Students’ Understanding of the Equal Sign: A Case in Suburban School

Reni Wahyuni

, Tatang Herman

Department of Mathematics Education, Universitas Islam Riau, Pekanbaru, Indonesia

Department of Mathematics Education, Universitas Pendidikan Indonesia, Bandung, Indonesia

Keywords:

Algebra, Equality, Middle School, Pre-Algebra

Abstract:

Mathematics has an essential role in education and to get employment opportunities then algebraic topics are

one of the fundamental topics of efforts to reform mathematics education. Understanding algebra is related

to understanding the basic concepts of equality. There are not many students understand the meaning of the

equal sign. Therefore, this article is aimed to examine students’ understanding of the meaning of the equal

the qualitative method. The research carried out the 22 students, the second-semester secondary school, who

have learned the material of algebraic form and its operations, the seventh grade of SMP in Pekanbaru, Riau.

The data provided in this study shows that many students do not have a sophisticated understanding of the

meaning of the equal sign. Most students understand the equal sign as a result of a problem, such as solving a

problem likely a count from the left side to the right side. The view of the students’ misunderstanding about

the equal sign causes that the students perceive the meaning of the equal sign as the result of operations, not a

symbol of mathematical equality.

1 INTRODUCTION

In recent years, many mathematics education

communities have been interested in investigating

students’ difﬁculties in learning algebra (Guti

errez

et al., 2016). Algebra is a fundamental concept for

understanding high and advanced school mathematics

either for their daily lives or for employment.

Much research has done in investigating students’

difﬁculties in learning algebra, especially for middle

school students, but algebra problems still provide

exciting opportunities to be examined in many

aspects. Lately, the theme of early algebraic thinking

has become an interesting one. This theme indicates

that learning algebra, not only for students in middle

school but also students, in primary school, can learn.

Research in early algebra began with the

movement research of Carraher and Schliemann

about how young students were able to start learning

algebra in elementary school (Carraher, 2007; Kieran

et al., 2018). Algebra in middle school usually

has been a difﬁcult subject for the students. The

transition from arithmetic to algebraic thinking give

the difﬁculties problem to the students. From the

research Carraher and Schliemann, they showed that

the young students could learn algebra earlier. The

students are given a problem in their learning topic,

like numbers, then they could solve the problem

algebraically. However, their idea research gave

difference against doing the concept of a traditional

curriculum in which states that algebra can be learned

if students already have sufﬁcient knowledge in

arithmetic. The students, in elementary school, learn

about numbers and operations, but, in secondary

school, they will learn about algebra (Van Amerom,

2003; Chimoni et al., 2018). If the students who are

already proﬁcient in mathematics, it will easy to learn

algebra, but the fact that occurs, the student’s prior

experience in arithmetic is not enough to help students

learn algebra, even make students afraid of learning

algebra.

There are two ideas for algebra, the basic concept

for developing algebraic reasoning, that is variable

and equality (Knuth et al., 2005; Ertekin, 2017). In

this article, we focus on the discussion of the concept

of equality. Equality is an essential concept for

algebra achievement (Van Amerom, 2003; Chimoni

et al., 2018). In elementary school, the students

have learned about equality. They learn about ﬁnding

the value of an unknown number. Then in the next

level, the student will learn how to connect the two

mathematical expressions that have the same value.

Wahyuni, R. and Herman, T.

Students’ Understanding of the Equal Sign: A Case in Suburban School.

DOI: 10.5220/0009057100240028

In Proceedings of the Second International Conference on Social, Economy, Education and Humanity (ICoSEEH 2019) - Sustainable Development in Developing Country for Facing Industrial

Revolution 4.0, pages 24-28

ISBN: 978-989-758-464-0

The symbol that became a necessary concept is the

equal sign. Most of the student has the complicated

meaning of the equal sign. They generally perceive

that the equal sign as a result of arithmetical operation

rather than mathematical equality (Kieran, 1981;

Stephens et al., 2013; Vermeulen and Meyer, 2017).

The famous research founded by Falkner, Levi, and

(Falkner et al., 1999) showed that most of the

students perceive the meaning of the equal sign as

an operational symbol, calculating the number from

left side to right side. Most of the students do not

recognize the meaning of the equal sign as a relational

symbol.

The usage of the equal sign in mathematics has

four primary categories, such as an answer of the

operation, quantities equality, an expression that is

right for all value of the variable, and an expression

that express the new variable (Warren, 2006; Ertekin,

2017; Vermeulen and Meyer, 2017).

In Indonesia, the research on the meaning equal

sign does not take in much discussion in the area

of middle school. Whereas the understanding of the

equal sign is an essential concept in algebra learning,

it is necessary to advance mathematical concepts.

Indonesia Curriculum does not mention explicitly

about the meaning of the equal sign. Though the

meaning the equal sign is the basic concept for

the learning mathematics. In this paper, we drew

the students’ understanding of the equal sign with

examine their worksheet and thinking by interview.

