Creative Education, 2019, 10, 2615-2630

https://www.scirp.org/journal/ce
ISSN Online: 2151-4771
ISSN Print: 2151-4755

Student Error Analysis in Learning Algebraic

Expression: A Study in Secondary School
Putrajaya

Md. Yusoff Daud, Ainun Syakirah Ayub

Centre of Innovation in Teaching and Learning, Faculty of Education, Universiti Kebangsaan Malaysia, Bangi, Malaysia

How to cite this paper: Daud, Md. Y., & Abstract

Ayub, A. S. (2019). Student Error Analysis
in Learning Algebraic Expression: A Study Algebraic expressions are an essential topic in the mathematics curriculum as
in Secondary School Putrajaya. Creative they are interrelated in the application of problem-solving in science and
Education, 10, 2615-2630.
mathematics. Previous studies showed that most students have a low level of
https://doi.org/10.4236/ce.2019.1012189
understanding of algebraic expressions. This study was conducted to identify
Received: October 16, 2019 the type of student error in learning Algebra Expression topics among Form 4
Accepted: November 25, 2019 students. A total of 67 students from Secondary School Putrajaya, comprising
Published: November 28, 2019
31 males and 36 females, were involved in the study. The instrument used in
Copyright © 2019 by author(s) and this study was the Algebraic Expression Diagnostic Test with a reliability val-
Scientific Research Publishing Inc. ue of α = 0.819. This diagnostic test covers two critical concepts in algebraic
This work is licensed under the Creative expressions, namely the development and functioning of algebraic expres-
Commons Attribution International
sions. The types of errors are classified according to Newman’s Error Hierar-
License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/ chical Model, such as the reading error, comprehension, transformation,
Open Access process skills, encoding or negligence. Data were analyzed using descriptive
statistics, namely mean, frequency, and percentage. Inferential statistics, name-
ly the Pearson r correlation. The findings show that students often make the
wrong kind of process skills. No students made any reading mistakes. The
number of students who form the wrong kind of comprehension, transfor-
mation, encoding, and negligence is small. Student’s error in the topic of al-
gebraic expressions is due to students’ weaknesses in finding the most signif-
icant common factor, cross-sectional processes, and weaknesses in mastering
fractions and negative numbers. It is hoped that the information obtained in
this study will help students and teachers to solve learning problems related
to the topic of algebraic expressions.

Keywords
Algebraic Expressions, Error Analysis, Transformation, Process Skills,
Encoding

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Md. Y. Daud, A. S. Ayub

1. Introduction
The Mathematics Curriculum in Secondary School has been formulated, refined,
and rearranged under the National Education Philosophy to provide students
with mathematical knowledge and skills aimed at developing systematic and
competent individuals who apply mathematical knowledge effectively and are
responsible for solving problems or making decisions. This motivates them to
address the challenges of everyday life in keeping with the latest science and
technology developments (MOE, 2013). Mathematics is an essential subject in
the curriculum of our country’s schools. These subjects are taught from all levels,
from kindergarten to higher education. Even in colleges and universities, ma-
thematics is still an important subject in most courses. Mastery in mathematics
requires students to understand and master basic concepts of computation.
Through conceptual understanding, directly train students to think constantly in
finding solutions to the problems they are facing. Through mastery of mathe-
matics, students’ thinking will grow and develop. Every problem encountered
will be investigated from various angles to find a solution. Therefore, a student
needs to develop an understanding of learning mathematics skills and concepts
to increase his/her desire and interest in learning mathematics and to improve
his/her ability to solve a problem especially in the 21st-century learning practice
(Saliza & Siti Mistima, 2019).

