Strategies Abstract Algebra This article is based on the study, which tries to unpack strategies of learning abstract algebra from learners’ perspective. Ethnography was used to collect the required information. The study found the strategies of learning abstract algebra are: to use idiosyncratic figure, analogical reasoning, particular concrete examples, and particular relation. This study can lead teachers of abstract algebra to a new awareness of their teaching strategies and their practices.
## 1. INTRODUCTION“Mathematics
is a product of the human mind and the possibilities of human rational
thinking” (Maasz & Schloeglmann,
2006, p.2). Devlin defined “modern Mathematics
is about abstract
pattern, abstract structure, and abstract relationships” (2000, p.136).
Similarly, abstract algebra
belongs to modern
mathematics because it requires thinking with a high level of
abstraction. Abstract algebra explores the possible relationship among abstractions. Many researchers report
on students‟ and teachers‟ difficulties with
learning and teaching abstract algebra that prevent them from fulfilling the
objectives of the course (Dubinsky et. al 1994, 1998, 2001, Asiala
et.al 1997, Burn 1996, Brown et.al 1997, Findell
2001, Fukawa
Connelly 2007, Hirsch 2008, Kontorovich & Zazkis, 2017). My own experiences as an Abstract
Algebra teacher of undergraduate and graduate
students in Nepal motivated me for
this study. Researches on how higher mathematics learning could be facilitated to the students
have not been conducted in Nepalese context yet. Then, I analyze
and unfold how mathematicians’ ways of learning mathematics can be
reconstructed from an andragogical perspective. My ultimate aim is to
understand how learning of higher mathematics could be facilitated to the
students in Nepalese context. ## 2. OBJECTIVE AND RESEARCH QUESTION OF THE STUDYLearning of abstract algebra
by undergraduate students
was analyzed on the basis of
cognitive learning theories. Dubinsky et.al (1994), Asila
et.al (1996,1997), Edward & Brenton (1996), Brown et.al (1997), Findell (2001), Mingus (2001), Fukuwa-connelly
(2007), Hirsch (2008) have done experimental studies in explaining the learning
of abstract algebra. Some of the aforementioned studies have utilized technology
as tools for meaningful mediation of algebra contents and the learning for
making meaningful understanding. The present study is however different than those, for it is based on traditional classroom setting with no computer
assisted teaching in Nepalese context.
The objective of the study was: to explain the learning strategies of undergraduate
students in relation to learning of abstract algebra. In this context the
following question arises as a research question. What
learning strategies do the undergraduate students employ in learning of abstract algebra? ## 3. THEORETICAL FRAMEWORKThe
conceptual framework of the study was built on the Dubinsky‟s
APOS theory, and Vygotsky‟s theoretical frameworks because to learn and teach abstract
algebra, we have to pay a great deal of attention
to the mental operations of the
learners, use and meaning of various
signs of the definitions, and purposeful interaction. In APOS theory, “genetic
decomposition is a hypothetical model that describes the mental structures and mechanisms that the students
might need to construct in order
to learn a specific mathematical concept” (Arnon et
al., 2014, p.27). According to APOS theory, individual makes sense of
mathematical concepts by using certain mental structures i.e. Action, Process,
Object, and schema (Piaget & Garcia,1983/1989). These structures arise
through instance of reflective abstraction or mental mechanism such as
interiorization, coordination, reversal, encapsulation, and generalization-
that lead to the construction of mental structures (Dubinsky, 1991). A major theme of Vygotsky‟s theoretical framework
is that social
interaction plays a fundamental role in the development of cognition. Vygotsky
(1978, p.57) believed that “everything is learned
on two levels”. That is: first, on the social
level, and later, on
the individual level;
first between people
(interpsychological) and then inside
the individual (intrapsychological). The
potential for cognitive development is limited
to a
“Zone of proximal
development” (ZPD). Vygotsky
believed that internalization of semiotic mediation: the processes by which social
processes are transformed into internal processes with the help of instruments of psychological
activities, led to higher thinking skills. To learn
an abstract concept, spontaneous and scientific concepts should play an
interdependent role. The concepts which are embedded in sense perception and
practical/everyday experiences are called spontenious
concepts. The concepts, which are acquired with conscious effort in the course
of formal instructions, are called scientific concepts. “...the development of
the spontaneous concepts proceeds upward and the development of scientific
concept downwards...” (Vygotsky, 1986, p. 193). Vygotsky‟s
(1986, p.157) mentioned spontaneous and scientific concepts “are part of single
process: the development of concept formation which is affected by varying
external and internal conditions but is essentially a unitary process, not a
conflict of antagonistic, mutually exclusive forms of thinking”. Interaction
between spontaneous and scientific concept within the Vygotsky‟s
Zone of Proximal Development (ZPD) fosters the development of ## 4. RESEARCH DESIGNThe study
used ethnographic design under qualitative approach in naturalistic setting to
unpack the strategies for learning abstract algebra. As research tools for this
study, the everyday journal writing by students (The journal of learning Group
Theory), Non participant observation diary, clinical interviews with students,
interviews guidelines for semi-structured interviews were used. ## 5. SAMPLEAll the students
enrolled in the undergraduate with major mathematics in B.Ed/undergraduate program,
from different colleges
of Kathmandu Valley
(private and constituent colleges were participants in the study for the
purposes of field observations. All the participants had previously taken
courses in Euclidean and non-Euclidean geometries, calculus, and real analysis.
