Grounded Theory Approach:
Conceptions of Understanding in Engineering Mathematics
Learning
Henry Khiat
Singapore Polytechnic, Singapore
Mathematics is of utmost importance in engineering courses but studies on
engineering students’ conceptions of understanding in mathematics
learning are found lacking in the literature. Therefore, this research
attempts to address the above issue by answering the research question:
“What are engineering students’ conceptions of understanding in
mathematics learning?” It employs the grounded theory methodology
(Strauss & Corbin, 1990, 1998) and data are collected from in depth
interviews with a total of 21 students and six lecturers. The substantive
theory of engineering mathematics understanding (comprising of
conceptual, functional, procedural, disciplinary and associational
understanding) emerges in this study. The emergence of functional,
disciplinary and associational understanding is unique in the context of
engineering mathematics learning and has implications on successful
engineering problem solving. Key Words: Engineering Mathematics,
Conceptual Understanding, Functional Understanding, Procedural
Understanding, Disciplinary Understanding, Associational
Understanding, and Grounded Theory
Introduction
Students’ conceptions of understanding in mathematics can change as their
learning contexts change (Kloosterman, 2002). Students may not have the same
conceptions of understanding in mathematics learning when they are studying primary,
secondary, or tertiary mathematics. At the tertiary level, such change of learning context
might include specialization in domain areas of mathematics learning such as engineering,
business or social sciences. This emphasis of the use of mathematics in specific domains
is different as compared to mathematics learning at pre-tertiary levels, where a more
general form of mathematics is learnt. The change of context in the uses of mathematics
can possibly influence students’ conceptions of understanding in mathematics learning.
In this study, the context of engineering mathematics learning in a Singapore
polytechnic is set. Engineering depends significantly on the mastery of mathematics.
Engineering students understand that mathematics is the foundation of engineering
(Berlinghoff & Gouvea, 2004; De Lange, 1996; Graves, 2005; Peterson, 1996; Putnam,
1986; Stanic, 1989; Steen, 1988). Students’ academic success in engineering courses
relies greatly on their ability in mathematics as many engineering concepts could not be
actualized without mathematics. Studies such as Bell (1993), Canobi (2005), Hiebert and
Carpenter (1992), Mason and Spence (1999) and Yager (1991) showed that students’
1460 The Qualitative Report November 2010
conceptions of understanding mathematics are important in their success in mathematics
learning. However, these studies are conducted in the context of general mathematics
learning and related studies on conceptions of understanding of engineering mathematics
learning are found lacking. Notwithstanding that, there may be a possibility that
conceptions of understanding in general mathematics and engineering mathematics may
differ. Some more, if engineering students are not able to learn mathematics effectively
and mathematics lecturers are not able to teach them competently because of the lack of
understanding of their perspectives in studying engineering mathematics, the quality of
engineers produced in the society may be compromised. Therefore, it is imperative that
the issue of the conceptions of understanding in engineering mathematics learning is
looked into.
To summarise, the research problem here is the lack of understanding of the
students’ conceptions of understanding in engineering mathematics that might give rise to
the possible consequences of students’ ineffective learning or/and lecturers’ incompetent
teaching and the eventual compromising of the quality of future engineers in Singapore.
Thus, this study would centre on understanding the students’ conceptions of
understanding in engineering mathematics as part of their engineering diploma courses in
a polytechnic in Singapore.
Purpose of the Study
The purpose of this grounded theory study is to uncover a central theory
explaining conceptions of understanding in mathematics learning, as perceived by the
engineering students and lecturers. In this regard, this study aims to answer the research
question: “What are engineering students’ conceptions of understanding in mathematics
learning?” This research question would be answered in the context of the conditions
below, as experienced by the participants:
1. The conceptions of understanding in engineering mathematics are
understood from the perspectives of engineering students and lecturers
in this study.
2. These students are currently enrolled in diploma level engineering
courses in a Singapore polytechnic and the lecturers are engineering
mathematics teachers.
A clear and in-depth understanding of the engineering students’ conceptions of
understanding in mathematics learning is important to the students themselves, their
mathematics lecturers, and engineering, and other related professions reliant on
mathematics. From the teaching viewpoint, the mathematics lecturers may be able to
adjust their instructional strategies accordingly to ensure their students learn effectively.
At the same time, the students can also alter their learning strategies usefully through
understanding their own, and others’ experiences in engineering mathematics learning.
More importantly, it seeks to understand the study characteristics of future engineers,
who form an important part of the economy. Lastly, this study contributes to the lack of
literature in engineering mathematics learning.
Henry Khiat 1461
Literature Review
Students seem to hold a variety of views of mathematics in the classroom. Across
the academic levels, elementary students see that effort, regardless of ability, is the key to
learning mathematics, but when they advance to high school level; they see the lack of
ability as a significant impediment in mathematics learning (Kloosterman & Cougan,
1994). In the classroom, some students may believe that a good grade is important in
mathematics assessment, while others may not (Hurn, 1985). Some students view
mathematics learning as interesting, others may believe that it is a form of tedious and
monotonous work (Cooney, 1992; Cotton, 1993). Others may even see mathematics as a
subject that causes them negative emotions such as fear, anxiety and anger during lesson
(Hoyles, 1982). Some students feel that they learn mathematics because of their intrinsic
interest in it (Kloosterman, 2002). At the same time, some students may view
mathematics learning as being forced on them by schools and teachers (Ainley, Bills, &
Wilson, 2005; Cotton, 1993). These students may possibly feel that they do not
understand the “purpose” of the mathematics tasks assigned to them and thus see no
meaning in doing these exercises (Ainley & Pratt, 2002). Students may also feel that
some of the so called real world contexts used by teachers to relate to mathematics
concepts may not be interesting to them and even create confusion in their problem
solving (Ainley, 2000; Silverman, Winograd, & Strohauer, 1992). In terms of learning
mathematics effectively, Kloosterman reported that students view procedures as more
important than concepts. They also feel that memorisation is an important part of
mathematics learning (Kloosterman). In relation to social influences in mathematics
learning, some students may believe that teachers make learning mathematics difficult to
understand and give little guidance to their mathematics learning (Kloosterman). On the
other hand, other students may view mathematics as a subject where failure to achieve
the right answers is usually met with disapproval and criticism by their teachers (Ernest,
2004). At the same time, some students may recognize that peers can help or impede
them in their mathematics learning (Perret-Clermont & Schubauer-Leoni, 1988;
Sternberg & Wagner, 1994; Zimmer & Toma, 2000). The various studies above showed
that students can hold a huge diversity of views about mathematics learning at personal,
social, conceptual, procedural, cognitive and emotional levels. Such views might
possibly influence how well they are able to do mathematics.
Attempts to relate the engineering mathematics learning to the above studies
might not be useful for two reasons. First, most of these views are only representative of
primary and secondary mathematics students and do not fully characterize those of
tertiary students studying engineering mathematics. Second, all these studies are not
conducted in the context of Singapore where this research is carried out. Thus, it may be
shown there is a lack of such research in Singapore with regard to understanding
engineering mathematics learning. This further emphasizes the importance of this study
in allowing practitioners (such as engineering students, teachers, researchers and policy
makers) to understand the perspectives of students in engineering mathematics learning.
Students’ success in mathematics learning can be influenced at personal, social,
conceptual, procedural, cognitive and emotional levels as the above studies have shown.
However, this study would focus on the level of conceptual understanding in engineering
mathematics learning as the meaning of understanding in general mathematics learning
1462 The Qualitative Report November 2010
has been extensively discussed in the literature (Canobi, 2005; Hiebert & Carpenter, 1992;
Kieran, 1992; Mason & Spence, 1999; Munakata, McClelland, Johnson, & Siegler, 1997;
Rittle Johnson, Siegler, & Alibali, 2001; Sierpinska, 1994; Skemp, 1978; Stigler &
Hiebert, 2004; Vincent & Stacey, 2008; Vinner, 1992; Yager, 1991). Besides, these
studies also suggested possible relationships between students’ understanding in
mathematics learning and their level of success in mathematics learning.
In one of the earlier studies that explored the meaning of understanding in
mathematics learning, Skemp (1978) first distinguished mathematics learning into
conceptual/relational understanding and procedural/instrumental understanding. He
believed that conceptual/relational understanding is achieved if a student comprehends
the underlying principles that create a particular formula or theorem and its relationship
with the other formulae or theorems. On the other hand, procedural/instrumental
understanding in mathematics is achieved as long as a student can execute the rules of a
particular formula in completing a mathematical task. In simpler terms, Mason and
Spence (1999) distinguished conceptual and procedural understanding in mathematics as
“knowing about” and “knowing to”. As for Stigler and Hiebert (2004), they perceived
understanding in mathematics in terms of concepts, connections and computational skills.
Sierpinska (1994) suggested three ways in comprehending the meaning of understanding
in mathematics – acts of understanding, the understanding that occurs from the acts of
understanding and the processes of understanding. These three ways are actually
processes of building, having and using mathematical knowledge respectively. Shaw and
Shaw (1999) provided an interesting description of the types of engineering students
learning mathematics. They classified them into five groups: Ambivalent with poor pre
university teaching, Downhillers, Haters, Ambivalent with good pre university teaching
and High fliers. These students perceive engineering mathematics differently in terms of
its difficulty, the helpfulness of the teachers and lecturers, their level of liking and
enjoyment of it, motivation in studying it and keenness in improving their mathematical
capability (Shaw & Shaw). However, the study did not investigate on the types of
understanding they achieve in engineering mathematics learning. A further search of
research and thesis database of the only teacher training institution (National Institute of
Education) and its subsidiary research centre (Centre for Research in Pedagogy and
Practice) where most local educational research is done, also yields no similar studies in
the local context. Since these studies are not conducted in the context of engineering
mathematics learning or investigate on conceptions of understanding in engineering
mathematics, this study can shed some light on how engineering students perceive
understanding in mathematics.