However, the students have learned algebra topics,

their thinking of the meaning the equal sign describe

how they were understanding of the equal sign.

2 RESEARCH METHOD

This research used a qualitative methodology

and descriptive analysis, based on the participant

observation, interview, and worksheet (Creswell and

Creswell, 2017).The task was about algebra domain,

and the participants have already learned in the ﬁrst

semester in grade seven. The participant was from a

private middle school in Pekanbaru. The school has

a well-resourced in the suburban area of Pekanbaru

City. Its teacher is a very enthusiastic person and

proper motivation. The position of the researcher

in this research was observer and researcher directly

while the research was doing. Because this research

was a qualitative approach, the researcher was a

dominant aspect to control the process of the research,

and teacher became a partner and together decided

on examining the work of students and selecting

students to be interviewed. This study took part in

22 students (aged 13 to 14). In general, the students

have a different academic, and their daily life was

surrounding in that school.

Data for this research was taken from students’

answer sheet, such as the written document that

the students’ produces, and the audio record while

interviewing the participants. The participants were

given the essay test; then, they wrote the answer in the

blank paper as a document for the researcher. After

they ﬁnished the answer, the researcher examined the

answer sheet and discussed with the teacher how the

result of the students’ worksheet. We chose three

participants who solve the problem correctly and

three participants who solve the problem incorrectly.

The participants also were students who were able

to communicate when the researcher conducted

interviews and evaluated students’ understanding in

solving problems.

The data were analyzed by description for the

document answer sheet. We collected all document

and the code based on the student answer and what

their understanding of the equal sign while solving the

equation. The audio record, the result of the students’

interviewing, were analyzed in transcripts, and we

chose what students’ the misunderstanding and the

thinking of the equal sign.

3 RESULTS AND DISCUSSION

As it already mentions the concept of the equal

sign, that is, the equality shows two mathematical

expressions that have the same value. In the

ﬁrst question in this study, we present simple

equations that are similar assignments in the students’

elementary school. The task is the following “Find

the value of this number sentence 30 = ∆ + 20”.

Most of the students solve the problem by subtraction

operation. The students claim that the ﬁrst time

while seeing this problem as strange. The students

are seldom solving this problem. They say that they

usually solve the problem like ∆ + 20 = 30. So, it is a

simple problem to solve.

However, it is a strange form for themselves; they

can solve and ﬁnd a solution. There are 19 students

get the correct answer of the 22 seventh grade students

in this class. Three students have an incorrect answer

in seeing the symbol triangle. The mistake of the

students is when they move the triangle symbols from

the right side to the left side then do the addition

30 + 20 = 50.

The correct students ﬁnd the solution by using

subtraction operation as 30 − 20 = ∆”. However,

initial their thinking is confused; they move the

Students’ Understanding of the Equal Sign: A Case in Suburban School

triangle from the right side to the left side. It emerges

that in their thinking, the symbol should put it on the

left side. The student is usual in solving the equation

by counting from the left side to the right side. The

student’s answer sheet was categorized as operational

level in which the student’s ides that do the operation

number like addition or subtraction (Knuth et al.,

2008), when the students are asked the meaning of

the equal sign in that problem, student answer that it

is as a sign the answer to the problem. The researcher

has indicated that the operational interpretation of the

students; it is mostly from their experience inﬂuences

in prior knowledge in elementary school.

The usage of the equal sign is in this case as an

answer to the operation. The students are solving

the problem in this case for one unknown number.

The students think that it is a simple problem and

more accessible to solve it. Most research shows that

the students are not troubled while solving equation

3 + 4 =? but trouble in solving in 8 = 4 + ... research

found that proportion students to understanding the

function of the equal sign as relational only 37% of

the students at the same time just 48% of the students

know the equal sign as an operational (Asquith et al.,

2007).

As a result of this case, the students link their

thinking about the equal sign with the operational

aspect in arithmetic. The students perform the equal

sign as a symbol to get a ﬁnal answer from their

calculation.

Another case that the student says while

interviewing. The student is confused while moving

the symbol and number from the left side to the

right side, and vice versa. If the positive numbers

move from the left side to the right side, the positive

numbers will become the negative numbers. They put

the triangle on the left side, but they do not give the

negative sign in the triangle symbol. They do the

addition of number ∆ = 30 + 20. It is incorrect one

in this task. The inaccuracy of the students is one of

which causes the wrong answer.

While interviewing happening, another student

tries to solve the other problem. We ﬁnd the unusual

case of his answer sheet and try to investigate what he

did. That is his handwriting.

Figure 1: The student put the equal sign as a result.

In ﬁgure 1 show that the student put the equal

signs as a result of 16 + 1. He does not think that the

meaning of the equal sign is the equivalence of two

mathematical expressions. In this case, the expression

of 8 × 2 and 16 + 1.

