Faculty of Cognitive Sciences and Human Development

STUDENTSt UNDERSTANDING OF EQUAL AND EQUALITY SYMBOLS IN MATHEMATICS

Izzatul Nabila Binti Sarbini

Bachelor of Education (Honours) QA 11.2

Mathematics 2009

198 2009

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membuat salinan untuk tujuan pengajian sahaja. 3. Pusat Khidmat Maklumat Akademik, Universiti Malaysia Sarawak dibenarkan

membuat pendigitan untuk membangunkan Pangkalan Data Kandungan Tempatan. 4. Pusat Khidmat Maklumat Akademik, Universiti Malaysia Sarawak dibenarkan

membuat salinan tesis ini sebagai bahan pertukaran antara institusi pengajian tinggi.

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menyatakan sekali sebab dan lempoh lesis ini perlu dikelaskan sebagai TERHAD.

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Pusa Khidmat Maklomat Ak mit UNJVE SlTr MAlAY ARAWAK.

STUDENTS' UNDERSTANDING OF EQUAL AND EQUALITY SYMBOLS IN MATHEMATICS

IZZATUL NABILA BINTI SARBINI

This project is submitted in partial fulfilment of the requirements for a Bachelor of Science with Honours

(Mathematics Education)

Faculty of Cognitive Sciences and Human Development UNIVERSITI MALAYSIA SARA W AK

(2009)

Statement of Originality

The work described in this Final Year Project, entitled "Students' Understanding of Equal and Equality Symbols in Mathematics"

is to the best of the author's knowledge that of the author except where due reference is made.

(Student's signature) IZZATUL NABILA BT SARBINI 14171

The project entitled 'Students' understanding of equal and equality in mathematics' was prepared by Izzatul Nabila Binti Sarbini and submitted to the Faculty of Cognitive Sciences and Human Development in partial fulfillment of the requirements for a Bachelor of Science with Honours (Mathematics Education)

Received for examination by:

(Dr Hasbee Hj U sop)

Date:

Gred

A-r

i ,.....

ACKNOWLEDGEMENT

First of all I would like to express my gratitude to my supervisor, Dr Hasbee Hj

Usop for his advise and support throughout the process of completing this study

which entitle "Students' Understanding of Equal and Equality Symbols in

Mathematics". I am sincerely thankful for his guidance especially his critical

comment that help me improved this study.

Moreover, I am deeply thankful to the Principal, the teachers and the students who

were involved in enabling the data to be collected for this study. Their

understanding and assistance are sincerely appreciated.

Also, special thanks to the Ministry of Education and Bahagian Perancangan dan

Penyelidikan Dasar Pendidikan for approving my study.

Lastly, not forgotten to my family and friends for their support and

encouragement throughout the completing of this study.

Thank you all.

I

Pusat Khidmat um. A1c~1lIp.mik

UNfVERSm MALAYSIA SARAWAK.

TABLE OF CONTENTS

Page

Acknowledgements Table ofContents 11

List ofTables v List of Figure vi Abstract V11

Abstrak viii

CHAPTER 1- INTRODUCTION

1.0 Introduction 1 1.1 Background of the study 2 1.2 Research problem 3 1.3 Objective of the study 4 1.4 Research question 5 1.5 Rationale of the study 5 1.6 Research framework 7 1.7 Significance of the study 8 1.8 Limitation of the study 9 1.9 Conceptual and operational definition of terms 9

CHAPTER 2 - LITERATURE REVIEW

2.0 Introduction 12 2.1 Algebraic thinking 13 2.2 Arithmetic 14 2.3 Misconceptual of equal and equality symbols

2.3.1 Misconception in solving question involving 15 equation in one side

2.3.2 Misconception in solving question involving 15 equation in both sides

2.3.3 Misconception in translating the mathematics sentence 16 into symbols

2.3.4 Definition of equal and equality symbols from the 17 students views

2.4 The importance of equal and equality symbols in algebra 17

11

l

I

I ,..