2. Background of the Study

Mathematics is a core subject at the primary or secondary level. However, stu-
dent achievement and interest in Mathematics is not very encouraging (Mazlan,
2002). Students still think that Mathematics is a difficult subject (Bed Raj, 2017).
Various changes in the educational world, including the teaching and learning
approach in the classroom, the use of computer technology and calculators, the
application of creative and critical thinking skills, and mastery learning have not
been able to erase students’ negative perceptions towards Mathematics. Mathe-
matics is an abstract subject. Therefore, the construction of a mathematical con-
cept will not be successful by memorization alone. Students often encounter
such problems with low levels of ability. They think that mathematics is a diffi-
cult subject to master and boring. This negative perception causes them to lose
interest in learning mathematics. As a result, mathematical achievement rela-
tively low and not satisfied by a particular individual.
Among the factors that influence students’ weaknesses in mathematics are
from the knowledge base or basic concepts and skills from previous learning
(Aini Haziah & Zanaton, 2018; Petterson, 1991). Students who fail to master the
necessary skills and ideas at any level of learning, can influence their achieve-
ment at the next level. Hence, it is vital for a teacher to make an assessment of
teaching and learning in the classroom and try to identify the difficulties and
mistakes that are often experienced by students.

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 Md. Y. Daud, A. S. Ayub

2.1. Problems Statements

Algebraic expressions are one of the most important topics in the school ma-
thematics curriculum. In general, algebraic applications can be found in all fields
of mathematics and science covering one-third of the secondary school mathe-
matics curriculum (Kementerian Pendidikan Malaysia, 2003). It is considered as
one of the important topics in the examination, namely the Lower Secondary
Assessment (PMR) and the Malaysian Certificate of Education (MCE). There-
fore, proficiency in solving algebra problems is very important to the overall
achievement of students’ mathematics in the national examination. Studies on
the development of understanding of topics in difficult mathematics have been
well documented (Warren, 2003). However, most of the studies emphasized the
topic of linear equations and students’ difficulty in solving linear equations.
However, studies related to the topic of algebraic expressions are still lacking and
need to be explored.
Algebra involves variables, while algebraic expressions contain variables, con-
stants, and operations symbols such as add, minus, multiply and divide. There-
fore, it is necessary for students to understand the concepts of the variables and
the meanings of algebraic terms in order to master algebra correctly (Filloy &
Rojano, 1989). Many studies have been able to identify some of the errors and
misconceptions among students in understanding algebra (Ling et al., 2016).
Many of the students do not understand the idea of a letter being used as a vari-
able (Booth, 1981). They tend to interpret a letter as just a specific number, and
different letters necessarily represent a varying number (Kuchemann, 1981).
Terms and regulations in algebra may also be a source of confusion for students.
Many of them find it difficult to follow abstract terms and to manipulate sym-
bols and numbers at the same time. Many laws or rules in algebra seem insigni-
ficant to students and this often causes them to create their own laws (Demby,
1997).
The majority of students in secondary schools still have a low level of under-
standing in algebra. This condition should be taken seriously by teachers who
teach mathematics. This is because mathematics teaching in secondary school is
at the highest level (Aida Suraya, 1991). For example, students will study the title
of Algebraic Expressions I during Form One. While in Form Two, students will
learn the topic of Algebraic Expression II where in this topic, the content of the
lesson will be deeper than the previous Algebraic Expressions I. Later, in Form
Three, students will also learn the topic of Algebraic Expressions III. If students
are not able to fully master the content of the lesson in the Algebraic Expression
I topic, it will be difficult for them to master the Algebraic Expressions II and
Algebraic expressions III.
In the topic of algebraic expressions, Saripah Latipah (2000) found that stu-
dents did not understand the basic concepts of algebraic expressions well, which
led to misconceptions in basic algebraic operations. Rosli (2000) also found that
students make mistakes in certain aspects of algebraic expressions such as sim-

