval (1994). Lumsdaine & Lumsdaine (1995b) state the CPS as five
distinct steps: (i) Problem Definition, (ii) Idea Generation, (iii) Creative Idea Evaluation, (iv) Idea Judgment, (v) Solution
Implementation and show the relations between these stages and the four-quadrant thinking of brain in Herrmann Model
(1988, 2001). They believe that the process of CPS involves all analytical, creative, and critical thinking and it can be used to
Hamidreza Kashefi et al. / Procedia - Social and Behavioral Sciences 56 (2012) 117 – 125
119
strengthen the quality of teamwork, thinking and communication skills of students in whole brain during of its stages
(Lumsdaine & Lumsdaine, 1995b).
3. Mathematical Thinking
Mathematical thinking is a dynamic process which expands our understanding with highly complex activities, such as
abstracting, specializing, conjecturing, generalizing, reasoning, convincing, deducting, and inducting (Mason, Burton &
Stacey, 1982; Tall, 1991; Yudariah & Roselainy, 2004). Tall in many researches (1986, 1989, 1990, 1993, 1998, 2003) used
an environment to support students’ mathematical thinking (quadrant A from left-brain) to overcome their difficulties in
calculus based on Socratic dialogue between teacher and students (quadrant C from right-brain) which is enhanced by the
addition of the computer facilities like visualization (quadrant D from right-brain). In fact, Tall try to support mathematical
thinking as a mode of quadrant A by using different thinking from other quadrants thinking concerned by visualization
(quadrant D) and communication (quadrant C).
In the earlier study (Yudariah & Roselainy, 2004; Yudariah, Roselainy & Mason, 2007; Roselainy, Sabariah & Yudariah,
2007; and Sabariah, Yudariah & Roselainy, 2008), in developing the mathematical pedagogy for classroom practice, they
adopted the theoretical foundation of Tall (1995) and Gray et al. (1999) and used framework from Mason, Burton & Stacey
(1982) and Watson & Mason (1998). They focused on three major aspects of teaching and learning: the development of
mathematical knowledge construction, mathematical thinking processes, and generic skills (see Fig 2). They highlighted some
strategies that can help students to empower themselves with their own mathematical thinking powers and help them in
construction new mathematical knowledge and soft skills, particularly, communication, team work, and self-directed learning.
Furthermore, the mathematical thinking activities can be taught of as powers were: specializing and generalizing, imagining
and expressing, conjecturing and convincing, organizing and characterizing (Yudariah & Roselainy, 2004; Roselainy,
Sabariah & Yudariah, 2007).
Competency in
procedures &
techniques
Construction of
mathematical
knowledge
KNOWLEDGE
DEVELOPMENT
Processes:
specializing,
generalizing,
sorting,
justifying,
reasoning, …
Structures:
facts,
properties,
representation,
notation, …
THINKING
FOCUS OF
LEARNING
GENERIC SKILLS
Communication
Self-directed
learning
Teamwork
Fig. 2. Focus of mathematical learning
Roselainy, Sabariah & Yudariah (2007) had developed and implemented their model of active learning in the teaching of
Engineering Mathematics at UTM. They considered the following aspects in the implementation of active learning in
Engineering Mathematics classroom (Roselainy, Sabariah & Yudariah, 2007; and Sabariah, Yudariah & Roselainy, 2008).
Classroom tasks- by categorizing book as Illustrations, Structured Examples and Reflection with Prompts and Questions.
Classroom activities (approaches)- by working in pairs, small group, quick feedback, students’ own examples,
assignments, discuss and share, reading and writing.
Encouraging communication- by designing prompts and questions to initiate mathematical communication.
Supporting self-directed learning- by creating structured questions to strengthen the students’ understanding of
mathematical concepts and techniques.
Identifying types of assessment- by incorporating both summative and formative types.
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Fig 3 gives a summary of their model for active learning (Roselainy, Sabariah & Yudariah, 2007; and Sabariah, Yudariah
& Roselainy, 2008).