CHAPTER 3 - METHODOLOGY

3.0 3.1 3.2 3.3 3.4

Introduction Research design Participant Instrument Procedure 3.4.1 Data collection 3.4.2 Data analysis

CHAPTER 4 - FINDINGS AND DISCUSSION

4.0 Introduction 4.1 Findings of the research

4.1.1 Misconception in solving questions involving equation in one side

4.1.2 Misconception in solving question involving equation in both sides

4.1.3 Misconception in translating the mathematics sentence into numbers and symbols and vice versa

4.1.4 Definition of equal and equality symbols from the students views

4.2 Discussion

CHAPTER 5 - CONCLUSION, RECOMMENDATION AND IMPLICATION

5.0 Introduction 5.1 Summary

5.1.1 Misconception in solving questions involving equation in one side

5.1.2 Misconception in solving question involving equation in both sides

5.1.3 Misconception in translating the mathematics sentence into numbers and symbols and vice versa

5.1.4 Definition of equal and equality symbols from the students views

5.2 Implication of the findings 5.3 Recommendation 5.4 Conclusion

III

19 19 20 21

26 26

28

29

30

32

35

36

41

42

43

43

44

45 45 46

,....

References 47

Appendix A Letter from Bahagian Perancangan dan Penyelidikan 50 Dasar Pendidikan

Appendix B Instrument 52

IV

LIST OF TABLES

Table 3.1 Categories of rule violation

22

Table 4.1 The answer for rules violation questions (equation in one side)

29

Table 4.2 . The answer for questions with unknown(s) (equation in one side)

29

Table 4.3 The answer for rules violation questions (equation in both sides)

30

Table 4.4 The answer for questions with unknown(s) (equation in both sides)

31

Table 4.5 The answer for problem involving translation of mathematics equation into words and sentences

32

Table 4.6 The answer for problem involving translation sentences into numbers sndsymbols

33

Table 4.7 The answer for problem involving translation of mathematics equation into words and sentences

34

Table 4.8 The answer for the specific problem

34

Table 4.9 The definition of equal and equality symbols from students view

35

v

7

LIST OF FIGURES

Figure 1.1 Research framework

VI

,.....

ABSTRACT

STUDENTS' UNDERSTANDING OF EQUAL AND EQUALITY SYMBOLS

Izzatul Nabila Bt Sarbini

This study aims to identify the Fonn 1 students' understanding of equal and equality symbols. The specific objectives are to investigate whether the fonn 1 students have problem in solving questions involving equations of one side such as "23+24 = 47"; to investigate whether the fonn 1 students have problem in solving. questions involving equations of both sides such as "23+24-16 = _+25-13"; to investigate whether the form 1 students have problem in translating the mathematic sentence into symbols and vice versa and to find out the definition of equal and equality symbols from the students view. In this study, instrument were taken from Oksuz,(2007) and Knuth et al, (2006). The instrument consists of 12 questions and the questions were divided into five sections. The instrument includes various types of questions including true/false question, multiple choice questions, fill out question, and open question sentences. In this study, the participants were 15 Fonn 1 students from the secondary school located in Petrajaya, Kuching. During the data collection, the students were given 30 minutes to complete the exercise. Then, the data were analyzed and the results were discussed. From the findings it is found that students do not have any problem in solving the equation with none rule violation which means the equation is on the left side while the equal symbol is on the right side for example "4 + 6 = 10". However, the students have a slight problem when given questions that violate the rules where the equation is on the right side and the equal symbol is on the left side for example "9 = 5 + 4". Apart from that, this study also found that the students has problem in solving equation that appear on both sides. When given question such as "4 + 6 = [ ] + 2", the students think that lOis the answer for the unknowns. It shows that students think that the answer should come after the equal symbol. Moreover, the students were not able to write the correct equation to represent the given situation. Most of the students write an equation that shows their understanding in arithmetic and not in tenns of algebraic thinking. It is also found that none of the students were able to use the equal symbol to show the relationship between the equations. The students think that the symbol is a symbol to show the answer or to represent the answer and this might be the cause of their misconception in the exercise. Therefore, further research on investigating the reason of each misconception should be done. In addition, a research on finding the appropriate instructional strategies to solve this problem can be done in the future. This will improve the students understanding in learning mathematics.