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Md. Y. Daud, A. S. Ayub

plifying algebraic fractures, factoring and developing two expressions. Azrul

Fahmi and Marlina (2007) point out that algebraic expressions are one of the
topics in mathematics that students often make mistakes. If students cannot
master the basic concepts of mathematics in primary school, students would face
problems in the study of mathematics in secondary school and subsequently at
the tertiary level (Wong, 1987). Among the main factors that cause a low level of
understanding of the topics in Algebra expression among students is a poor
mastery of fundamental concepts and abstract algebraic expressions.
Based on the problem statement discussed, the researchers would like to con-
duct a study to identify and determine the extent to which the basic concepts of
mathematics in the Algebraic Expression topic among Form 4 students by con-
ducting the analysis of the types of errors made by the students in the Algebraic
Expressions topic since this topic is relevant and closely related to other topics
such as Functions, Expressions and Quadratic Equations and Concurrent Equa-
tions.

2.2. Research Questions

Based on the objectives of the study, this study was conducted to answer the fol-
lowing research questions:
1) What are the types of student errors in solving algebraic expressions?
2) What are the types of student errors in solving the problem of algebraic ex-
pressions factorization?
3) To what extent there is a relationship between student achievement in the
topic of algebraic expression and student mathematics achievement in the Lower
Secondary Assessment (PMR) examination?

2.3. Research Design

This study was a descriptive study conducted to identify the mistakes made by
Form 4 students in solving problems related to algebraic topics. The sample con-
sisted of 67 Form 4 students (science stream), different gender from Secondary
School Putrajaya. The sampling method used in this study was a simple random
sample method. This research instrument consists of a set of test questions that
focus on the topic of Algebraic Expressions, namely the Algebraic Expression
Diagnostic Test. The items contained in this instrument were modified from the
research instruments of Azrul Fahmi and Marlina (2007). Meanwhile, scoring
was based on modifications of Charles and Lester’s (1987) analytical scoring
scheme to assess respondents’ answers. The instrument was divided into two
sections, Part A and Part B. Part A is the demographic information of the res-
pondents. Meanwhile, Part B is an Algebraic Expression Diagnostic Test that
contains 20 subjective questions.
This diagnostic test question was given to respondents to be answered within
an hour. Table 1 is the subtopic table tested and item order in the Algebraic Ex-
pression Diagnostic.

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 Md. Y. Daud, A. S. Ayub

Table 1. Algebraic expression diagnostic test subtopic.

Algebraic Expression Diagnostic Test Subtopic Item No. Item Order No.

Development of Algebraic Expression:

Determining the development of the product for an expression

5 1, 2, 3, 4, 5
with a term

Determining the development of the product of two expressions 5 6, 7, 8, 9, 10

Algebraic Expressions Factorisation:

Converting the algebraic expressions that contain two terms to

5 11, 12, 13, 14, 15
the product of the term with an expression

Converting the algebraic expressions that contain two terms to

2 16, 17
the product of two expressions

Converting the algebraic expressions that contain three terms to

3 18, 19, 20
the product of two expressions

In this study, two field experts were consulted to determine the validity of the
study tool. Subsequently, a pilot study was conducted on 32 Form 4 students
who had similar characteristics to the sample in the actual study. The purpose of
this pilot study was to test the suitability of items used in terms of validity and
reliability. The Cronbach’s alpha value obtained was 0.819. According to Majid
Konting (2000), the alpha coefficient value exceeds 0.60 indicates that the in-
strument has high reliability.

3. Finding and Discussion

The discussion and findings of this study will focus on aspects related to the
respondents’ demographics and analysis of error types in the topic of Algebraic
Expressions based on Frequency of Errors. Newman’s Error Hierarchy Model
consists of six aspects, namely: 1) Reading, 2) Comprehension, 3) Transforma-
tion, 4) Process Skills, 5) Encoding and 6) Negligence.

3.1. Respondent Demographics

The respondent demographics consisted of 31 (46.3%) male students and 36
(53.7%) female students. In terms of Mathematics achievement at the Lower
Secondary Assessment (PMR) level, 43 (64.2%) of students received grade A, 19
(28.3%) of them received grade B, and 5 (7.5%) of students received grade C as
shown in Table 2.