Rittle Johnson et al. (2001) believed that a student’s acquisition of conceptual
understanding in mathematics can influence their level of acquisition of mathematical
understanding at procedural level, or vice versa. On the other hand, Canobi (2005) also
reported there is a relationship between the students’ conceptual understanding of
mathematics and their perceived proficiency of related mathematical procedures in
problem solving. According to Vincent and Stacey (2008), having procedural
competency in mathematics learning does not help to enhance conceptual understanding
in learning mathematics. Kieran (1992) reported that students who failed to understand a
mathematics formula conceptually would rely on memorising the procedures and thus
could not see the essence of that formula. It would be interesting to investigate: If
Henry Khiat 1463
conceptual and procedural understandings do exist in engineering mathematics learning,
what would be their relationship?
Yager (1991) saw the potential of improving students’ conceptual understanding
by teaching them using problem solving contexts. The formation of conceptual contexts
in which mathematics learner can use the mathematics learned in different working
contexts is mentioned in Hiebert and Carpenter (1992). If the learners are able to apply
the mathematical concepts in as many as contexts as possible, their conceptual
understanding of the learned mathematics would be sounder (Hiebert & Carpenter;
Vinner, 1992). The above conclusion is supported by Bell (1993) who proposed that the
lack of connection between conceptual and procedural understanding in terms of real
world context, can be an issue in mathematics learning. Munakata et al. (1997) proposed
that the ability to present concepts correctly is context dependent. The study would also
help to investigate whether understanding in engineering mathematics learning is context
dependent.
With respect to the fact that mathematics is indispensable in engineering
education (De Lange, 1996;; Peterson, 1996; Putnam, 1986; Stanic, 1989; Steen, 1988), it
would be timely to understand how engineering students perceive the concepts of
understanding in mathematics learning and their possible relationships. This study can
help to fill up the relevant gaps in the literature.
Role of the Researcher
Researcher’s Context
It is the polytechnic context of engineering courses in which I work as a
mathematics lecturer that provides the background for this research. The unique
relationship between mathematics and engineering provides the foundation for this study.
Mathematics is important to engineers as it is “a tree of knowledge: formulae, theorems,
and results hang like ripe fruits to be plucked” (Steen, 1988, p. 611) or “a well stocked
and vital warehouse” (Peterson, 1996, p. 1) where the formulae, theorems and results are
at the disposal for their uses in their work solving engineering problems. As a lecturer
teaching mathematics to engineering students in one of the five polytechnics in
Singapore, I observe that not all engineering students do well in engineering
mathematics. And the common comments given by students to explain their bad
performance in engineering mathematics are related to the concept “Understanding”.
(Such as “I do not understand”; “I don’t know how to do”, “What is the use of the
formulae?; “I cannot see any meaning in the formulae”, etc.). Therefore, I decided to
conduct this research to explore the concept of “Understanding” in engineering
mathematics learning and its relationship to successful engineering mathematics learning.
Limitations
As I am a lecturer in the department in which students are the participants for this
study, I am able to observe the different ways they approach engineering mathematics
learning. In other words, I have prior views on the subject. I have my own perspectives
about how engineering students think and feel about mathematics learning and, their
1464 The Qualitative Report November 2010
subsequent behaviours towards it due to my proximity to the participants in my position
as a lecturer. Such prior perspectives can bias data collection and analysis. At the same
time, due to our lecturer-student relationship, some of the students are more apprehensive
and reserved in sharing with me about their experiences in mathematics learning. I
believe our unique lecturer-student relationship might have created a power imbalance
between us. Therefore, the participants may not want to divulge honest information to
me in fear of affecting their academic progress or offending me. Such power imbalance
may bias the data collection and analysis stages too. In conclusion, as an insider
researcher in this study, I have to acknowledge the methodological limitations caused by
my preconceived conceptions about the participants and the power imbalance between us.
While recognising such methodological limitations due to my proximity to the
participants could not be totally eliminated, I took steps to minimise them so as to
improve the rigor of the findings.
Insider research is very useful for teachers in exploring and developing their own
practices in mathematics education (Jaworski, 2004; Jaworski & Goodchild, 2006).
However, there are some research limitations that come with it. These limitations need to
be tackled effectively to improve the rigor of this study. The first important limitation of
insider research that was addressed here is the pre-understanding/familiarity of the
researcher in the research setting (Brannick & Coghlan, 2007; Mercer, 2007). As I have
been teaching in this polytechnic for a number of years, I have experience in dealing with
different types of mathematics students and their behaviours. This is beneficial as it
makes me more theoretically sensitive and I am also able to elucidate the meanings of
data analysed more effectively. However, such pre- understanding/familiarity can create
preconceptions about the researched phenomenon and this can bias the study (Brannick &
Coghlan; Mercer). Such bias can be created in the data collection and analysis stages
through the researcher’s failure to probe areas outside their scope of research knowledge,
unsubstantiated assumptions of the types of data to be collected and unwillingness to
reframe their current thinking in data analysis (Brannick & Coghlan; Mercer).
To reduce such limitation, I was consciously aware of my beliefs, values and
possible biases due to my familiarity with, and experiences in, the research context in the
data collection and analysis processes by recording them in my memos. By being
constantly reflexive, I maintained a self awareness of such pre-understanding bias by
continually appraising the subjective responses and intersubjective relationships within
the data analysed in relation to my potentially biasing experience (Brannick & Coghlan,
2007; Finlay, 2002; Nightingale & Cromby, 1999).
The next limitation discussed here is the power imbalance between the insider
researcher and the participants. As a lecturer in the same polytechnic as the students, I
have ready access to potential student participants. Besides, I am also able to build
rapport with them more easily (Mercer, 2007). However, disadvantages in terms of
power and trust relations between the participants and the researcher may be created as a
result of such proximity (Mercer; Wengraf, 2001). Power imbalance between them may
result in the reluctance of the participants to divulge important information or to question
the validity of the research and its analytic framework and results (Hall & Callery, 2001).
This is because they may not want to offend the researcher and/or affect their academic
progress due to the researcher’s higher authority in academic settings. Thus, the
participants may not trust the researcher with information that they feel may disadvantage
Henry Khiat 1465
them. As a result, some participants may not want to divulge information that is against
the social norms and may instead give socially desirable responses. This might well
greatly affect the rigor of the research. Therefore, to remove such limitations caused by
power and trust relations, I ensured that the participants were not my former students, nor
would they be my future students. This helped prevent any bias due to role duality (the
situation where I teach and research on the same set of students) in the research process
(Brannick & Coghlan, 2007). At the same time, I shared my personal and professional
values with the participants so as to gain their trust, which in turn, enabled more honest
disclosures from them. They were also assured that the research would not affect them
academically or personally in any way. Such honest communication is known as the
processes of relationality and reciprocity (Mercer; Hall & Callery).
Methodology
The interpretive research paradigm can answer the research question posed in this
study more appropriately as compared to the positivist research paradigm as the questions
aim to understand the students’ perspectives of studying engineering mathematics within
a context instead of seeking causal relationships among variables related to it (Cohen,
Manion, & Morrison, 2000; Donmoyer, 2001; Shaughnessy & Zechmeister, 1997).
Besides, few if any study of students’ perspectives of studying engineering mathematics
has been conducted in Singapore and the inductive nature of the interpretive paradigm is
best suited for such studies with no or few existing theories (Cohen et al.; Nassar, 2001).
There is a wide selection of approaches in the interpretive research paradigm to
use in this study (such as grounded theory, ethnomethodology, life history, inductive
analysis, phenomethodology and case study). In the selection of any research
methodology in a study, Field and Morse (1985) highlighted the importance of its
methodological relevance to the research problem and the congruence of the researcher’s
epistemological focus with it. On the other hand, Nassar (2001) emphasised the type of
training a researcher has and his/her research experience also influence the choice of the
research methodology in a study.
This study attempts to investigate the experiences of students studying
engineering mathematics through understanding their thoughts, feelings and
actions/interactions. Therefore, the symbolic interactionist aspect of grounded theory
which assumes that one’s communications and behaviours express meanings as
influenced by social conditions and interactions around them, suits it well (Blumer, 1969;
Grbich, 1999; Mead, 1964). In short, grounded theory allows processes, meanings and
relationships in the phenomenon of studying engineering mathematics to be investigated.
On the other hand, there are few if any theories that are associated with the students’
experiences of learning engineering mathematics. As grounded theory is known for
building theories inductively from data in under-researched areas, it is suitable for the
exploratory nature of this study (Grbich). Besides, as a researcher who prefers to work
within clearly determined processes, grounded theory suits me as it is more rigorously
systematic in its data collection and analytical procedures. In addition, I have always
been more familiar and confident with grounded theory methodology due to my
postgraduate training in it. Thus, in line with Field and Morse (1985) and Nassar (2001),
I have chosen to use the grounded theory methodology in this study.