Vll

ABSTRAK

Kajian ini bertujuan untuk mengenalpasti pemahaman pelajar tingkatan satu dalam penggunaan simbol sama dengan (=). Tujuan spesijik kajian ini adalah mengkaji kesalahan pemahaman dalam persamaan yang mengandungi operasi di satu bahagian seperti 23+24 = 47; mengkaji kesalahan pemahaman dalam persamaan yang mengandungi operasi di kedua-dua bahagian seperti 4+1 = 2+3; mengkaji kesalahan pemahaman pelajar dalam menterjemahkan persamaan matematik kepada perkataan dan ayat dan sebalihzya dan akhir sekali mengkaji definasi simbol sama dengan dari perspektif pelajar. Dalam kajian ini, satu latihan matematik telah digunakan untuk memperolehi data. Latihan matematik ini diambil daripada kajian­kajian terdahulu iaitu Oksuz, (2007) dan Knuth et ai, (2006). Latihan ini mengandungi 12 soalan yang dibahagi dalam 5 bahagian. Pelbagai jenis soalan telah digunakan termasuk soalan betullsalah, soalan terbuka dan soalan mengisi tempat kosong. Setiap soalan didalam latihan ini mengkaji setiap objektif. Kajian ini melibatkan J5 pelajar tingkatan satu dari sebuah sekolah menengah yang terletak di kawasan Petrajaya, Kuching. 30 minit telah diperuntukan untuk pelajar yang terlibat menjawab soalan didalam latihan yang disediakan. Sejurus data diperolehi, pengkaji telah menganalis data dan membuat analisa bagi menjawab setiap soalan kajian yang dikehendaki dalam kajian ini. Daripada data yang telah dianalisa, pengkaji telah berjaya mencapai objektif kajian yang telah ditetapkan. Antaranya ialah pelajar tidak menghadapi sebarang masalah dalam memahami simbol sama dengan sekiranya simbol berkenaan digunakan dalam persamaan yang mengandungi operasi di satu bahagian seperti 4+1 = 5. Walaubagaimanapun, pelajar menghadapi masalah dalam memahami persamaan yang ditulis dengan cara terbalik seperti 5 = 4+1. Pelajar juga mengahadapi masalah dalam menjawab soalan seperti 18 + 7 = [ 1 + 4. Terdqpat pelajar yang mengisi 25 didalam tempat kosong berkenaan. Ini menunjukkan pelajar mengganggap simbol sama dengan sebagai simbol · untuk member jawapan. Selain itu, kajian ini mendapati bahawa pelajar menghadapi masalah dalam menterjemah satu situasi dalam betuk persamaan matematik. Pelajar juga gagal menunjukkan simbol sama dengan sebagai simbol yang menghubungkan satu persamaan dengan persamaan yang lain. Pelajar menggambarkan simbol sama dengan sebagai simbol untuk menunjukkan hasil operasi atau jawapan. Masalah ini haruslah ditangani pada peringkat awal agar pelajar dapat memahami pembelajaran matematik dengan lebih mudah. Oleh itu, kajian selanjutnya bagi menyiasat sebab­sebab kesalahan pemahaman ini berlaku amatlah diperlukan. Selain itu, kajian untuk mencari kaedah pengajaran yang sesuai bagi mengatasi masalah ini juga perlu dilakukan.

Vlll

Chapter 1

Introduction

1.0 Introduction

This chapter provides an overview of the aims of the study, the related research

questions, and discusses their significance in the light of the changing educational

context of Sarawak. This chapter is divided into nine sections: the background of

the study, the research problem, objectives, the research questions, the rationale of

the study, the research framework, significance, the limitation of the study and the

conceptual and operational definitions in terms.

The topic of this study is "Students understanding of equal and equality symbols".

The understanding of equal and equality means the students conception and

definition of the symbols. It is how the students view the symbols and interpret

the meaning in their mind and ways of thinking. Their understanding is very

important for them to learn higher level mathematics in the future. There might be

some misconception on the meaning of the symbols. Moreover the students might

have the problem to translate a sentence to symbols. It means that they have

problem using the symbols to explain the mathematics sentence given. This may

affect their learning development on mathematics.

