Undergraduate Students’ Conceptual Understanding on Abstract

Algebra

Risnanosanti, Yuriska Destania

Mathematics Study Program in Muhammadiyah University of Bengkulu Indonesia

Keywords: Conceptual Understanding, Abstract Algebra

Abstract: This study aims to describe students’ conceptual understanding on abstract algebra at mathematics education

study program. The concept of binary and group operations is a fundamental issue in abstract algebra. This

current study is a qualitative research that aims to determine level of students’ understanding. Subjects of the

study were students of mathematics education study program of semester 7 who were taking course of algebra

structure in academic year of 2017/2018. The data were collected by using two instruments; test and

interviews. After the data obtained then it was categorized into appropriate levels of understanding of students.

The categories used in this study are the low, medium and high level of conceptual understanding. Based on

the results of data analysis, the result of study indicates that more than 50% of students still have low level of

understanding. It can be concluded that there are still many students who lack of conceptual understanding on

binary and group operations. Students define binary and group operations based solely on their current

knowledge. Therefore, there must be a special action done by lecturers to connect the concepts learned with

it precondition concepts.

1 INTRODUCTION

Mathematical branches studied in mathematics

education study program include algebra, analysis,

applied mathematics, geometry, number theory, basic

introduction to mathematics, and mathematical logic.

Algebra is a course that contains basic skills expected

for those who study mathematics. This means that

basic knowledge of mathematics can be obtained by

studying algebra (Kaput, 2000). In higher education,

algebra is also called 'Abstract Algebra'. The subject

of abstract algebra aims at finding the same picture of

the algebraic structure, obtaining additional results

based on existing results, and making the

classification of operations on the structure of the

material being studied. Learning a structure means

studying the interrelationships among concepts

involved in the structure. According to Arikan, et.all.

(2015) the difficulties’ experienced in a concept will

cause difficulties in learning the next concept. There

are several reasons for the difficulty in learning

abstract algebra one of which is the lack of mastery

on concepts. The less mastered concepts make

students are not able to think abstractly, cannot depict

verbal expressions or cannot make mathematical

formulas. Therefore, in order to have good mastery in

abstract algebra, a good conceptual understanding is

highly required.

Procedural skill and conceptual understanding are

two types of knowledge that everyone needs in

learning mathematics. Procedural skill refer to the

ability of a person in solving problems by using a

coherent algorithm (Byrnes and Wasik, 1991; Bisson

et.all, 2016). In addition, procedural skill can also be

identified through the ability to change the notation

used in solving a mathematical problem. On the other

hand, conceptual understanding refers to an

understanding of concepts, relationships, and

principles in mathematics (Rittle, Siegler and Alibali,

2001; Crooks and Alibali, 2014). National Council

Teaching of Mathematics (2014) states that learning

mathematics should give emphasis at conceptual

knowledge by reducing attention to procedural

learning. Paying more attention to executional

procedures and ignoring conceptual understanding

will have an impact on students' success in learning

the topic (Ocal, 2017). Tatar and Zengin (2016)

explain ones of the causes of students having

difficulty in learning mathematics especially on the

topic of calculus is the lack of conceptual knowledge

on the topic. A learning process that emphasizes

conceptual knowledge, means providing an activity

438

Risnanosanti, . and Destania, Y.

Undergraduate Students’ Conceptual Understanding on Abstract Algebra.

DOI: 10.5220/0008523304380443

In Proceedings of the International Conference on Mathematics and Islam (ICMIs 2018), pages 438-443

ISBN: 978-989-758-407-7

that makes meaningful interrelationships between

concepts in mathematics, emerging understanding,

and giving chance for applying concepts. Arslan

(2010) states that conceptual knowledge involves a

student's understanding of interpreting a concept and

establishing a connection between the concepts.

People who have high conceptual understanding

ability can make the interrelationship between

sections in mathematics better than those with low

understanding of concepts. The results of Hallet,

Nunes and Bryant (2010) study found that students

will be more successful in mathematics learning when

a conceptual approach is used more than procedural

approaches. The use of a conceptual approach in a

learning activity may provide students with correct

procedures and better capacity in transferring

knowledge (Rittle and Alibali, 1999). Joersz (2017)

explain that students whom learning activities use a

conceptual approach will both improve their

conceptual understanding and establish correct and

flexible troubleshooting procedures. The term

conceptual understanding or knowledge includes

what has been known about the concepts and the way

in which the concept is found. Conceptual knowledge

is defined as knowledge of abstract concepts and

principles in general (Rittle, Siegler and Alibali,

2001). The increase of understanding about concepts

in mathematics means enhancing the ability of

conceptual understanding. In principle stated by the

National Council of Teachers of Mathematics (2014),

it is firmly stated that conceptual understanding is

followed by the establishment a basic for the smooth

procedure. That is why conceptual understanding

becomes foundation and need for a good mastery on

utilizing procedure to solve mathematical problems.