3.2. Type of Error Analysis in Algebraic Expression Topics

The types of errors were classified based on student written work analysis. The
types of errors were identified based on the first breakdown point performed by
students. This type of error diagnosis was based on Newman’s Error Hierarchy
Model. The types of errors were classified either from reading type, comprehen-
sion, transformation, process skills, encoding, and negligence. This type of error
was verified through interviews conducted after the analysis of written work done.

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Md. Y. Daud, A. S. Ayub

Table 2. Respondent demographics of the study.

Statement Frequency Percentage (%)

Gender:

Male 31 46.3

Female 36 53.7

Total 67 100.0

Mathematics Achievement (PMR):

A 43 64.2

B 19 28.3

C 5 7.5

Total 67 100.0

The findings of the students’ error analysis of the subject of Algebraic expres-
sions as a whole indicate that the student’s achievement of the subtopics tested
was satisfactory, especially in the subtopics involved in the concept of algebraic
expressions. This can be seen by the percentage of respondents who answered
correctly for each subtopic tested above 70%. However, for the conceptual solu-
tion involving algebraic expressions factorization, there is one subtopic that in-
dicates that the percentage of respondents who answer correctly is less than 70%,
namely, the subtopic that converts algebraic expressions contains three terms to
the product of two expressions.
Table 3 shows the percentage of students who responded correctly to the
sub-topics tested based on the development of algebraic expressions and alge-
braic expressions factorization. Further description is based on error analysis
based on development factors and algebraic expressions factorization.

3.3. Types of Errors in the Development of Algebraic Expressions

The analysis of the types of errors in the subtopics of Algebraic Expression De-
velopment is described in detail as follows:
1) Type of error in determining the expansion of the product of an expression
with a term
The most common types of mistakes made in this sub-topic are the types of
negligence and the error types of process skills. For the first item, which is 2(x +
5y), 4 (6.0%) of the students made mistakes in the process skills in developing
operations involving algebraic expressions. For example, they solve 2(x + 5y) as
2x + 5xy and 2x + 5y. There was also one (1.5%) student who made a mistake by
giving the final answer as 2x + 10xy. The student can answer correctly when
asked to answer for the second time.
For the second item, m(m + 8), the most common type of error is negligence.
There were 19 (29.4%) students who made the mistake of negligence. Of these,
13 (19.4%) of the students answered m2 + 4. They got the right answer when
asked for a second time. Other errors made for this item were that 9 (13.4%) of

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 Md. Y. Daud, A. S. Ayub

Table 3. Percentage of correct answers according to Sub-Topic.

Percentage of
Algebraic Expression Diagnostic Test Subtopic
Correct Answers

Development of Algebraic Expression:

Determining the development of the product for an expression with a term 86.9%

Determining the development of the product of two expressions 76.1%

Algebraic Expressions Factorisation:

Converting algebraic expressions that contain two terms to a product of one

74.6%
pronunciation with one expression

Converting algebraic expressions that contain two terms to the product of

72.4%
two expressions

Converting algebraic expressions that contain three terms to the product of

66.2%
two expressions

the students made mistakes in the process skills. Students know how to use the
correct operations, but fail to develop algebraic fragments and mostly give the
final answer m2 + 4. Also, 2 (3.0%) students made the wrong type of encoding
and transformation for the item.
The most common type of error for the next item, uv(v + w), is also the negli-
gence type. A total of 4 (6.0%) students answered uv2 + uw. Students were care-
less when multiplying the uv pronunciation with the second pronunciation in
the expression v + w. When asked for the second time, they can respond to the
item correctly. In addition, one (1.5%) of the students made the wrong kind of
process skills and gave the final answer as u2v + uw. Students know how to use
proper operations and methods, but fail to do the calculations correctly.
The most common mistake for items −3x(2y − z) is the type of process skills.
There were 2 (3.0%) of the students made this type of error. The most common
error for items −3x(2y − z) is the type of process skills. 2 (3.0%) of the students
made this type of error. This error occurs when the student fails to handle the
negative sign when multiplying algebraic expressions and giving the final answer
as −6xy − 3xz. There was one (1.5%) student who made the mistake of saying
algebra and giving the wrong answer −6y + 3xz.
For the last item in this subtopic, −r(2q + r), the error made by the student is
the type of process skill. 2 (3.0%) of the students made this type of error due to
failure to handle algebraic pronunciation with negative sign and error while cal-
culating. The wrong final answer is given −2rq + r2. An analysis of the types of
errors for each item in this subtopic is shown in Table 4 as follows.
Overall, the most common types of errors made in this section are the types of
negligence and process skills. Students were careless in developing algebraic ex-
pressions, especially those involving fractions. For errors in process type skills,
most students know how to perform operations and can use correct operations
and methods, but fail to perform proper calculations. This type of negligence
and process skills process is most common when involving algebraic fractures.

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Md. Y. Daud, A. S. Ayub

Table 4. The type of error in determining the expansion of a product from an expression.

Frequency of Errors According to the

Item Transformation Process Skills Encoding Negligence Total**

1 4 1 5

2 1 9 1 19 30

3 1 4 5

4 2 1 3

5 2 2

Total** 1 18 1 25 45

Note: (**) The sum indicates the number of times an error was detected by item (row) and by error (col-
umn).

However, there are no reading and comprehension errors made by students in

this subtopic.
2) Type of error in determining the development of the product of two ex-
pressions
For the subtopic of determining the development of the product of two ex-
pressions, the most common type of student error is the encoding and process
skills. The solution to the first item of this section, (x + 2)(x + 2) indicates that 3
(4.5%) of the students made some encoding error. Students know how to use the
proper operations and methods and can perform the right calculations, but
failed to identify and write the final answer correctly. So, the final answer was
only up to x2 + 2x + 2x + 4. There were 2 (3.0%) of the students who made the
wrong type of process skills due to incorrect calculation work and gave wrong
final answers such as x2 + 2x and x + 4x + 4.
Process skills are the most common type of error for the next item, namely (x
− 2)(x − 3). A total of 7 (10.4%) students made an error in the process skills for
failing to make the correct calculations for similar pronunciations and involving
negative marks. For example, the most common type of error in process
processing was (x − 2)(x − 3) = x2 − x + 2x + 6. The student also did the type of
error in coding for this item. 2 (3.0%) of the students who made this type of er-
ror gave x2 − 3x − 2x + 6 answers and failed to write the final answer correctly.
The same type of encoding error was repeated by the same student for the
item (2x + 7)(2x − 7). A total of 8 (11.9%) students made an error in the coding
and failed to identify and write the final correct answer. Most of the final an-
swers given were 4x2 − 14x + 14x − 49. They did not continue the settlement
process after the development. This type of error in process skills was also a ma-
jor issue when students failed to develop items that involve negative markings
and errors in computation. A total of 9 (13.4%) students made this mistake.
Among the wrong answers given were 4x2 + 49 and 4x − 14.
The same type of encoding error was repeated by the same student for the
next item (a + b)(a + b). A total of 13 (19.4%) students made this type of error
and all of them gave the final answer as a2 + ab + ab + b2. There were also 11