The symbol of equal and equality is frequently used in mathematics especially in

algebra. It is one of the basic symbols in mathematics. Therefore students

understanding of equal and equality symbols are very important in studying

mathematics especially algebra. According to Oksuz, (2007, pg 1), "Algebraic

thinking begins to develop in children in their early grades, before they learn

formal algebra. Their knowledge of the properties of numbers and operations

1

creates a basis upon which to learn algebra". Therefore the understanding of equal

and equality in early stages is very crucial in learning algebra and for the higher

level mathematics.

The lower secondary students should know how to translate the question "what

number when added to 22 gives 48?" into algebra as "if 22 + Y= 48, what is y?".

However, "over 20 years of research in developmental psychology and

mathematics education has indicated that many elementary school students (ages

7 to 11) have an inadequate understanding of the equal sign (Baroody &

Ginsburg, 1983; Behr, Erlwanger,&Nichols, 1980; Carpenter, Franke,&Levi,

2003; Kieran, 1981; McNeil & Alibali, 2005a; Rittle-Johnson & Alibali, 1999

cited in McNeil et ai, 2006). Therefore, the students understanding of equal and

equality needs to be investigated. Through this investigation, their problem on the

understanding will be known and solved.

This study will provide a lot of advantages to the teaching and learning of

mathematics. It will allow the teacher to understand their student perspectives on

equal and equality perspectives. This will give the teacher awareness on the

students' misconception of the symbols. Therefore, the teaching process will be

improved and so do the learning of mathematics. This will contribute to the

development of mathematics among the students.

1.1 Background of the study

This study had been done before by other researcher in other countries. According

to Oksuz, (2007), the reason for him to conduct this study is to answer these

questions "Why do so many students have trouble with the equality concept? How

do students view or understand the equal symbol? What are their misconceptions

regarding the concept of equality? What kind ofunderstanding do students need to

develop to have an accurate meaning of equality and the equal symbol? Finally,

what kind of instructional activities would make it easier to reach this

2

understanding?" (OksllZ, 2007). By answering all the questions, the teaching and

learning process of mathematics especially algebra can be modified in order to

make it better. Moreover it can maximize the students understanding in

mathematics.

From the previous study, there are some misconceptions on the understanding of

equal and equality symbols found among the students' ages 12 and 13. The

previous researcher focuses on understanding of this misconception in order to

find the instructional strategies that can be used to overcome this

misunderstanding.

1.2 Research problem

This study focus on the students' problem in understanding the equal and equality

symbols. From the previous study, "many elementary and middle school

demonstrate inadequate understanding if the meaning of equal sign, frequently

viewing the symbol as an announcement of the results of an arithmetic operation

rather than as a symbol of mathematical equivalence (Baroody & Ginsburg, 1981;

Behr et al., 1980; Kieran, 1981 ; Rittle-Johnson & Alibali, 1999, cited in Knuth et

al., 2006). The previous study is done in the international context. However, this

study has not been done locally and therefore this study will be conducted in the

Malaysian context.

The problem lies in the lack of understanding of equal and equality among the

lower secondary students. This will affect their understanding towards the

algebraic equations. Most of the students in lower secondary cannot define the

symbol of equal and equality. There are misconceptions among the students

toward the symbols. According to Knuth et al., (2006), "third- and fourth-grade

students' attempts to solve equations such as 4+3+5 = _+5 (e.g, Alibali, 1999)".

•It is found that many students provided answers of 12, 17 or 12 and 17- answers

that are consistent with an understanding of the equal sign as announcing a result"

3

(Falkner et al. cited in Knuth et al., (2006). This shows that the student sees equal

and equality symbols as a sign for giving the answers.

Students also have problem in translating the mathematic sentence in the form of

symbols. Take for example of the sentence ''what is the number that added with

28 and subtract 12 gives 13 added with 3?". Most of the students might have

problems in translating this sentence to "x+28-12 = 13+3, fmd x". Therefore this

will create problem in their learning process.