1466 The Qualitative Report November 2010
As grounded theory is popularised, differences in its methodological assumptions
and methods arise. There are different variations of grounded theory methodology 40
years after its birth (Mills, Bonner, & Francis, 2006). According to Creswell (1998),
there are three main groups of grounded theorists – Glaser (1992, 1998), Strauss and
Corbin (1990, 1998) and the social constructivists such as Charmaz (1990, 2006) and
Annells (1996). However, it is not feasible to discuss their differences here due to space
constraint. Nevertheless, notwithstanding the differences in the methodological
assumptions and methods among the different schools of grounded theory, its
philosophical roots and main canons remain the same (Creswell, 1998). In this study, I
used the methods proposed by Strauss and Corbin (1990, 1998) because its detailed and
systematic methods are more user friendly to me.
Ethical Issues
Informed process consent, confidentiality and emotional protection were ethical
issues addressed in this research (Kvale, 1996).
Informed consent. Before the start of the interview process, there is a need to let
the participants know the purposes and implications of the research (Bogdan & Biklen,
2003; Kvale, 1996). Therefore from there, it is only possible to get the informed consent
of the participants. Evans and Jakupec (1996) and Kvale perceived informed consent as
one of the most critical issues in interpretive research although there are others such as
Fine (1992) who felt that covert research is acceptable as long as the participants are not
harmed. I do not agree with Fine as it is the rights of the participants to be in the know.
There is a need to confirm informed consent during the different stages of the research
(Bartunek & Louis, 1996). As the events unfold during different stages of research, the
participants need to be consulted for consent if it is deemed that any new development
may harm them. Lastly, participants must be given ample opportunities to withdraw
from the research proper without any explanation.
In this study, the participants were informed of the research’s purposes and
implications. They were then asked to consent to the study without coercion. They were
given one week to consider the request. During the interview process, they were allowed
to terminate it anytime without reasons. At the same time, I ensured that the content and
analysis of the interview transcript of a participant go through him/her for checking
before being released to others. If the participant did not agree to release the analysis for
any reason (his/her reason would not be asked), his/her decision would be respected
unconditionally. In this study, no participants terminated the interviews halfway or
objected to my analysis in the report.
Confidentiality. The confidentiality of the participants must always be protected
(Bogdan & Biklen, 2003; Fraenkel, 1990; Kvale, 1996). In this study, no participant was
identified without their approval. The interview transcripts were returned to the
participants and the subsequent analysis report were sent to them for review in case that
any part of the analysis might identify them (Rowling, 1994). This protects the
participants from any form of embarrassment or harm (Bogdan & Biklen). However,
Berg (1998) stated that maintaining the anonymity of the participants may not be able to
Henry Khiat 1467
protect the confidentiality of their identities if other information provided may identify
them indirectly. Therefore, I took care in not revealing other information that may
divulge their identities in my report. In this aspect, I had given pseudonyms to the
participants’ school, courses, names and gender. Their tutors were not identified too.
Emotional protection. The researcher has to be aware about the issue of
sensitivity when dealing with emotional research questions (Bogdan & Biklen, 2003). In
short, the interview process must not cause any emotional distress to the participants.
The researcher must actively watch out for signs of such emotional distress and terminate
the interview when necessary. The nature of this study will not cause any physical harm
to the participants directly from the interview process or the release of the research
findings. However, it may elicit memory of bad experiences of mathematics learning.
Therefore, in this study, I constantly watched for such emotional distress from the
participants and would terminate the interview sessions when necessary. However, there
was no instance where I needed to terminate the session due to the interviewee’s
emotional stress.
Context of Study
In the case Polytechnic, all students, most of whom, are aged between 17 and 20
years old, undergo two years (two semesters per year) of compulsory engineering
mathematics learning (comprising of four modules) as part of their engineering courses.
They learn mathematics concepts that include algebra, trigonometry, complex numbers,
calculus, Laplace transforms, statistics, methods of integration, differential equations,
infinite series and Fourier series. Then the students need to apply all these mathematics
concepts extensively in their other compulsory engineering modules required in the
completion of their engineering diploma courses. At the same time, these mathematics
concepts tend to be more difficult for the students to master than those they learnt in
secondary mathematics education, due to their mathematical complexity and relationships
to engineering. Also, the focus of engineering mathematics education centres on skills
acquisition in relation to engineering theories. Such a unilateral approach to the
engineering mathematics curricula can influence the students’ perspectives of
engineering mathematics learning.
Data Collection Methods
This section will detail the various stages of data collection in this study.
Setting and participants. This research was conducted in a polytechnic in
Singapore. As there are many types of engineering courses (with different mathematics
syllabi) in this polytechnic, any attempt to investigate all courses is futile due to the
possible extensive range of experiences that cannot be adequately covered within the
confines of time and space of this study. Thus, the sample for this study was drawn from
selected courses in the School of Electrical and Electronics Engineering. The sampled
students in the four diploma courses – Electrical Engineering, Aerospace Engineering,
Bioelectronics Engineering and Communication Engineering, needed to take a centralised
1468 The Qualitative Report November 2010
programme (that consists of three modules) in engineering mathematics. Although the
students’ experiences of mathematics learning in other engineering disciplines are not
covered in this study, they should serve as possible areas of inquiry in forming more
substantive theories in future.
Potential participants were first briefed on the purpose of the study - to understand
their experiences of learning engineering mathematics. They were also told that the
consequences of the research could improve teaching and learning of engineering
mathematics and at no time, their grades or performance in their study here would be
affected by it. They were then informed that the interviews would be confidential and
they were allowed to discontinue at any point of the interview without giving any reason.
These measures thus ensure the ethics of interview research (informed consent,
confidentiality and its consequences on the participants) are adhered to (Kvale, 1996).
Each participant was given a week to consider if he/she wanted to take part in this study.
A total of 24 students were invited to participate in this study and 21 students agreed to
take part. The three students who did not wish to participate in the study cited that they
did not have the time for the interviews.
The interviews of these 21 students served as the main source of data for this
study. Data were collected through interviews that centred on the four fragmented
specific research questions for this study. These questions were based on the concepts
derived from the literature review and my personal experience relevant to the
phenomenon studied. However, as these concepts did not evolve from the data, they
were considered as provisional and could be discarded if they were not theoretical
relevant eventually (Strauss & Corbin, 1998). During the initial open sampling stage, a
semi structured interview guide that focused on the research question was used. The
initial interview guide changed four times for the purpose of theoretical sampling (Strauss
& Corbin, 1998).
All interviews were conducted in the privacy of soundproof classrooms or library
special rooms. Throughout the interviews, I conscientiously attempted to improve my
interviewing skills by following the desirable qualities (knowledgeable, structuring, clear,
gentle, sensitive, open, steering, critical, remembering and interpreting) of an effective
interviewer as suggested by Kvale (1996). A digital voice recorder was used in recording
each interview. Each interview lasted from 15 minutes to 35 minutes. There was a total
of 499 minutes of interviews in all. All interviews were recorded, transcribed verbatim
and stored in a computer database. There were also five follow up interviews which were
done through email correspondence. They were conducted to clarify and validate certain
categories and their relationships. Email correspondence was used as it was considered
more efficient than fixing another face to face interview with them.
Another source of data was obtained through the informal conversational sessions
(each lasting less than ten minutes) with the 21 interviewed students’ tutors. Two tutors
who were approached in the study did not agree to be involved in the sessions. I did not
ask them their reasons of declining the interviews. In the end, a total of six tutors (three
of them were the ones who recommended me the interviewees) were involved in these
sessions. Some of them were either the current or former tutors to a few of the
interviewed students. Questions asked in such sessions were essentially related to their
perceptions of their students’ attitudes in engineering mathematics learning. These
sessions were not audio-taped as four of them were uncomfortable about the recording
Henry Khiat 1469
but notes were taken during the interviews. However, this should not affect the rigor of
this study as the information obtained was considered complementary since this study
focuses primarily on the perspectives of the students. Thus, this source of data from the
tutors was mainly used to triangulate the main sources of data collected through the
students’ interviews.
Sampling method. In grounded theory, sampling is conducted according to the
principle of theoretical sampling where sampling choices are dictated by the categories of
the emerging theory. In theoretical sampling, the researcher jointly collects codes and
analyses his/her data. He/she then further decides what the data to be collected next are
and where to find them, so as to develop the theory as it emerges (Glaser & Strauss,
1967; Strauss & Corbin, 1998). Thus, the function of theoretical sampling is to aid in the
selection of participants who will yield data that produce categories of a phenomenon
until no new categories are found. At the same time, it also helps to develop, elaborate
and refine existing categories through searching their other uncovered properties and
dimensions, until none is found. In other words, the researcher is “sampling along the
lines of properties and dimensions, varying the conditions” (Strauss & Corbin, 1998, p.
202). Therefore, theoretical sampling is not searching for repeatedly occurring properties
or dimensions of a category (Charmaz, 2006). In short, theoretical sampling increases the
range of variation and analytic density of the categories and puts them on firmer and
stronger theoretical grounds (Strauss & Corbin, 1998). In this study, open sampling was
utilised (Strauss & Corbin, 1998).