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 Md. Y. Daud, A. S. Ayub

(16.4%) students who made a mistake in the process skills because they could
not perform the correct operation when adding two identical algebraic expres-
sions, namely ab. Examples of final answers given were a2 + a2b2 + b2, a2 + ab2 +
b2 and a2 + 2a + 2b + b2. In addition, 5 (7.5%) of the students made a mistake by
solving (a + b)(a + b) = a2 + ab + b2 for being careless when adding two similar
ab terms. However, they were able to respond correctly after being asked to try
out for the second time. Only one (1.5%) student made the type of transforma-
tion error.
For the last item in this subtopic, which is (2p − q)2, the most common error
is the type of process skills. A total of 12 (17.9%) students made this type of error
for failing to develop the expression properly and gave the final answer like 4p2 −
q2. In addition to the process type error, there were 3 (4.5%) students who made
the error due to negligence, 3 (4.5%) students made the error type of encoding,
and only one (1.5%) student made the error type of transformation. Analysis of
the types of errors in determining the development of the product of two expres-
sions is detailed in Table 5.
On the whole, the most common error made by the students in subtopics of
determining the development of the product of two expressions is the type of
process skills. Errors of the process skills are often encountered, especially for
items with negative terms and expressions that do not involve any figures or
numbers such as expressions (a + b)(a + b). Another common type of error is
encoding. There are no reading and comprehension errors made by students in
this subtopic.

3.4. Types of Errors in Algebraic Expressions Factorization

The types of errors analysis in the subtopics related to Algebraic Expressions
Factorisation are described in detail as follows:
1) Type of error in converting algebraic expressions that contain two terms to
the product of one term with one expression
One type of error that students often make in converting algebraic expressions
that contain two terms to the product of one term with one expression is the

Table 5. Type of error in determining the development of the product of two expressions.

Frequency of Errors Based on Newman’s Error Hierarchy Model

Item Transformation Process Skills Encoding Negligence Total**

6 3 2 5

7 7 2 9

8 9 8 17

9 1 11 13 5 30

10 1 12 3 3 19

Total** 2 42 28 8 80

Note: (**) The sum indicates the number of times an error was detected by item (row) and by error (col-
umn).

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Md. Y. Daud, A. S. Ayub

process skills. The findings show that for 8pq − 12q items, 6 (9.0%) of the stu-
dents failed to get the correct answer due to the error type of process skills. Stu-
dents know how to use the correct operations and methods for factoring in these
expressions but failed to compute the calculations properly. Examples of solu-
tions shown were 2q(4p − 6) and 2(4pq − 6q). There were also 2 (3.0%) students
who made an error due to negligence by giving answers such as 4q(2 – 3). This
error may have been caused by the student’s inability to answer the question.
Only one (1.5%) of the students made the error type of transformation.
For the next item, factoring in the expression 3mn2 + 21m, the most common
error made is the process skills. However, the number of students who make this
type of error is small, only 3 (4.5%). Errors were also caused by the failure to find
common factors for both expressions. Some of the wrong answers given were
3mn(n + 7), 3m(2mn + 7) and 7mn(n + 3). In addition, only one (1.5%) student
made an error due to negligence and one (1.5%) student made the error type of
transformation. An example of wrong answers given for negligence was 3m(n +
7).
The type of process skills error increases when it comes to expression, where
one of the terms does not have any coefficients such as the 3kp − k2p item. A
total of 18 (26.9%) students made this type of error. Of these, 9 (13.4%) of them
gave answers k(3p − kp) and the rest gave answers such as 3kp(1 − k), 3k(p −
kp), 3p(k − k2) and others. There were 2 (3.0%) of the students who made this
type of transformation error incorrectly because they failed to describe the ques-
tion into a form that allowed them to use the appropriate operation for the alge-
braic expressions factorization given.
The most common type of error in the next item, 21ab2c + 14bcd is also the
process skills. A total of 16 (23.9%) of the students made the error type of
process skills for this item. This is also due to the students’ weakness in finding
common factors for both terms in a given expression, and this further increases
their difficulty in finding solutions when involving more variables in a term. In
addition, 5 (7.5%) of the students made the error type of negligence by providing
solutions such as 7bc(3a + 2d) and 7b(3ab + 2d). They can give the right answer
when they try again. Another type of error was the transformation, where 3
(4.5%) of the students made this error. Students understand the requirements of
the question, but cannot find and formulate a method for the algebraic expres-
sions factorization.
Students also make the same type of error on items 48m2n + 12mn2, which is
process skills. A total of 21 (31.3%) students made this type of error. Most of
them give answers such as 4mn(12m + 3n), 6mn(8m + 2n) and 12(4m2n + mn2).
This error also occurs due to students’ inability to find the largest common fac-
tor for each term in a given expression. On the other hand, the type of errors
made was negligence and transformation. Among the answers given for negli-
gence was 12mn(4 + n). Students will be able to provide the correct answer when
asked to try again. Analysis of the type of errors for this subtopic is summarized
in Table 6 as follows:

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 Md. Y. Daud, A. S. Ayub

Table 6. A type of error in converting algebraic expressions that contain two terms to a
product of one term with one expression.

Frequency of Errors Based on Newman’s Error Hierarchy Model

Item Transformation Process Skills Encoding Negligence Total**

11 1 6 2 9

12 1 3 1 5

13 2 18 20

14 3 16 5 24

15 2 21 4 27

Total** 9 64 25 85

Note: (**) The sum indicates the number of times an error was detected by item (row) and by error (col-
umn).

The findings show that the type of errors in process skills is often made when
involving expressions or terms that have a variable with the highest power
square. Hal This causes students to become confused in the process of finding
common factors that may cause them to make mistakes inadvertently. However,
there was no error in the reading, comprehension, and encoding of the students
for this subtopic.
2) Type of error in converting algebraic expressions that contain two terms to
the product of two expressions
Items in the subtopic: converting algebraic expressions that contain two terms
to the product of two expressions: item 16 and item 17. Items in this subtopic
require students to use their identities in algebraic expressions, namely a2 − b2 =
(a + b)(a − b). The most common type of errors is comprehension and process
skills.
For item 16, namely x2 − 64, there were 3 (4.5%) students who made the type
of error in process skills. Most of them may not even know the identity of the
algebraic expression. So, students do not know how to perform operations and
only make random guesses. Examples of random response answers given were (x
+ 16)(x − 16). In addition, 2 (3.0%) of the students made an error in compre-
hension and 2 (3.0%) of the students made an error in transformation. The type
of error in comprehension occurs because students do not understand the ques-
tion and they failed to understand the term “the product of two expressions”. So
they don’t know how to implement a solution and just make a random guess.
The type of error in transformation occurs because students use algebraic ex-
pressions r incorrectly, by solving x2 − 64 = (x − 8)2. Only one (1.5%) student
made an error in encoding. For example, students do not proceed with the solu-
tion after obtaining x2 − 64 = x2 − 82.
The students also made the same type of error for the next item, 4y2 − 36,
namely process skills. A total of 22 (32.8%) of the students made this type of er-
ror also because they did not know how to use the algebraic expressions and
gave incorrect answers such as (2y2 + 6)(2y2 − 6). There were also 7 (10.4%) stu-