To select sample of the study, first I visited some private and constituent
colleges. Having observations of some classes, interaction with students and teachers, studying
students‟ written text and
informal discussion on the basis of the written text from those colleges, I found,
more or less students‟ learning
styles, their difficulties regarding learning algebraic concepts, types of learners,
teachers‟ delivered ways are similar. The key participants were selected
by purposive sampling method. For that class observations, written tests
(related to definition writing, to create examples and non-examples, proof
writing), interviews were conducted. For interviewing students, issues were
generated from preliminary analysis of the three sources of information:
diaries, observation notes, and written tests. Later using all these
information I came to know the limitations and potentials of the students which
provided the tentative ZPD of each student.
From this group of students
I chose five students as key participants, who had given permission for
full participation for the study, The key participants have been given pseudonyms. ## 6. ANALYSIS AND INTERPRETATIONAs
analysis techniques and interpretation, it was done from the very beginning of
the field work focusing on the best solutions, common errors and the most
frequently occurred errors, unanticipated and exceptional errors of students
were sorted from the entire texts prepared from field study. The strategies of
learning abstract algebra were analyzed and interpreted by using theoretical
framework. Instantly some outstanding excerpts are:
Ruma:
How did you formulate the definition of a Normal Subgroup? Bahadur
[340]: 1 (iii) Here,
students do not necessarily make the same distinctions as those made by mathematicians
and mathematics educators. Students
cut up experiences in different ways, both indicating and further establishing
a collection of concepts that are substantially and structurally different
concepts that are used in mathematical community. Example 2 Ruma:
What do you mean by binary operation? Anu [51]: Binary Operation means and Bahadur
[6]: Binary operation means but for group and
Ruma:
Is is subgroup of ? Nabin
[22]: From this question it is clear that Z
Here, all
the elements of Z
To prove
particular theorems of prerequisites theorems of Lagrenges
Theorems, students focused on particular relations. For instance RME[308]:
Could you explain how iff belongs to ? Bharat [305]: Let be an arbitrary element of . Now by definition of I mean definition of left coset, . Therefore . Here, we know Thus, . Again, Let then we can show and we can say . So, . Actually, I remember this step
(laughs). But as the result of this step we will get From and we can say .
This completes the proof. Bharat’s line [305] reflects
that to prove, he remembered some steps as formula. For example, here to show ,
he focused on two relations: and Other-part,
just he manipulated the symbols for these relations without having any clear
reasoning. As someone looks the product (the proof), it seem that Bharat has
understood the theorem. But in reality, due to this tactical technique
only, in surface it looks like that he understood
but indeed he did not know the real reasons about how these important steps
appeared. This is very dangerous in the learning process of abstract algebra.
Since, he recalled the important steps, he felt he knew all the things (he does not feel that he has to think again deeply),
at the same time he did not know the reason behind each and every
step. Here, for other steps of the theorem just he manipulated the symbols for
the relations without any clear logic. Example-4 Sometimes
students focus on the “symbolic pattern” rather than reflecting thinking. [251] Bharat:
I think the inverse of 2 is 2 This wrong analysis is due to analogous to expecting '1' is the identity for operation
and using the symbols looking
its form rather
than searching the meaning/sense for the
symbols. So, in he could not see the inverse of 2 is 4 and the inverse
of 3 is 3. And he
just searching
and
in whilst .
Another more important
fact is that his sentence “...because we know the inverse of b is b ## 7. FINDINGS AND CONCLUSIONSThis
study made significant contributions to unpack the strategies of learning
Abstract Algebra. On the completion of analyzing aforementioned examples this
study found: One of
the learning strategies of undergraduate students to learn abstract algebra is
to use the idiosyncratic figures like Bahadur’s line [340], which is more or
less similar to Findell’s (2001) study where he found
students do not necessarily make the same distinctions as those made by
mathematicians and mathematics educators. Another students’ strategy of
Abstract algebra is to stuck on the particular concrete examples. For instance,
Anu’s line [51], Bahadur’s line [6], and Nabin’s line
[22]. To use analogous reasoning: for instance, Bharat’s line [251]. Students
used different proof writing strategies for different contents. Strategy
focused on figure: to prove two groups are Isomorphic student preferred the
group table but when the order of group
is large, changed
the strategy: strategy
focused on particular relation. For instance, Bharat’s line [305] These
findings are not intended to stand as an indictment of the existing teaching
and learning process of abstract algebra in undergraduate level, particularly
in Nepal. Rather, the teaching learning of abstract algebra practices exhibited
by students and teachers, outlined above, suggest that the concept of the
combination of APOS theory and the ZPD of Vygotsky in teaching learning process
of abstract algebra must take account of the existing
practices and hidden
barriers that facilitate and constrain both teachers
and students in their teaching and learning process in course of scaffolding
students' learning of abstract algebra. Consequently, the conclusions continue
with the suggestion that all these findings are distinct and fundamental
aspects of teaching and learning of advanced mathematics in general and
particularly in abstract algebra. ## SOURCES OF FUNDINGThis research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. ## CONFLICT OF INTERESTThe author have declared that no competing interests exist. ## ACKNOWLEDGMENTNone. ## REFERENCES [1] Arnon I., Cottrill
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