The principle of NCTM shows that students must

develop their basic conceptual understanding first

before they have procedural knowledge. Procedural

knowledge was not developed early as an action to

expand the development of conceptual knowledge

(Rittle, Schneider, and Star, 2015). To achieve an

increased conceptual understanding of learning in

classroom, a valid and reliable measuring tool is

needed (Bisson, et.all., 2016) Conceptual

understanding can be measured by using a variety of

tasks, starting from the task of evaluating the

correctness of an example or a procedure of defining

and explaining the concepts learnt by students

(Crooks and Alibali, 2014; Rittle, Schneider, Star,

2015). The virtues of conceptual tasks are relatively

unknown or have not been encountered by the

students. So in order to solve them, students must

have the conceptual knowledge ability (Bisson, et.al.,

2016)

Based on the above descriptions, this study aims

to describe students’ conceptual understanding at

mathematics education program of FKIP

Muhammadiyah University of Bengkulu in abstract

algebra course, especially on the topic of group

theory.

2 METHOD

2.1 Type of Research

This research is qualitative descriptive study using

described qualitative data to produce clear and

detailed description about students’ conceptual

understanding on topic of group theory at

Mathematics Education Study Program of

Muhammadiyah University of Bengkulu. Qualitative

data in this study is the result of students’ answers on

test. After the test was conducted, then the subjects

were interviewed in order to describe and dig up in-

depth informations those were not obtained from

student test.

2.2 Subject of the Research

Subjects of this study were 63 seventh semester

students of mathematics education of FKIP of

Muhammadiyah University of Bengkulu 2017/2018

academic year who were taking the course of

Algebraic Structure. Based on the results of the test,

6 students had been selected to be interviewed. The

students who were interviewed represented low,

medium, and high conceptual ability groups.

2.3 Research Intrument

The main instrument used in this study was the

researcher herself by conducting interviews to

explore in-depth information on mathematics

education students’ understanding on the concept of

binary and group operations based on test results. The

instrument used in this study was a test on conceptual

understanding of binary and group operation material.

The questions of this test were in the form of

description consisted of two questions based on the

indicator. In addition, an interview guide was also

used to know in-depth information about the occuring

processes in term of students’ algebraic thinking in

solving algebra problems.

Undergraduate Students’ Conceptual Understanding on Abstract Algebra

439

2.4 Data Collection

Data collection of this research was done by using

two techniques, written test and interview. Written

Test is a test of understanding on the concept of

binary and group operations to obtain data on

students' conceptual understanding. Meanwhile, the

interview used in this study was a task-based

interview so that interview guidelines only contained

with outline of questions. It was conducted to obtain

clearer data on conceptual understanding of binary

and group operating materials.

2.5 Data Analysis Tehcnique

The data analysis on conceptual understanding of

binary and group operation was done in depth after

the students were divided based on their ability

category. The process of data analysis began by

reviewing all available data from interview and

observation that has been written in field notes. Then

the researcher performed data reduction, data

exposure, and conclusion drawing and verifications.

3 RESULT AND DISCUSSION

3.1 Result

The problem in this research was “𝐿𝑒𝑡 G = {a +

√

2, a, b ∈ Q, a ≠ 0 or b ≠ 0} Prove that G is a

Group under regular multiplication” . We wanted to

analyze whether the students learnt the group

principle in this problem. We marked that most of the

students memorized the group principles (closure,

associativity, identity element, inverse element) but

they could not interpret them.

Table 1. Conceptual Understanding of the Subject in The

First Problem

Number of The Problem

Percentage of Correct Answered in Closure

Property

Percentage of Correct Answered in

Associative

Percentage of Correct Answered in Identity

Element

Percentage of Correct Answered in Inverse

Element

In accordance with table 1, it appears that for the

problem, only 54% of the subjects answered correctly

in closure property. Most of the students of

mathematics education in FKIP Muhammadiyah

University of Bengkulu (64%) have an error in

determining inverse of each element in a group. Table

1 also depicts that for the understanding on terms of a

set to be a group, less than 50% (47.75%) of subjects

answered correctly.

Results of this study aim to determine Mathematics

Education students’ conceptual understanding on

binary and group operations in the subject of

Structure of Algebra.