1.3 Objectives of the study

The general objective of the study is to investigate the students understanding of

equal and equality symbols. This means to investigate whether they truly

understand the concept and the definition of equal and equality. The specific

objective is to investigate the type of misconception that students have in solving

problems that involve the equal and equality symbols. From the research problem

mentioned, there are several kinds of misconception among the lower secondary

students in Malaysia. The specific objective includes:

a) To investigate whether the form 1 students have problem in solving

questions involving equations of one side such as "23+24 = 47"

b) To investigate whether the form I students have problem in solving

questions involving equations of both sides such as "23+24-16 =

+25-13".

c) To investigate whether the form 1 students have problem in translating

the mathematic sentence into symbols and vice versa.

d) To find out the definition of equal and equality symbols from the

students view.

4

1.4 Research questions

This study will be conducted in order to answer four research questions that are

related to the objectives. The questions of the study are:

a) What are the problems that the fonn I students face when solving

questions that involves equation ofone side?

b) What are the problems that the fonn I students face when solving

questions that involves equation ofboth sides?

c) What are the problems that the fonn 1 students face in translating the

mathematics sentence into symbols and vice versa.

d) What are the definitions of equal and equality symbols from the students

view?

I.S Rationale of the study

This study will be focused on the Fonn 1 students because the Fonn I students are

baving the transition from the primary school to secondary school. They probably

have some problems in understanding the symbols. According to Oksuz, (2007,

pg 2), "Kieran found that 12- and 13 years old had difficulty assigning meaning to

expressions such as 3a, a+ 3 and 2a+5 because one of the students said that "there

is no equal sign with a number after it". Their conception of equal and equality

ymbols from the primary school will be investigated and their misconception can

be defined.

Moreover, they will begin to learn the first part of algebra and their adequate

understanding of equal and equality symbols is very crucial. According to Oksuz,

5

(2007, pg 3), "Children must understand that equality is a relationship expressing

the idea that two mathematical expressions hold the same value". "A lack of such

an wtderstanding is one of the major stumbling blocks for students when they

move from arithmetic towards algebra" (Falkner, Levi & Carpenter, 1999; Kieran,

1981; Matz, 1982 cited in Oksuz, 2007). Therefore early detection of their

misconception is important in order to find the instructional strategies to

overcome this problem.

Other than that, this study is important in improving the mathematics among

students. Students' interest in mathematics can be improved when their problem

ofunderstanding can be overcome. The subject such as algebra will become easier

to wtderstand and master by the students. Therefore, this study will improve the

students understanding and achievement in the future.

6

Pusat Khidmat MakJumat Akademjl: lJNJVERSm MALAY.~IA SARAWAK.

1.6 Research framework

What are the What are the What are the problems that the problems that the defmitions of form 1 students are form 1 students equal and equality having when having in symbols from the solving questions translating the students view? involves equations mathematics at the both sides? sentence into

symbols and vice versa?

Data representing students' understanding of equal and equality symbols

Coding: form codes

Listings: codes

Summarizing: codes

Categorizing: codes

Students' understanding of equal and equality symbols

Figure 1.1

7

1.7 Significance of the study

From the teaching perspective, this study will help to increase the teacher

understanding of their students. This study will provide infonnation on the

students' idea about the symbols to their teacher. If there is any misconception, all

of it will be known by the teacher at the end of this study. Therefore, this study

will affect the teaching strategies ofmathematics.

Meanwhile, this study will also affect the learning perspectives in a way where

learning algebra will become less difficult. The result of the study and the

strategies taken can be used to overcome any misconception. As we know, by

understanding how the students think will give us the infonnation to built

strategies to help their thinking of equal and equality symbols.

Through this study, research on the understanding of equal and equality can

continuously be developed. Since this study is done in Malaysia and specifically

in Sarawak, it will give comparison between other research that has been done on

this topic in other countries. This study will give us awareness on how students

IDlderstanding in Malaysia differ from and similar to students in other countries.

This will contribute to the research development in Malaysia.

Teacher training also can gain advantages later from the result of this study. The

new instructional strategies will be suggested at the end of the study based on the

problem detected during the study. These strategies can be implemented during

the teacher training to improve education in Malaysia.