They think the equal sign is the result of what they

are doing and ﬁnishing the problem. When we are

expanding the students writing, that is 8 × 2 = 16 +

1 = 17, of course, it became the wrong expression.

This way is that most of the students write while

solving the problem. The impact of these mistakes

that these students always make will give forthcoming

aspects in understanding the meaning of the equal

sign (Vermeulen and Meyer, 2017).

Next, we look at the third equation that it is

given to the students. The task is, how do we

solve this statement to be true? This is the number

sentence problem ♦ + 4 = ∇ + 1. This problem is

two mathematical expressions that have two unknown

numbers. The student should ﬁnd the two unknown

numbers, then the sum of the numbers should equal

between in the left side and the right side.

The student answer sheet seems to contradict from

the researcher thinking. He gives a unique answer

from that equation. He put ﬁve on the left side and

two on the right side. Firstly when we examine the

answer sheet, we do not understand the meaning of

this answer. After that, we interview the student for

knowing this reason.

Figure 2: The Student is misunderstanding while solving

equality task.

From ﬁgure 2, we directly say that it is a wrong

statement. We will not think why the student chooses

those numbers. In interviews, the student shows the

reason that he writes in solving this equation. The

transcript between researcher and students will be

shown in this following (R is a researcher, and S is

a student) :

R: Is 5 + 4 the same as 2 + 1?.

S: No.

R: Why do you put 5 and 2 in that the geometric

form?

S: Because? (silence and scratch the paper)

S: I think that 5 add 1 equal to 6 and 4 add 2 equals

to 6. So, the answer is 5 and 2.

ICoSEEH 2019 - The Second International Conference on Social, Economy, Education, and Humanity

R: Ok, let us see the question again!

S: (See the problem and try to change the answer).

R: Why do you change this position?

S: Because 5 + 1 = 6 and 2 + 4 = 6.

R: Are you sure? Can you explain it?

S: Hmm, I think this a correct one (he point his

script)

R: What do you think about the equal sign?

S: The equal sign stands for the result.

R: Can you explain it?

S: Yach, it is adding 5 + 1 = 6 dan 2 + 4 = 6. So,

the left side and the right equal to 6

R: Is there other meaning that you know about the

equal sign?

S: No.

This student sees the equal sign as the result of the

operation of the number. Nevertheless, in his answer

sheet, initially, he does not make aware of the equal

sign between two mathematical expressions. He does

the calculation to ﬁnd the value; then he decides that

the task is a true statement.

From the results of this research, Many students

view the equal sign as an operational symbol and the

result of the calculation of numbers (Stephens et al.,

2013) They fail to see the equal signs as the symbols

of the equivalence between the left side or the right

side (Kieran, 1990; Knuth et al., 2008). The other

student thinks that there is no relation between the

two statements either in left or right. The students

cannot be able to explain why this number sentence

is correct or not, and why they do the calculation by

adding between two number.

Concerning the correct answer of the students,

they are correctly in solving the equation, but they are

not correct the meaning of the equal sign (Knuth et al.,

2008). Most students did not seem to understand the

usage of the equal sign. The concept of the equal sign

is necessary to lead the student in learning algebra

either in middle school or the advance school. The

problem seems the simple equation, but it has a goal

to ﬁnd the students’ understanding. The students

are easier to ﬁnd the one unknown number, but it is

challenging to ﬁnd more unknown numbers (Falkner

et al., 1999; Warren, 2003; Ertekin, 2017).The limited

experience of the student in solving the problem

makes this task to be a difﬁculty. For a reason, the

students should typically use the meaning of the equal

The next research of this topic, the researcher

thinks about the students in the city area. The question

for the researcher is, “Is there the same problem for

the city school in Riau province?”. It will become the

next research for us to know the problem in Pekanbaru

city. The larger target for the research could help to

ﬁnd more problem.

4 CONCLUSIONS

Based on the discussion that students show in their

responses while solving the problem, indicate that

the student has learned and see the equal sign that

is a result activity from a statement in left to right.

Most of the student does not see the relation between

two statements in the right and left. In this research,

we cannot conclude that the students in the suburban

cannot understand the equality domain. In this aspect,

it is just looking at the view of how the understanding

of student while solving the equation.

As we reﬂect that in our background in this paper,

this is the notion of equality and the equal sign in

the middle classroom, and we should continue the

other aspect what the student interest and excited

while solving the task and bring in-depth discussion.

The students understanding the equal sign should

be developed as they learn about numbers and

operations. This understanding will give the student

the reﬂection about the equation and will lead them to

learn learning algebra in the next grade.

ACKNOWLEDGEMENTS

The author wants to express the special thank of

gratitude to the institution that is Universitas Islam

Riau (UIR), who gives the funding fee to following

this event.

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