1. Result of analysis on subject 1 (represent students

with high conceptual ability), based on result of

the test and interview:

a. Closure property: subject 1 could explain

well the property on the multiplication of

positive rational numbers derived from the

closure properties of multiplication with a

detail explanation at each explanatory step.

b. Associative property: subject 1 could

describe the associative property by giving

explanation about the connection the

multiplication operation property.

c. Identity element: subject 1 managed to

define the identity element which is also an

element of a set of rational numbers with a

clear and detailed explanation on each step

taken.

d. Every element has inverse: subject 1 could

coherently determine the inverse element by

using the identity element specified in the

previous step. Subject 1 explained in

sequence and detail the steps to determine

the inverse element and show that the

inverse is also an element of the set of

rational numbers.

2. Results of analysis on subject 2

(represented students with medium

conceptual ability)

a. Closure property:Subject 2 did not

mention that the set was not an empty set

that has binary operations. The subject

directly mentioned that the set meets the

closure property and other properties.

Subject 2 could show that the set of

positive rational numbers has closure

property to binary operation on

multiplication. But in each step, subject

2 did not explain the used properties.

b. Associative property: Subject 2 described

the associative property from the left-to-

right side in detail using other variables

and described the inherent property of

each written step. However, subject 2

ICMIs 2018 - International Conference on Mathematics and Islam

440

was less precise in restoring the

examplified variables.

c. Identity element: Subject 2 was able to

determine the element of identity in the

set of positive rational numbers.

However, written test results indicated

that the subject was not understood the

property he wrote. Subject 2 also

hesitated in explaining which definitions

are meant in each explanatory step.

d. Identity element: Subject 2 showed that

every element in the set of positive

rational numbers has an inverse element

by asserting the inverse with another

variable. In the explanation of this

property subject 2 was more flexible

when it came to substituting the given

variables. Eventually, subject 2

reasonably showed that the inverse is

also an element of the set of positive

rational numbers. The subject also

explained definition of a group by

repeating the explanation of each trait.

Unfortunately, subject 2 was not careful

enough in determining inverse.

3. Results of analysis on subject 3

(represented students with low conceptual

ability)

a. Closure property: Subject 3 managed to

write the definition of closure property.

However, the subject failed to show

closure property on the given binary

operation. Subject 3 did not put meaning

on the definition of binary operations

properly and could not mention the

notation of the set of numbers.

b. Associative property: Subject 3 could

write the definition of associative

properties down. However, the subject

could not show that the given binary

operation meets the associative

property. Subject 3 copied from sample

of previous problems that had different

binary operations. The explanation of

the left-hand side to the right-hand side

did not match the definition of the

binary operation assigned to the

problem that was being worked on.

Subject 3 has not understanding on the

application of binary operations in the

term of associative property.

c. Identity element: Subject 3 wrote the

definition of identity element correctly.

However, it could not elaborate the

property using the definition of given

binary operation. The subject was

mimicking from binary operations has

ever been undertaken without sufficient

understanding on the definition itself. In

addition, subject 3 assumed that the

element of identity is always zero. So

the operation performed was a

reduction operation which is the inverse

of the addition operation.

d. Inverse element: Subject 3 could denote

the inverse element and wrote down the

definition that characteristic of each

element is having an inverse element.

However, the subject did not succeed to

show the inverse of the element because

it wrote the wrong identity element.

Furthermore, the decomposition of the

binary operation definition was still

incorrect. So that the inverse element

obtained was also incorrect. Subject 3

was not able to link any written steps

with the definitions already written at

the beginning. Subject 3 mentioned that

the problem that was being solved was

a group but could not explain the reason

correctly.

3.2 Discussion

Based on the results of the data analysis it can be

concluded that subject 1 could define the binary and

group operation in detail starting with a non-empty

set, having binary operations, and then showing one

by one closure and associative properties, having an

identity element, each element has an inverse, and are

distributive by using inherent properties in operation.

Based on the data of Subject 2, it can be stated that

subject 2 was able to identify a binary and group

operation or not binary and group operations by

showing one by one closure and associative

properties, having an identity element, each element

has an inverse and distributive. The subject did not

mention that binary and group operations must be

non-empty set. Subject 2 did not include attributes

attached to the operation. In addition, Subject 2 was

also rather careless in completing the test. Based on

the data analysis on subject 3 it can be concluded that

Subject 3 was able to write the definition of binary

and group operation. However, it was less fluent and

incomplete in showing one by one the properties.