Open sampling. At the start of any study, open sampling has to be employed as
no theoretically relevant categories are as yet uncovered (Strauss & Corbin, 1998). Open
sampling aims to discover initially emerging theoretically relevant categories that may
serve as the basis for theoretical sampling later. In open sampling, the participants are
selected based on their expert knowledge of the phenomenon to be studied, that is,
students who can share information about their perspectives of studying engineering
mathematics in this study. There are different variations in open sampling (Strauss &
Corbin, 1998). In this study, the initial open sampling of engineering students was based
on the literature review and my professional experience. I observed that proficiency in
mathematics can significantly determine the types of experiences of mathematics
learning. Proficiency in mathematics is especially relevant in the Singaporean context
where mathematics education is a highly competitive and stressful journey which affects
the students academically, economically and socially in their future lives (Fan, Quek,
Zhu, Yeo, Lionel, & Lee, 2005; TIMMS, 2003).
I involved three of my colleagues in this study and enlisted their help in
recommending students who satisfied the initial sampling criteria. Thus, the selected
students were not my current or former students. At the same time, I would not have a
chance to teach them as they had either completed their three compulsory mathematics
modules recently or were doing their last mathematics module when I interviewed them.
This greatly reduces the power imbalance between the interviewer and interviewee which
can affect the trustworthiness of the study (Wengraf, 2001). This should also enable
them to speak openly and honestly without the fear of affecting their current or future
grades, or offending me. In this study, eight students were initially selected according to
1470 The Qualitative Report November 2010
their differences in academic abilities in mathematics (good [A-C grades] and mediocre
[D-F grades] in their past engineering mathematics modules) across the four diploma
courses. This enabled maximum variation in the initial open sampling in this study.
Before the next round of interviews was carried out, data transcription and analysis
were completed for the previous interviews. This is to prevent the researcher from
missing the chance to sample and collect more information regarding new emerging
concepts (Strauss & Corbin, 1998). There were two more groups of interviewees that
consisted of eight and five students respectively.
Data Analysis Methods
The eventual generation of original, and yet justifiable, theories depends on the
rigorous operationalisation of the data analysis phase in grounded theory (Strauss &
Corbin, 1998). The data analysis phase is the stage where the data are broken down,
conceptualized and creatively put back in new ways. From there, new concepts are built
from the data that contribute to the creation of new theory or theories. Data analysis is
achieved through the process of coding as dictated in grounded theory methodology
(Charmaz, 2006; Glaser, 1992; Strauss & Corbin, 1998). Grounded theory method relies
on the method of constant comparison where the analysis of data involves the data
interacting with one another through comparison.
Constant comparative method. In the constant comparative method, each piece
of relevant data is continually compared with every other piece of relevant data to
generate theoretical concepts that encompass as much behavioral variation as possible
(Glaser & Strauss, 1967). It involves four stages: (a) comparing incidents applicable to
each category; (b) integrating categories and their properties; (c) delimiting the theory
and (d) writing the theory (Glaser & Strauss, p. 105). It is important to compare every
incident and category with one another (Strauss & Corbin, 1998). This is done through
asking questions of the information provided by each incident or category to identify
whether any two are similar. Through this comparison process, the collection, coding,
and analysis phases work in tandem from the start to the end of the investigation. This
allows the gradual development of the data from the lowest level of abstraction to a
higher one of theoretical conception (Strauss & Corbin, 1998). At the same time,
theoretical sensitivity, which is important in the data analysis stage, is fostered in the
constant comparison phase (Strauss & Corbin, 1998).
Theoretical sensitivity. Theoretical sensitivity is very important in the analysis
stages of grounded theory methodology. Theoretical sensitivity is the ability of the
researcher to identify the important features of the collected data, perceive the concepts,
categories, properties and their interrelationships that arise and finally give meanings to
them (Glaser, 1992; Glaser & Strauss, 1967; Strauss & Corbin, 1998). In the initial stage
of data analysis, certain events may be overlooked, but as theoretical sensitivity
increases, they can be recoded and reanalyzed (Strauss & Corbin, 1998). Glaser and
Strauss suggest that the researcher’s personal inclinations and experience are helpful in
creating theoretical sensitivity to the ongoing research. In this study, I was able to draw
on my former experiences as a secondary school mathematics teacher and in my current
Henry Khiat 1471
appointment as a polytechnic lecturer, teaching engineering mathematics. This helped
me to attain an acceptable level of theoretical sensitivity in dealing with the initial data
collection and analysis. At the same time, the reading of literature also helps in
enhancing the theoretical sensitivity of the researcher as it exposes him/her to the
different dimensions of a phenomenon (Glaser, 1978). Thus, the literature review
conducted was very helpful in developing my theoretical sensitivity. As the study
progressed, the data analysis phase became another source for increasing my theoretical
sensitivity due to my exposure to more aspects of the investigated phenomenon (Strauss
& Corbin, 1998).
Although it is recognized that pre-existing perspectives and knowledge can
develop theoretical sensitivity to enhance theory development, their meanings may also
be forced into the data and limit theory development (Charmaz, 2006; Glaser, 1978).
This is because a researcher’s background perspectives and knowledge play the role of
“points of departure” (Charmaz, 2006, p. 17) where they set the grounds for asking
interview questions, the way that the data are collected, organized and analyzed. Because
of the presence of pre-existing conceptions from his/her background perspectives and
knowledge, certain data may be filtered out consciously or subconsciously. Then the
researcher will not be able to identify some of salient features of the data. This can result
in an inaccurate portrayal of the phenomenon investigated.
To prevent any filtering of data through such pre-existing conceptions, I have
constantly evaluated the fit between my preconceptions and the emerging data by looking
at data from multiple angles, making constant comparisons, going towards any possible
new directions and building on any feasible ideas (Charmaz, 2006). This is also known
as the process of theorizing (Strauss & Corbin, 1998).
Open coding. In this study, three types of coding are utilized: open coding, axial
and selective coding (Strauss & Corbin, 1998). Open coding involves the labelling and
categorization of the phenomenon as indicated by the data. Coding does not entail the
mini-descriptions of the different blocks of data but it works in capturing the meanings of
theirs instead (Charmaz, 2006; Strauss & Corbin, 1998). The end products are concepts
which are the building blocks that will help build up the grounded theory. The
comparative method that employs the procedures of asking questions and making
comparisons is being utilized in the open coding process (Glaser, 1978). By asking
simple questions such as who, why, what, how, when, where etc, every word, phrase, or
sentence in each line of data is analyzed. Each analyzed line is then broken down
microscopically into different discrete events called codes. All codes are assigned
individual incident labels. Such line by line microscopic analysis serves to prevent
researcher from making biased analyses due to preconceived ideas about the data or
theory as microanalysis forces the researcher to be exposed to the complete range of the
data (Strauss & Corbin, 1998). Open coding then progresses to the platform where the
codes are compared and similar codes expressing the same incidents are grouped together
under the same conceptual label. Each such group thus becomes a concept. These
conceptual labels are then contrasted again and further clustered into a higher and more
abstract level known as categories (Strauss & Corbin, 1998). After a category is formed,
there is a need to identify its properties and dimensions (Strauss & Corbin, 1998). The
properties of a category can be obtained from the same principle of asking who, why,
1472 The Qualitative Report November 2010
what, how, when, where questions within the category. The dimensions of the category
include the range of situations with such properties have occurred for the subjects.
Examples of opening coding where the eventual category of associational understanding
is first named under different codes, are shown below in Table 1:
Table 1. Examples of open coding
Interviewee Transcripts Open Coding
E01 Somehow, it does lah. After all, in electronics, it still covers
things like complex numbers and it is already covered before
mathematics. So that we have a more clear picture of what we
studied in electronics, sine waves, cosine waves, all these
kinds.
Acknowledging the importance of
EM; Understanding the Relations of
maths concepts in engineering
concepts
E05 Newton’s Law lah, Ohm’s Law lah, and things, and things
like, and things like, laws came in you need to apply
mathematics into, into it lor, when you calculate formulas lah,
or we, we found out what’s the current, things like that. We
actually use Mathematics to solve, circuit problems lor… Ah.
Physics related to maths formulae
E06 Ah… It is link in a way that. Ah… Sometimes when, the
lecturers teach us Engineering Mathematics… Link some of
the…, some of the cases the problem to, to eng-, to the
engineering right, like how you analysis the circuit.
EM related to engineering concepts
Axial coding. As for axial coding, those assembled data are put back together in
fresh ways by making associations between a category and its subcategories (Strauss &
Corbin, 1998). This is to bring together the categories and subcategories in explaining
the phenomenon that is embedded in the data. The development of main categories and
subcategories is central to the process. The first step in axial coding is the identification
of the properties and dimensions of each higher order category or subcategory as
compared to those in open coding (Strauss & Corbin, 1998). The second stage involves
the exploration of the relationships between them and uncovering the conditions, actions
and consequences for the phenomenon through these relationships. Using the same
examples in open coding, the process of axial coding is demonstrated below in Table 2.
Selective coding. Selective coding refers to the integration of the categories to
structure the initial theoretical framework so as to analytically come up with the
grounded theory from the data. The first step involves the identification of the core
category. Strauss and Corbin (1998, p. 146) stated that the core category “is a conceptual
idea under which all the other categories can be subsumed.” The core category is, in fact,
the conceptualization of the storyline about the central phenomenon of the research study.
The core category is the main theme of the data such that it can explain the whole
phenomenon investigated. It is also important that the other categories must be able to
relate to the core category in the description or explanation of the whole phenomenon.
Therefore, the auxiliary categories may be linked to the core category in complex and
intertwining ways. There is a need to note that there may be more than one core category
that represents the phenomenon investigated. At the same time, the data form each
category must not be forced into forming a relationship with the core category/categories.