Fig. 3. Model of active learning
In other words, they had provided and promoted a learning environment where the mathematical powers are used
specifically and explicitly, towards supporting students (i) to become more aware of the mathematics structures being learned,
(ii) to recognize and use their mathematical thinking powers, and (iii) to modify their mathematical learning behavior
(Yudariah & Roselainy, 2004; Roselainy, Sabariah & Yudariah, 2007; and Sabariah, Yudariah & Roselainy, 2008). Their
model of active learning environment involves components that are approximately from whole brain such as communication
and discussion; however, they did not invoke strong tools to support them. Moreover, in this method is not used the potentials
of other thinking like visual thinking by using computer facilities.
It seems that each methods of supporting students’ thinking powers to overcome their difficulties in mathematics did not
use all potentials of whole brain. Then we need a learning environment that not only has the benefits of both Tall and
Roselainy & her colleagues methods but also uses different activities from four quadrants of whole brain.
4. Blended Learning
There are many definitions of blended learning in the literature review; however, the term is still vague (Oliver &
Trigwell, 2005; Graham, 2006; Hisham Dzakiria et al., 2006). The three common definitions of blended learning are: the
integrated combination of instructional delivery media, the combination of various pedagogical approaches, and the
combination of online and F2F instruction (Oliver & Trigwell, 2005; Graham, 2006; Huang, Ma & Zhang, 2008). The third
definition indicate that the blended learning is an opportunity to use synchronous and asynchronous e-learning tools including
chat rooms, discussion groups, podcasts and self-assessment tools to support traditional teaching methods including lectures,
in-person discussions, seminars, or tutorial (Reay, 2001; Thorne, 2003; Graham & Valsamidis, 2006; Allan, 2007).
Blended learning has gained considerable interest in recent years as an environment for supporting learning and teaching
of mathematics (Iozzi & Osimo, 2004; Groen & Carmody, 2005; Harding et al., 2006; Hinch, 2007; Sojka & Plch, 2008).
Carman (2002) noted that five important elements of blended learning process are:
(i) Live Events: Synchronous, all learners participate at the same time in a live virtual classroom or traditional F2F classroom
as instructor-led learning events.
(ii) Online Content: Learning experiences that the learner completes individually such as interactive, Internet-based or CDROM training.
(iii) Collaboration: Environments in which learners communicate with peers and lecturer by: e-mail, threaded discussions, and
online chat.
(iv) Assessment: A measure of learners’ knowledge. For determining prior knowledge pre-assessments can come before live
or self-paced events, and also post-assessments can occur following scheduled or online learning events to measure
learning transfer.
(v) Reference Materials: Job aids materials that enhance learning retention and transfer, including PDA downloads, and
PDFs.
5. Discussion and Conclusion
According to the theory of three modes of representation of human knowledge (Bruner, 1966), enactive, iconic and
symbolic are the three forms of representation in mathematics. Furthermore, the various forms of symbolic representation also
are: verbal (language, description), formal (logic, definition), and proceptual (numeric, algebraic etc) (Tall, 1995). This
Hamidreza Kashefi et al. / Procedia - Social and Behavioral Sciences 56 (2012) 117 – 125
121
representation leads to the idea that there are not only three distinct types of mathematics worlds; there are actually three
significantly different worlds of mathematical thinking as: conceptual-embodied, proceptual-symbolic, axiomatic-formal (Fig
4) (Tall, 2003, 2004, 2007).
Fig. 4. The relation between three Bruner’s modes and three worlds of mathematical thinking
On the other hand, the theory of Skemp (1979) identifies three modes of building and testing conceptual structures as
shown in Table 1 (Tall, 1989, 1993).
Table 1 Reality construction
According to Skemp (Tall, 1989, 1993), pure mathematics relies on Mode 2 and 3, but it is not at all based only on Mode 1
(Tall, 1986). On the other hand, computer environment brings a new refinement to the theory of Skemp (Tall, 1986) and Tall
(1989) extended this theory to four modes: Inanimate, Cybernetic, Interpersonal, and Personal. The last of these corresponds
to Skemp’s Mode 3.The interpersonal mode of building and testing concept also corresponds to Skemp’s Mode 2, whilst the
first two are a modification of Skemp’s Mode 1 (Tall, 1989, 1993). In fact, the computer provide an environment and that
give us a new way for building and testing mathematical concept by supporting all these modes. Therefore, computer
environment can be used in all these modes and learner also may build mathematical concepts by considering examples (and
non-examples) of process in interaction with this environment especially in embodiment world (Tall, 1986).