8

not

.8 Limitation of the study

'lben: are some limitations in doing this research. One of the problems is the

possibility that mathematics teacher may resist in letting the researcher their

teaching period to collect the data since the exercises need approximately 30

nrinntes to complete. Since the time taken for the exercises are too long, it might

affect the teaching process of the class.

Moreover, the researcher may need to revisit the school to collect more data or to

recollect the data. This would need permission and consideration from the school.

The researcher might have problems in getting the permission to come back to the

school. However, this can be solved by asking permission directly from the

principal and request for her consideration on this matter.

13 Conceptual and operational definitions of terms

According to Stanford Encyclopedia of Philosophy (2007), "The terms "equality"

(Gr. isotes, Lat. aequitas, aequalitas, Fr. egalite, Ger. Gleichheit), "equal," and

"equally" signify a qUalitative relationship. 'Equality' (or 'equal') signifies

correspondence between groups of different objects, persons, processes or

circumstances that have the same qualities in at least one respect, but not all

1'CI8peCtS, i.e., regarding one specific feature, with differences in other features.

Equality' needs to thus be distinguished from 'identity' - this concept

'gnifying that one and the same object corresponds to itself in all its features: an

object that can be referred to through various individual terms, proper names, or

descriptions. For the same reason, it needs to be distinguished from 'similarity' ­

the concept of merely approximate correspondence (Dann 1975, p. 997; Menne

1962 p. 44 ff.; Westen 1990, pp. 39, 120). Thus, to say e.g. that men are equal is

to say that they are identical. Equality implies similarity rather than

sameness"'. In tenn of mathematics, equality is "the paradigmatic example of the

more general concept of equivalence relations on a set: those binary relations that

9

of

polynomials,

specified

curriculum for

UDderstand

are reflexive, symmetric, and transitive. It goes beyond the other equivalence

matioos by also being anti symmetric. In fact, these four properties uniquely

determine the equality relation on any set S and render equality the only relation

on S that is both an equivalence relation and a partial order. It follows from this

that equality is the smallest equivalence relation on any set S, in the sense that it is

a subset of any other equivalence relation on S' (Wikipedia, 2008).

According to Vance, (1998), "Algebra is sometimes defined as generalized

arithmetic or as a language for generalizing arithmetic. However, algebra is more

than a set of rules for manipulating symbols; it is a way of thinking". Meanwhile,

Wikipedia (2008) define algebra as a branch of mathematics concerning the study

structure, relation and quantity". "Together with geometry, analysis,

combinatorics, and number theory, algebra is one of the main branches of

mathematics. Elementary algebra is often part of the curriculum in secondary

edocation and provides an introduction to the basic ideas of algebra, including

effects of adding and multiplying numbers, the concept of variables, definition of

along with factorization and determining their roots"

(Wikipedia,2008). Moreover, "algebra is a branch of mathematics in which

symbols, usually letters of the alphabet, represent numbers or members of a

set and are used to represent quantities and to express general

relationships that hold for all members of the set. A set together with a pair of

binary operations defined on the set. Usually, the set and the operations include an

idcDtity element, and the operations are commutative or associative" (The

American heritage dictionary ofthe English language).

In this study, algebra is one of the sections in mathematic syllabus. Students will

ICart to study algebra when they are in form 1 in chapter 7 (Algebraic expression).

will be continued in fonn 2 and form 3 whereas the topic of the chapter is called

Algebraic expression II" and "Algebraic expression III". Based on the Integrated

secondary school (mathematic fonn 1), (2002), the learning

~ectives of algebraic expression are to understand the concept of unknowns,

the concept of term and understand the concept of algebraic

10

UDderstanding of it.

Sammary

expression. Meanwhile, equal and equality symbols (=) is the symbols that shows

the right side is the same with the left side such as 3+x = x+13, 4=4 and so on.

'Ibc topic is all about knowing the true meaning of the symbols and the students

In this chapter, the researcher explained about the study that will be conducted.

'Ibis chapter allowed the researcher to explain more about the research topic and

related issues. Some of the result from other related study also has been included

in this chapter in order to support this study.

11