There are four methods to identify a core category, namely writing a storyline,
conceptualization, use of diagram and review of memos (Strauss & Corbin, 1998). An
example of selective coding in the use of diagram is shown below in Figure 1.
Henry Khiat 1473
Table 2. Example of axial coding
Open Coding Axial Coding
Category
Name
Properties
Mathematical knowledge
Engineering concepts
(Physics)
Relating
Dimensions
Types of formulae
Level of understanding of formulae
Types of engineering concepts
Level of understanding of engineering
concepts
Types of relationship of formulae to
engineering concepts.
Level of understanding of formulae
and engineering concepts
Level of exposure to engineering
related mathematics problems
Context
Lectures, tutorial, self
studying
Intervening conditions
Lecturers, peers, self
Acknowledging
the importance of
EM;
Understanding the
Relations of maths
concepts in
engineering
concepts
Physics related to
maths formulae
EM related to
engineering
concepts
Associational
Understanding
Action / interaction
strategies
Students aim to achieve
the different levels of
associational
understanding through
different strategies.
Consequences
They may achieve different levels of
associational understanding.
Figure 1. An example of selective coding
Conceptual Understanding
Functional Understanding
Procedural Understanding
1474 The Qualitative Report November 2010
Rigor
According to Lincoln and Guba (1985), the basic question addressed by the
notion of rigor in interpretive research is "How can an inquirer persuade his/her
audiences that the research findings of an inquiry are worth paying attention to?" (p.
290). The rigor of an interpretive study is examined by the notion of trustworthiness.
Trustworthiness is defined as the conceptual soundness of the research results and is
influenced by the notions of credibility, transferability, dependability and confirmability
(Lincoln & Guba).
Strauss and Corbin (1998) set the criteria for judging the rigor of a grounded
theory as plausibility, reproducibility, generalizability, concept generation, systematic
conceptual relationships, density, variation, and the presence of process and broader
conditions. These criteria translate into eight conceptual questions to judge the
trustworthiness of the study (Strauss & Corbin, 1998, pp. 270-272):
1. Are concepts generated?
2. Are concepts systematically related?
3. Are there many conceptual linkages and are the categories well
developed? Do categories have conceptual density?
4. Is variation within the phenomena built into the theory?
5. Are the conditions under which variation can be found built into the
study and explained?
6. Has process been taken into account?
7. Do the theoretical findings seems significant and to what extent?
8. Does the theory stand the test of time and become part of the
discussions and ideas exchanged among relevant social and
professional group?
These criteria in determining the rigor of a grounded theory can be complemented
by ensuring the credibility, transferability, dependability and confirmability of the study
are satisfied (Guba & Lincoln, 1989).
Credibility. In terms of interpretive inquiry, internal validity is almost
impossible to achieve (Lincoln & Guba, 1985). This is because one needs to know the
one objective reality that represents the phenomenon that is investigated, which is
unattainable in interpretive research. Therefore, an interpretive study will not set to
prove if this represented reality of the phenomenon is the one objective truth in positivist
tradition. Instead, it should strive to describe accurately the reality of a phenomenon it
intends to represent within the research context. This is thus known as the credibility of
the study.
Theory/perspective triangulation in this study was achieved through the literature
comparison stage as proposed by Glaser (1992). By tying the emergent theory with
extant literature after the analysis, the credibility of the theory building can be enhanced
(Glaser, 1992). Triangulation of sources was achieved here through ensuring maximum
variation in the sampling of interviewees and types of data sources (Hammersley &
Atkinson, 1995; Patton, 1990). This also improves the credibility of the study.
Henry Khiat 1475
Transferability. Transferability refers to the applicability of the research
findings to other similar settings (Lincoln & Guba, 1985; Merriam, 2002). In this study,
a reader can achieve the transferability of the data through reading the thick descriptions
(Geertz, 1973) of how engineering students perceived understanding in mathematics
learning in the context of a polytechnic (Lincoln & Guba). This will allow others to
make accurate and informed judgements if the findings of this study are applicable to
their contexts.
Dependability. Dependability refers to the consistency between the data
collected and the findings (Lincoln & Guba, 1985; Merriam, 2002). An audit trail that
consisted of a detailed documentation of the methods and the collection and analysis of
data was maintained to ensure the dependability of this study (Merriam; Seale, 1999).
This audit trail included the list of interviewees, interview guide, audio records and
transcripts of interviews, data collection and analysis procedures, memos and results.
Confirmability. Confirmability refers to the degree the findings can be
corroborated by other researchers (Merriam, 2002). A researcher can improve
confirmability of a study by being reflexive (Merriam). Reflexivity is a process of
conscious self awareness where a researcher continually appraises the subjective
responses and intersubjective relationships within the data in relation to his/her values,
experiences, interests and beliefs (Finlay, 2002; Nightingale & Cromby, 1999). My
reflexivity was maintained by being consciously aware of my epistemological
preferences, beliefs, values, theoretical orientations, bias, experiences and recording them
in my memos. In addition, the audit trail mentioned above helped in enhancing the
confirmability of this study.
Findings
Before the various types of understanding are discussed, the meaning of theorems,
formulae and procedures unique in engineering mathematics will be defined.
Mathematical theorems are statements that can be proven on the basis of agreed
mathematical assumptions and can be represented in the form of formulae. In
engineering mathematics, theorems are usually represented as formulae. Procedures refer
to the steps in operationalizing the theorems or formulae. In this study, it is discovered
that understanding in engineering mathematics learning can be conceptual, functional,
procedural, disciplinary and associational.
Conceptual Understanding
Conceptual understanding refers to the participants’ ability to understand how the
mathematical formulae are derived. In other words, it indicates the basis of the
mathematical assumptions and proofs of the mathematical formulae. The students’
comments below describe the notion of conceptual understanding when they were asked
about their perceptions of the formula, d x5 5x4
dx
:
1476 The Qualitative Report November 2010
It is good to know the theories or proof behind the formula and why it is
like this. (E06)
If I know why this formula comes about [the proof] why it is put this way,
I can understand it better. (E08)
T01, T02, T03 and T06 agreed that their students could not understand how the formulae
are derived although the lecturers demonstrated their proofs.
Functional Understanding
Functional understanding refers to the participants’ actual comprehension of the
functions of the mathematical formulae and procedures in the mathematics domain. The
participants’ remarks below demonstrate the meaning of functional understanding when
they were probed about what they think of the formulae, d x5 5x4
dx
and
cos xdx sin x respectively:
…it shows how much x5 changes when x is changing every second. (E13)
In maths, this also means we find the area under cos x, if two points are
given. (E06)
In mathematical terms, some of the students do understand the use of these
mathematical formulae; for example, they know that a definite integral
allows them to find the area under the curve. (T04)
Procedural Understanding
As perceived by the participants, procedural understanding refers to the ability to
model the steps in the mathematical formulae in solving mathematical problems. The
category of procedural understanding is shown by the participants’ comments below with
regards to the formulae, d x5 5x4
dx
and 1 2 cos( ) sin( )
4 4
j j
respectively:
Yes, this is the differentiate xn thing, if I see it in exams, I just need to put
in the formula and fill up the values, I think… (E07)
To get the answer (Right hand side) just got to key the values (Left hand
side) into the calculator. (E19)
All six tutors unanimously concurred that their students are proficient in modeling steps
of formulae in solving mathematics questions.
Henry Khiat 1477
Disciplinary Understanding
Disciplinary understanding refers to the students’ ability to understand
engineering concepts and theories in their respective disciplines in engineering they are
enrolled in. As engineering students, they need to know the various engineering concepts
and theories in solving engineering problems. From the data, it is perceived that such
disciplinary understanding might be important in engineering mathematics learning (it
will be elaborated in the latter sections). Two illustrations are given below.
I have to know the concept of electrical circuits in learning engineering
mathematics too. (E01)
I always remind my students that engineering mathematics also entails
them understanding physics concepts and they also acknowledge it too.
(T03)
Associational Understanding
Associational understanding refers to the participants’ ability to relate and utilize the
mathematical formulae in the engineering problems they are tackling. This means that
the participants convert mathematical formulae into engineering terms and apply the
formulae in solving engineering problems. Associational understanding is mentioned in
the participants’ accounts below with regard to the equations of 2sin
3
y x
and
d x5 5x4
dx
respectively:
We make use of the sine wave to calculate our AC curve, AC power
supply, the components that make the AC supply. (E08)
…(the formula) help you calculate in the design of circuit or stuff like
that.… (E11)
To be able to understand which formulae to use in solving engineering
problem is important. (E06)
Relationship among the Five Types of Understanding
Relationships among conceptual, functional and procedural understanding.
From the analysis of the students’ own perceived levels of accomplishment of the
different forms of understanding and the tutors’ feedback, the relationship between
conceptual, functional and procedural understanding is further conceptualized.
Participants who have achieved conceptual understanding can naturally attain both
functional and procedural understanding. Below are the comments of T02 and E21 that
supported the conclusion above.
1478 The Qualitative Report November 2010
My observations of my students throughout these years tell me that those
who understand the proof [conceptual understanding] behind the formulae
are always able to know the contexts they [formulae] work in. And for the
steps of the formulae [procedural understanding], they have totally no
problem in it. But I cannot say for students who cannot grasp the proofs
of the formulae.