In other words, computer environment provides not only a numeric computation and graphical representation; it also
allows manipulation of objects by an enactive interface (Tall, 1986) that by using them we can improve students’ difficulties
in embodiment world. To achieve these goals Tall (1989) defined generic organiser as an environment to build an embodied
approach to mathematics. However, the generic organiser does not guarantee the understanding of the concept and there were
some cognitive obstacles that aroused using generic organiser by students. To overcome these obstacles Tall (1986)
suggested that teachers can play a role as an organizing agent. Teachers as organizing agent do not have a directive role and
they only answer questions which may arise in the course of the student investigations through a Socratic dialogue with them
(Skemp’s Mode 2) which is enhanced by the presence of computers (see Fig 5) (Tall, 1986).
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Fig. 5. The relation between the theories of Bruner, Tall, and Skemp to promote mathematical thinking by using computer and teacher
Defining of blended learning as the combining synchronous physical formats (such as instructor-led classrooms and
lectures) and self-pased as asynchronous formats (such as online or offline learning) identifies an environment including two
important components of Tall’s method that are organizing agent (teacher) and computer. In fact, this environment has rich
facilities to extend of Tall’s approach for using of computer to promoting mathematical thinking. So this environment has also
potential to use some relevant strategies in F2F engineering mathematics through mathematical thinking approach.
On the other hand, Fahlberg-Stojanovska & Stojanovski (2007) noted that the best learning can takes place when all three
primary senses of seeing (visual), hearing (audio) and doing (enactive) are involved in an interactive environment. They
linked between these senses and two components of blended learning as the following (Fig 6):
Fig. 6. The relation between three primary senses and blended learning
Therefore, due to the relation between Bruner’s modes and primary senses on one hand we can see a link between the
components of blended learning and Bruner’s theory and also the relation between primary senses and blended learning on the
other hand (Fig 7).
Fig. 7. The relation between three Bruner’s modes and blended learning through primary senses
Fig 8 shows the relation between mathematical thinking and blended learning through Bruner and Skemp theories.
Hamidreza Kashefi et al. / Procedia - Social and Behavioral Sciences 56 (2012) 117 – 125
123
Fig. 8. The relation between Mathematical thinking and blended learning
Lumsdaine & Lumsdaine (1995b) explained how five stages of CPS from their approach employ other thinking especially
mathematical thinking and some components of four quadrants of brain based on Herrmann model (1988, 2001) such as team
work and communication. On the other hand, computer as the best analogy of the functioning of the human brain can be used
at least in four distinctly different ways: database and data processor (calculator), teaching machine, communication tool,
simulator and visualizer (graphics) (Lumsdaine & Lumsdaine, 1995b). In the context of teaching, learning, and thinking the
four different ways of using computers have relations in order with four quadrants of brain A, B, C, and D. Lumsdaine &
Lumsdaine (1995b) also explained that how computer facilities are used during the process of solving problems in CPS based
on their approach. Then, blended learning as an environment that has online and offline tools such as software, email, chat
room, and bulletin boards can help some components of four quadrants of brain such as visualization and communication for
better supporting of mathematical thinking through CPS.
The following chart (Fig 9) is a whole picture of a framework perspective that identifies blended learning is a relevant
environment to support students’ mathematical thinking powers and generic skills such as communication in mathematics
through CPS.
Fig. 9 The relation between blended learning, mathematical thinking, and CPS
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Hamidreza Kashefi et al. / Procedia - Social and Behavioral Sciences 56 (2012) 117 – 125
Acknowledgments
The authors acknowledge Universiti Teknologi Malaysia for the financial support given in making this study possible.
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