Subject 3 assumed that the identity of multiplication

is always 1 and the identity of addition is always 0.

Undergraduate Students’ Conceptual Understanding on Abstract Algebra

441

Subject 3 did not include attributes attached to the

operation. In addition, Subject 3 was also less

thorough in conducting the operation.

This research wanted to survey whether the

students learnt the group axioms in this question. The

researcher observed that most of the students

memorized the group axioms (closure, associativity,

neutral element, inverse element) but they could not

analyze them. We categorized the mistakes into two

groups on this question. In the 1st Category, students

listed the group axioms but they accepted them

“Correct” without sufficiently analysing it. In short,

they accepted that associativity was proved without

analysing. In the 2nd Category, the result we found is

that the students could not comprehend the required

associativity.

It could be said that the students preferred to copy

rather than to think abstractly when we consider that

they attended to the university as a result of test exam,

i.e. the central exam system (Soylu, 2008). It sounds

believable that they could have just memorized the

rules of theory without internalizing the descriptions.

Trying to proving the group axioms without thinking

on descriptions is a sign of rote learning based

education system. Whether a cognitive teaching has

been done on the algebraic structures or not has not

been known. Unless we internalize the meanings of

the concepts covered with the different learning

methods, mastering on a subject by rote will come

into the question. Using computer programmes, e.g.

computer algebra system CBS), will provide

convenience but, are there any academic members

applying to the computer programmes or how are

their perspectives to these embodying processes?

Doing a scientific research by academic members

about this matter, their opinions and their approaches

could be significantly useful (Tatar and Zengin,

2016). The questions, which measure whether

definitions and features of algebraic structures are

learnt, are generally measure proving, reasoning, and

discernment ability. Individuals experience many

problems in their daily life and they think

mathematically to solve their problems. Actions like

explaining a proposition, saying why it is right or

wrong and choosing and using different logical

thinking ways and proving types, present individual’s

ability on mathematical thinking. In this sense, the

students of the mathematics department are supposed

to use their ability of mathematical thinking and to let

the operations they do make sense. Mistakes made by

students, who participated in the study, came up as a

result of either misunderstanding the conditions of

group theory or examining these conditions wrong

(Arikan, et.all., 2015). Some challenges could be

experienced during the learning process but the

matter is to identify them correctly and to enhance

various methods to deal with them. Having

difficulties at the learning abstract concepts is the

most important one. Students can apply to rote

learning in order to overcome this difficulty, but they

can have difficulty in practice at this time. For

example, student lists the axioms (closure,

associativity, inverse element, neutral element) while

controlling the set whether it is group or not, but he

or she makes the operation supposing that the set is

closed (Arikan, et.all., 2015). In other words, student

cannot practice what he or she memorized or could

not know what to do in other cases. We have been

thinking the fact that this problem traces to the gaps

of education which was received in both high school

and university years. Students’ infrastructure they set

up with math training, which they had during their

education life up to attending university, has

inadequate mathematics they meet at the university.

They assume that the success at this lesson to be able

to perform the operations without using calculator

and dealing with just practical solutions in the math

exams. However, they meet theoretical mathematics

after the graduation from a high school before the

college and as a natural result, they are afraid of

another learning difficulty, which we thought it arises

from the same reasons, is the one which is dealing

with proving the theories (Ocal, 2017). While it is

rehearsing as if definitions and proofs have no

significance at secondary education, the theoretical

side of the mathematics is at the forefront at the

university, especially at the Algebra Math-1 Class.

Students even do not know how to study for this

lesson and they are having enormous learning

difficulties. Our suggestion to minimize this wavers

during this gradation process is to lecture the abstract

mathematics, such as logic, proving methods before

Linear Algebra and Mathematics Analysis I class, in

which the main subjects of the theoretical

mathematics have been taken into account.

Conceptual learning has much higher degree of

importance in the mathematics education for the

students who study at the mathematics department.

Unless the students can successfully comprehend

algebraic definitions, concepts and structures, they

will try to memorize these phenomenon’s(Soylu,

2008).

4 CONCLUSIONS

Based on the results of data analysis, it can be

concluded that the conceptual understanding of

ICMIs 2018 - International Conference on Mathematics and Islam

442

students on the topic of binary operations and group

theory is not satisfactory. Less than 50% of students

with good conceptual knowledge. It can be concluded

that there are still many students who lack of

conceptual understanding on binary and group

operations. Students define binary and group

operations based solely on their current knowledge.

This leads to the consequence that lecturers should

have new strategies to improve student conceptual

understanding.

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