As long as I make sure I know the proofs of the formulae, I usually have
no problem with its uses and steps. (E21)
On the other hand, participants who have attained functional or procedural
understanding might not be able to reach conceptual understanding as they do not
understand explicit assumptions and proofs behind the formulae. Participants who have
accomplished functional understanding are more likely achieve procedural
understanding. However, participants who have only achieved procedural understanding
usually fail to accomplish conceptual or functional understanding as they do not
understand the circumstances a formula is created and used. The comments by E16 and
T06 below illustrated the above conclusion.
I do not see the need for me to know the proofs of the formulae as I still
can easily understand and carry out their steps [functional and procedural
understanding]. (E16)
During lessons, I see that students who find it difficult to see the proofs
behind the mathematics formulae still can compute their steps without
mistakes. (T06)
The diagram below illustrates the relationship among conceptual, functional and
procedural understanding.
Figure 2. Conceptual understanding in relation to functional and procedural
understanding
Conceptual Understanding
Functional Understanding
Procedural Understanding
Henry Khiat 1479
Relationship between associational, functional and disciplinary
understanding. With regard to the relationship between the above first three types of
understanding (conceptual, functional and procedural) and associational understanding,
participants who achieve conceptual or functional understanding may be able to achieve
associational understanding. This is because the pre-requisites needed in achieving
associational understanding are the mastery of functional understanding and related
engineering concepts or theories (disciplinary understanding. Thus, the absence of the
mastery of disciplinary understanding means that the students may not be able to achieve
associational understanding even if they are conceptually and functionally adept in the
formulae learnt. However, participants with only procedural understanding will not be
able to achieve associational understanding due to their lack of functional understanding
in engineering mathematics.
Sometimes, I cannot link the formulae to engineering questions
[associational and disciplinary understanding] even though I can use the
formulae to solve mathematical questions [procedural understanding].
(E09)
Although I can do and understand the proof [formulae] all of the time,
there are many times I cannot apply it in engineering situations, like my
problem based assignments. (E02)
The diagram below shows the relationship among associational, functional and
disciplinary understanding.
Figure 3. Relationship among associational, functional and disciplinary understanding
Associational Understanding
Functional Understanding Disciplinary Understanding
1480 The Qualitative Report November 2010
Relationship between solving engineering problem and associational
understanding. Berlinghoff and Gouvea (2004), De Lange (1996), Ernest (2004),
Peterson (1996), Putnam (1986), Stanic (1989), and Steen (1988) have underlined the fact
that engineering and mathematics education have become indispensable in the social
needs and economic survival of any modern society. Engineers solve real life problems
in the society and mathematics is crucial in mastering it. Therefore, it is also essential
that that the relationship between the five types of understanding and engineering
problem solving is uncovered. From the data, it is uncovered that the key to the success
of solving engineering problem solving is the achievement of associational understanding
of mathematics learning. The comments below supported this conclusion.
Our engineering mathematics syllabus aims to impart mathematical skills
that the engineering students can use in solving problems in engineering
scenarios. They can know the steps or proofs of any formulae taught but
if they cannot apply them in engineering contexts, these formulae are
useless to engineers. (T02)
Our other engineering modules’ lecturers always expect us to be able to
use what we have learnt in engineering mathematics in their modules
because they said that it is this skill that will help us to solve the problems
in their modules. (E20)
A lot of times, when if we want to solve engineering related problem
effectively, we need to know how to apply the math into the context of the
problem. (E04)
The diagram below shows the important relationship between associational
understanding of engineering mathematics and successful engineering problem solving.
Figure 4. Relationship between associational understanding and successful engineering
problem solving
Theory of Engineering Mathematics Understanding
Through the relationships among the various types of understanding and successful
engineering problem solving, the substantive theory of engineering mathematics
understanding emerges. In the theory of engineering mathematics understanding,
students understand engineering mathematics conceptually, functionally, procedurally,
disciplinary and associationally. At the same time, these different types of understanding
are closely related to one another in the students’ understanding process in influencing
Associational Understanding
Successful Engineering
Problem Solving
Henry Khiat 1481
the success of engineering problem solving to demonstrate the relationship between these
five types of understanding, the example below is used as an illustration:
Differentiation is an important mathematical concept and is represented by
the formula:
0
lim ( ) ( )
h
dy f x h f x
dx h
. The participants who achieve
conceptual understanding are able to understand the explicit
assumptions and proof behind this formula. They also know that this
formula measures the rate of change of y with respect to x and they will be
usually utilised when rate of change is concerned in any mathematical
problem (functional understanding). More importantly, they are able to
compute a mathematical problem using the procedures involved in this
formula such as d xn nxn 1
dx
if the value of n is given (procedural
understanding). The differentiation formula can be applied in
engineering scenario where the relationship between current and
capacitance in an electric circuit (disciplinary understanding) is
represented by formula, i = C dv
dt
(i, C and v represent current, capacitance
and voltage respectively). If they know this formula, they have achieved
associational understanding as dv
dt
is an engineering concept that is
converted from a differentiation function.
The diagrammatic representation below shows the theory of engineering
mathematics understanding in this study.
Figure 5. Theory of Engineering Mathematics Understanding
Conceptual Understanding
Functional Understanding
Procedural Understanding
Associational Understanding
Disciplinary Understanding
Successful Engineering
Problem Solving
1482 The Qualitative Report November 2010
Discussion
The literature review has shown that most studies that investigated on
mathematical understanding have focused on Skemp’s (1978) conceptual/relational and
procedural/instrumental understanding. The theory of engineering mathematics
understanding formed in this study also includes the categories of conceptual and
procedural understanding as other studies claimed. Other than conceptual and procedural
understandings, the theory of engineering mathematics understanding include three other
forms of understanding that are uniquely present in engineering mathematics learning.
They are the categories of functional, associational and disciplinary understanding in
engineering mathematics learning. The presence of these three unique forms of
understanding in engineering mathematics learning might be due to the fact that
engineering mathematics focuses specifically on the learning of applied mathematical
skills in the context of solving engineering problems. Engineers need to understand the
functions of mathematical formulae (functional understanding) so that they are able to
apply them in solving engineering problems (associational understanding) in relation with
their disciplinary understanding of engineering concepts.
In this study, functional understanding refers to the ability of the students to
understand the functions of the formulae in general mathematical terms. To some extent,
functional understanding can be subsumed under Skemp’s (1978) conceptual
understanding that also encompasses the process of understanding the uses of the
concepts. In this study, both conceptual and functional understanding are detached into
separate entities, as it is revealed that only functional understanding is needed in the
important associational understanding process. In order to utilise mathematics in
engineering scenarios, students have to achieve disciplinary understandingthat give
meanings to the functions of the mathematical concepts they have learned. This ability to
convert mathematical formulae into engineering forms to solve engineering problems is
termed as associational understanding in this study. It is different from that of Skemp as
it is relating mathematical concepts and formulae to solving engineering problems while
the one proposed by Skemp refers to the relationships between mathematical concepts.
The findings of the study reveal that the students who participated in this study
generally focused on achieving procedural and associational understanding (that in turn,
needs functional and disciplinary understandings) of engineering mathematics. Thus,
conceptual understanding in engineering mathematics learning seems to be ignored. The
possible omission of the learning of conceptual knowledge of engineering mathematics
may well be related to the nature of mathematics as discussed by Schneider and Stern
(2005). In general mathematics, learning, the abstract nature of mathematics requires the
learners to constantly reflect on, and make inferences from, a mathematical formula if
they are to understand it conceptually (Schneider & Stern). Thus, conceptual
understanding may not be easily taught or learnt and it is exacerbated by the higher level
of difficulty in engineering mathematics concepts (as compared to pre-tertiary
mathematics) learnt in the case Polytechnic. In contrast, procedural understanding of
mathematics demands rather less attention and cognitive input from the learners
(Schneider & Stern). Thus, it may be perceived by these participants to be easier to teach
and learn.
Henry Khiat 1483
However, in the context of this research, the reasons why conceptual
understanding is not pursued by most of the students in this study may be influenced by
the wider social and educational context. First, the students recognized that attaining
procedural, functional and associational understanding is sufficient for them to perform
well in their tests and examinations. This is because the format of their examinations
were based on the students’ ability to understand associationally and procedurally. Thus,
conceptual understanding is of no academic value to the students in examinations. In
addition, they generally feel that conceptual understanding is irrelevant to them as they
are not required to prove or justify the formulae they are expected to learn in their future
engineering careers.
Nonetheless, there are some limitations to the research approach adopted in this
study regarding the conceptualisation of understanding in engineering mathematics.
First, although the research reported why the various types of understanding were
pursued in the students’ learning, it did not explain how these were formed. This was
because the investigation of the category of understanding did not center on a micro,
cognitive, process-based approach, which would have focused on how students perceived
the actual ordered step-by-step process of mathematics problem solving. Thus, the
research results could not clearly reveal the forms of understanding the students might
potentially have achieved in the different engineering mathematics topics. It also could
not explain why conceptual understanding in particular, and functional and/or
associational understandings to a lesser extent, were not attained by some of the students.
Neither did this study sufficiently clarify how the social contexts of the school (such as
the lecturers’ beliefs, pedagogies, and affective support, and peers’ academic and
affective support) as influenced by the curriculum, pedagogical or time constraints, might
influence the development of the various kinds of understanding. Lastly, it is uncertain
whether the five subcategories of understanding would be found in other domains of
learning such as arts, humanities, business or languages. The confines of this study
meant that the above issues fell outside its boundaries. However, these issues might well
serve as possible topics to research in future.
Conclusion
The purpose of this grounded theory study is to uncover a central theory
explaining engineering conceptions of understanding in mathematics learning, as
perceived by the students and lecturers. In this regard, this study aims to answer the
research question: “What are engineering students’ conceptions of understanding in
mathematics learning?” The research question was answered by the formation of the
substantive theory of engineering mathematics understanding (comprising of conceptual,
functional, procedural, disciplinary and associational understanding) in this study.
This study has deviated from the conventional forms of understanding of
mathematics (conceptual and procedural) proposed by earlier studies. It has proposed the
presence of three more types of understanding (functional, disciplinary and
associational), especially in the context of engineering mathematics learning at diploma
level. It has also shown how each form of understanding in engineering mathematics
learning is related to one another. The data also shows that associational understanding is
of utmost importance in solving engineering problems. To order to enable engineering
1484 The Qualitative Report November 2010
students achieve a high standard of associational understanding in engineering
mathematics learning, I would propose that the curricula should engage engineering
problems as the basis where the concepts and formulae are taught. This will allow these
prospective engineers to engage purposeful engineering tasks as part of their learning.
This will thus enable them to build up their associational understanding of engineering
mathematics. The affordances of tools such as technology can also be incorporated
effectively in the engineering mathematics curricula to improve engineering students’
associational understanding through effective visualisation and modelling of
mathematical concepts.
I also do understand that changes in educational institutions are constrained policy
issues in the wider political and social contexts. The pedagogical, affective and social
influences from teachers and the types of engineering students in mathematics learning
are also complex issues that need to be considered. Nonetheless, regardless of whether
the proposals above are eventually adopted, in the spirit of social justice of education, at
least, the awareness of the theory of engineering mathematics understanding can be
created among the relevant educators.
References
Ainley, J. (2000). Constructing purposeful mathematical activity in primary classrooms.
In C. Tikly & A. Wolf (Eds.), The maths we need now (pp. 138-153). London:
Bedford Way Papers.
Ainley, J., Bills, L., & Wilson, K. (2005). Designing spreadsheet-based tasks for
purposeful algebra. International Journal of Computers for Mathematical
Learning, 10, 191–215.
Ainley, J., & Pratt, D. (2002). Purpose and utility in pedagogic task design. In A.
Cockburn & E. Nardi (Eds.), Proceedings of the 26th Annual Conference of the
International Group for the Psychology of Mathematics Education, University of
East Anglia, Norwich, 2 (pp. 17-24). ERIC.
Annells, M. (1996). Grounded theory method: Philosophical perspectives, paradigm of
inquiry and postmodernism. Qualitative Health Research, 6(3), 379-393.
Bartunek, J. M., & Louis, M. R. (1996). Insider/outsider team research. Thousand Oaks,
CA: Sage Publications.
Bell, A. (1993). Principles for the design of teaching. Educational Studies in
Mathematics, 24(1), 5-34.
Berg, B. L. (1998). Qualitative research methods
More of Information's, Educational materials, both on project, journals, seminars, term papers., politics, advert etc.
Thursday, November 21, 2013
electricity
Electricity
Energy in Brief articles
What is the electric power grid?
The grid of electric power lines has evolved into three large interconnected systems that move electricity around the country. Standards have been developed by the electric power industry to ensure coordination for the linked operations.How much of our electricity is generated from renewable sources?
U.S. power plants used renewable energy sources — water (hydroelectric), wood, wind, organic waste, geothermal, and sun — to generate about 13% of our electricity in 2011.See all Energy In Brief articles ›
Electricity Explained
Electricity in the United States
Most of the electricity in the United States is produced using steam turbines. Coal is the most common fuel for generating electricity in the United States. In 2011, 42% of the country's nearly 4 trillion kilowatthours of electricity used coal as its source of energy.Use of electricity
Electricity consumption totaled nearly 3,856 billion Kilowatthours (kWh) in 2011. U.S. electricity use in 2011 was more than 13 times greater than electricity use in 1950.See more electricity topics ›
Features
Short-Term Energy Outlook
Released November 13, 2013EIA's monthly energy projections through 2014 includes projections for electricity generation and prices.
Electricity Monthly Update
Released October 24, 2013The Electricity Monthly Update is a new EIA product that replaces the Monthly Flash Estimates for Electric Power Data, containing the most recently monthly data.
Electric Power Monthly
Released November 20, 2013Provides monthly statistics at the state, Census division, and U.S. levels for net generation, fossil fuel consumption and stocks, quantity and quality of fossil fuels, cost of fossil fuels, electricity sales, revenue, and average revenue per kilowatthour of electricity sold.
Annual Energy Outlook 2013
Released April 15, 2013–May 2, 2013 Recently released content includes a Legislation and Regulations section that discusses evolving issues, a Market Trends section that highlights projections for energy markets, and a comparison of AEO2013 with projections from other organizations.
Monday, November 18, 2013
Table of Content
Introduction
Chapter 1: Energy - What Is It?
Chapter 2: Electricity
Chapter 3: Static Electricity & Resistance
Chapter 4: Electrical Circuits
Chapter 5: Stored Energy & Batteries
Chapter 6: Generators, Turbines and Power Plants
Chapter 7: Electricity Transmission System
Chapter 8: Fossil Fuels - Coal, Oil and Natural Gas
Chapter 9: Natural Gas Distribution System
Chapter 10: Biomass Energy
Chapter 11: Geothermal Energy
Chapter 12: Hydro Power
Chapter 13: Nuclear Energy - Fission and Fusion
Chapter 14: Ocean Energy
Chapter 15: Solar Energy
Chapter 16: Wind Energy
Chapter 17: Renewable vs. Nonrenewable - Environment & Air Quality
Chapter 18: Energy for Transportation
Chapter 19: Saving Energy and Energy Efficiency
Chapter 20: Hydrogen and Energy In Our Future
Conclusion
Chapter 3: Resistance and Static Electricity
6643783
As we have learned, some kinds of atoms contain loosely attached electrons. Electrons can be made to move easily from one atom to another. When those electrons move among the atoms of matter, a current of electricity is created.
Take a piece of wire. The electrons are passed from atom to atom, creating an electrical current from one end to the other. Electrons are very, very small. A single copper penny contains more than 10,000,000,000,000,000,000,000 (1x1022) electrons.
Electricity "flows" or moves through some things better than others do. The measurement of how well something conducts electricity is called its resistance.
Resistance in wire depends on how thick and how long it is, and what it's made of. The thickness of wire is called its gauge. The smaller the gauge, the bigger the wire. Some of the largest thicknesses of regular wire is gauge 1.
Different types of metal are used in making wire. You can have copper wire, aluminum wire, even steel wire. Each of these metals has a different resistance; how well the metal conducts electricity. The lower the resistance of a wire, the better it conducts electricity.
Copper is used in many wires because it has a lower resistance than many other metals. The wires in your walls, inside your lamps and elsewhere are usually copper.
A piece of metal can be made to act like a heater. When an electrical current occurs, the resistance causes friction and the friction causes heat. The higher the resistance, the hotter it can get. So, a coiled wire high in resistance, like the wire in a hair dryer, can be very hot.
Some things conduct electricity very poorly. These are called insulators. Rubber is a good insulator, and that's why rubber is used to cover wires in an electric cord. Glass is another good insulator. If you look at the end of a power line, you'll see that it is attached to some bumpy looking things. These are glass insulators. They keep the metal of the wires from touching the metal of the towers.
Static Electricity
But What Is Static Electricity?
Content on this page requires a newer version of Adobe Flash Player.
n various forms. One way to store it is in the form of chemical energy in a battery. When connected in a circuit, a battery can produce electricity.
Stored Energy and Batteries
Energizer Bunny
Energy cannot be created or destroyed, but it can be saved in various forms. One way to store it is in the form of chemical energy in a battery. When connected in a circuit, a battery can produce electricity.
If you look at
If you look at a battery, it will have two ends &emdash; a positive terminal and a negative terminal. If you connect the two terminals with wire, a circuit is formed. Electrons will flow through the wire and a current of electricity is produced.
Inside the battery, a reaction between the chemicals takes place. But reaction takes place only if there is a flow of electrons. Batteries can be stored for a long time and still work because the chemical process doesn't start until the electrons flow from the negative to the positive terminals through a circuit.
How the Chemical Reaction Takes Place in a Battery
A very simple modern battery is the zinc-carbon battery, called the carbon battery for short.
This battery contains acidic material within and a rod of zinc down the center. Here's where knowing a little bit of chemistry helps.
When zinc is inserted into an acid, the acid begins to eat away at the zinc, releasing hydrogen gas and heat energy. The acid molecules break up into its components: usually hydrogen and other atoms. The process releases electrons from the Zinc atoms that combine with hydrogen ions in the acid to create the hydrogen gas.
If a rod of carbon is inserted into the acid, the acid does nothing to it.
But if you connect the carbon rod to the zinc rod with a wire, creating a circuit, electrons will begin to flow through the wire and combine with hydrogen on the carbon rod. This still releases a little bit of hydrogen gas but it makes less heat. Some of that heat energy is the energy that is flowing through the circuit.
The energy in that circuit can now light a light bulb in a flashlight or turn a small motor. Depending on the size of the battery, it can even start an automobile.
Eventually, the zinc rod is completely dissolved by the acid in the battery, and the battery can no longer be used.
For a "great" on-line page about batteries, visit the Energizer Learning Center.
Sidebar
Different Types of Batteries
Food – Another Method of Storing Energy
Batteries store energy in a chemical process, but there are other ways of storing energy. Consider the "food chain" on our planet.
Plants, like grass in a meadow, convert the sun's energy through photosynthesis into stored chemical energy. This energy is stored in the plant cells is used by the plant to grow, repair itself and reproduce itself.
Cows and other animals eat the energy stored in the grass or grain and convert that energy into stored energy in their bodies. When we eat meat and other animal products, we in turn, store that energy in our own bodies. We use the stored energy to walk, run, ride a bike or even read a page on the Internet.
Get Adobe Flash player
Electrons with a negative charge, can't "jump" through the air to a positively charged atom. They have to wait until there is a link or bridge between the negative area and the positive area. We usually call this bridge a "circuit."
When a bridge is created, the electrons begin moving quickly. Depending on the resistance of the material making up the bridge, they try to get across as fast as they can. If you're not careful, too many electrons can go across at one time and destroy the "bridge" or the circuit, in the process.
In Chapter 3, we learned about electrons and the attraction between positive and negative charges. We also learned that we can create a bridge called a "circuit" between the charges.
We can limit the number of electrons crossing over the "circuit," by letting only a certain number through at a time. And we can make electricity do something for us while they are on their way. For example, we can "make" the electrons "heat" a filament in a bulb, causing it to glow and give off light.
Content on this page requires a newer version of Adobe Flash Player.
Get Adobe Flash player
When we limit the number of electrons that can cross over our circuit, we say we are giving it "resistance". We "resist" letting all the electrons through. This works something like a tollbooth on a freeway bridge. Copper wire is just one type of bridge we use in circuits.
Before electrons can move far, however, they can collide with one of the atoms along the way. This slows them down or even reverses their direction. As a result, they lose energy to the atoms. This energy appears as heat, and the scattering is a resistance to the current.
Think of the bridge as a garden hose. The current of electricity is the water flowing in the hose and the water pressure is the voltage of a circuit. The diameter of the hose is the determining factor for the resistance.
Current refers to the movement of charges. In an electrical circuit – electrons move from the negative pole to the positive. If you connected the positive pole of an electrical source to the negative pole, you create a circuit. This charge changes into electrical energy when the poles are connected in a circuit – similar to connecting the two poles on opposite ends of a battery.
Along the circuit you can have a light bulb and an on-off switch. The light bulb changes the electrical energy into light and heat energy.
Circuit Experiment
Introduction
Chapter 1: Energy - What Is It?
Chapter 2: Electricity
Chapter 3: Static Electricity & Resistance
Chapter 4: Electrical Circuits
Chapter 5: Stored Energy & Batteries
Chapter 6: Generators, Turbines and Power Plants
Chapter 7: Electricity Transmission System
Chapter 8: Fossil Fuels - Coal, Oil and Natural Gas
Chapter 9: Natural Gas Distribution System
Chapter 10: Biomass Energy
Chapter 11: Geothermal Energy
Chapter 12: Hydro Power
Chapter 13: Nuclear Energy - Fission and Fusion
Chapter 14: Ocean Energy
Chapter 15: Solar Energy
Chapter 16: Wind Energy
Chapter 17: Renewable vs. Nonrenewable - Environment & Air Quality
Chapter 18: Energy for Transportation
Chapter 19: Saving Energy and Energy Efficiency
Chapter 20: Hydrogen and Energy In Our Future
Conclusion
Chapter 3: Resistance and Static Electricity
6643783
As we have learned, some kinds of atoms contain loosely attached electrons. Electrons can be made to move easily from one atom to another. When those electrons move among the atoms of matter, a current of electricity is created.
Take a piece of wire. The electrons are passed from atom to atom, creating an electrical current from one end to the other. Electrons are very, very small. A single copper penny contains more than 10,000,000,000,000,000,000,000 (1x1022) electrons.
Electricity "flows" or moves through some things better than others do. The measurement of how well something conducts electricity is called its resistance.
Resistance in wire depends on how thick and how long it is, and what it's made of. The thickness of wire is called its gauge. The smaller the gauge, the bigger the wire. Some of the largest thicknesses of regular wire is gauge 1.
Different types of metal are used in making wire. You can have copper wire, aluminum wire, even steel wire. Each of these metals has a different resistance; how well the metal conducts electricity. The lower the resistance of a wire, the better it conducts electricity.
Copper is used in many wires because it has a lower resistance than many other metals. The wires in your walls, inside your lamps and elsewhere are usually copper.
A piece of metal can be made to act like a heater. When an electrical current occurs, the resistance causes friction and the friction causes heat. The higher the resistance, the hotter it can get. So, a coiled wire high in resistance, like the wire in a hair dryer, can be very hot.
Some things conduct electricity very poorly. These are called insulators. Rubber is a good insulator, and that's why rubber is used to cover wires in an electric cord. Glass is another good insulator. If you look at the end of a power line, you'll see that it is attached to some bumpy looking things. These are glass insulators. They keep the metal of the wires from touching the metal of the towers.
Static Electricity
But What Is Static Electricity?
Content on this page requires a newer version of Adobe Flash Player.
n various forms. One way to store it is in the form of chemical energy in a battery. When connected in a circuit, a battery can produce electricity.
Stored Energy and Batteries
Energizer Bunny
Energy cannot be created or destroyed, but it can be saved in various forms. One way to store it is in the form of chemical energy in a battery. When connected in a circuit, a battery can produce electricity.
If you look at
If you look at a battery, it will have two ends &emdash; a positive terminal and a negative terminal. If you connect the two terminals with wire, a circuit is formed. Electrons will flow through the wire and a current of electricity is produced.
Inside the battery, a reaction between the chemicals takes place. But reaction takes place only if there is a flow of electrons. Batteries can be stored for a long time and still work because the chemical process doesn't start until the electrons flow from the negative to the positive terminals through a circuit.
How the Chemical Reaction Takes Place in a Battery
A very simple modern battery is the zinc-carbon battery, called the carbon battery for short.
This battery contains acidic material within and a rod of zinc down the center. Here's where knowing a little bit of chemistry helps.
When zinc is inserted into an acid, the acid begins to eat away at the zinc, releasing hydrogen gas and heat energy. The acid molecules break up into its components: usually hydrogen and other atoms. The process releases electrons from the Zinc atoms that combine with hydrogen ions in the acid to create the hydrogen gas.
If a rod of carbon is inserted into the acid, the acid does nothing to it.
But if you connect the carbon rod to the zinc rod with a wire, creating a circuit, electrons will begin to flow through the wire and combine with hydrogen on the carbon rod. This still releases a little bit of hydrogen gas but it makes less heat. Some of that heat energy is the energy that is flowing through the circuit.
The energy in that circuit can now light a light bulb in a flashlight or turn a small motor. Depending on the size of the battery, it can even start an automobile.
Eventually, the zinc rod is completely dissolved by the acid in the battery, and the battery can no longer be used.
For a "great" on-line page about batteries, visit the Energizer Learning Center.
Sidebar
Different Types of Batteries
Food – Another Method of Storing Energy
Batteries store energy in a chemical process, but there are other ways of storing energy. Consider the "food chain" on our planet.
Plants, like grass in a meadow, convert the sun's energy through photosynthesis into stored chemical energy. This energy is stored in the plant cells is used by the plant to grow, repair itself and reproduce itself.
Cows and other animals eat the energy stored in the grass or grain and convert that energy into stored energy in their bodies. When we eat meat and other animal products, we in turn, store that energy in our own bodies. We use the stored energy to walk, run, ride a bike or even read a page on the Internet.
Get Adobe Flash player
Electrons with a negative charge, can't "jump" through the air to a positively charged atom. They have to wait until there is a link or bridge between the negative area and the positive area. We usually call this bridge a "circuit."
When a bridge is created, the electrons begin moving quickly. Depending on the resistance of the material making up the bridge, they try to get across as fast as they can. If you're not careful, too many electrons can go across at one time and destroy the "bridge" or the circuit, in the process.
In Chapter 3, we learned about electrons and the attraction between positive and negative charges. We also learned that we can create a bridge called a "circuit" between the charges.
We can limit the number of electrons crossing over the "circuit," by letting only a certain number through at a time. And we can make electricity do something for us while they are on their way. For example, we can "make" the electrons "heat" a filament in a bulb, causing it to glow and give off light.
Content on this page requires a newer version of Adobe Flash Player.
Get Adobe Flash player
When we limit the number of electrons that can cross over our circuit, we say we are giving it "resistance". We "resist" letting all the electrons through. This works something like a tollbooth on a freeway bridge. Copper wire is just one type of bridge we use in circuits.
Before electrons can move far, however, they can collide with one of the atoms along the way. This slows them down or even reverses their direction. As a result, they lose energy to the atoms. This energy appears as heat, and the scattering is a resistance to the current.
Think of the bridge as a garden hose. The current of electricity is the water flowing in the hose and the water pressure is the voltage of a circuit. The diameter of the hose is the determining factor for the resistance.
Current refers to the movement of charges. In an electrical circuit – electrons move from the negative pole to the positive. If you connected the positive pole of an electrical source to the negative pole, you create a circuit. This charge changes into electrical energy when the poles are connected in a circuit – similar to connecting the two poles on opposite ends of a battery.
Along the circuit you can have a light bulb and an on-off switch. The light bulb changes the electrical energy into light and heat energy.
Circuit Experiment
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