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Md. Y. Daud, A. S. Ayub

dents who made an error in comprehension because they failed to understand

the term “the product of two expressions”. Analysis of the type of errors for this
subtopic is summarized in Table 7 as follows:
On the whole, the most common type of error in converting algebraic expres-
sions that contain two terms to the product of two expressions is process skills.
This error may occur because students were unclear about the use of identities in
algebraic expressions. Another type of error made was comprehension, trans-
formation, and encoding.
3) Type of error in converting algebraic expressions that contain three terms
to the product of two expressions
The type of errors made by students in this section consists of process skills
and comprehension. Process skills were the type of error made by 13 (19.4%)
students for item x2 − 7x + 12 and 12 (17.9%) students for item x2 + 2x − 15.
Most students made an error in finding the right pair of factors and wrongly
placing positive and negative marks on the selected factor pairs such as x2 − 7x +
12 = (x + 3)(x + 4) and x2 + 2x − 15 = (x − 5)(x + 3). For item 3x2 + 9x + 6, 40
(59.7%) of the students also made the same type of process skills due to the same
factors.
For all three items, only one (1.5%) student made the same type of compre-
hension error, and the other student made the same error. Students gave ran-
dom responses that do not mean any solution to the three items. Examples of
wrong answers shown were factorisation x2 − 7x + 12 as x(−7x + 12), x2 + 2x −
15 as x(2x − 15) and 3x2 + 9x + 6 as 3x2(9x + 6). Analysis of the type of errors for
this subtopic is summarized in Table 8 as follows:
Overall, the most significant type of error in this subtopic is process skills.
This type of error occurs because the student fails to perform the cross-sec-
tional process to find the appropriate factor pairs for the given expression. In
addition, some students made a comprehension error. However, no student
has made an error in reading, transformation, encoding and negligence in this
subtopic.
A summary of the types of errors for the entire topic is given in Table 9.
The Relationship Between Achievements In The Topics of Algebraic Expres-
sions With Lower Secondary Assessment (PMR) Achievements.

Table 7. A type of error in converting algebraic expressions that contain two terms to the
product of two expressions.

Frequency of Errors Based on Newman’s Error Hierarchy Model

Item Comprehension Transformation Process Skills Encoding Total**

16 2 2 3 1 8

17 7 22 29

Total** 9 2 25 1 37

Note: (**) The sum indicates the number of times an error was detected by item (row) and by error (col-
umn).

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 Md. Y. Daud, A. S. Ayub

Table 8. Type of error in converting algebraic expressions that contain three terms to the
product of two expressions.

Frequency of Errors Based on Newman’s Error Hierarchy Model

Item Comprehension Process Skills Encoding Negligence Total**

18 1 13 14

19 1 12 13

20 1 40 41

Total** 3 65 68

Note: (**) The sum indicates the number of times an error was detected by item (row) and by error (col-
umn).

Table 9. The whole type of errors in the topic of algebraic expressions.

Item Reading Comprehension Trans-formation Process Skills Encoding Negligence Total*

1 4 1 5

2 1 9 1 19 30

3 1 4 5

4 2 1 3

5 2 2

6 3 2 5

7 7 2 9

8 9 8 17

9 1 11 13 5 30

10 1 12 3 3 19

11 1 6 2 9

12 1 3 1 5

13 2 18 20

14 3 16 5 24

15 2 21 4 27

16 2 2 3 1 8

17 7 22 29

18 1 13 14

19 1 12 13

20 1 40 41

Total* 12 14 214 30 45 315

The relationship between respondents’ achievement in algebraic expressions

and the mathematical achievement of respondents in the Lower Secondary As-
sessment (PMR) examination was determined by Pearson correlation analysis, r
= 0.618. Thus, respondents’ achievement in the topic of algebraic expression has
a strong correlation with the mathematics achievement of the respondents in the

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Md. Y. Daud, A. S. Ayub

PMR examination.

4. Conclusion
The findings showed that the most common type of error made by students is
process skills. Some students made other types of errors such as comprehension,
transformation, encoding, and negligence. However, the number of students
who made this type of error is small. There are no errors in reading shown by
the students on this topic. The most significant errors made by the students are
algebraic expressions factorization, especially for an expression with three terms.
In addition, the most significant errors can also be seen when it comes to items
that contain algebraic fractions. The findings also showed that when students fail
to master a certain level of learning in algebraic expressions, students will have
difficulty mastering the next level of learning or skills in the topic. The quality of
education that teachers provide to students is dependent upon what teachers do
in their classrooms (Zakaria & Iksan, 2007). Execution of duties in cooperative
learning can develop self-confidence in pupils. A study by Zakaria, Chin, &
Daud (2010) found that cooperative learning improves students’ achievement in
mathematics. Further, cooperative learning is an effective approach that mathe-
matics teachers need to incorporate into their teaching. Lately, one of the initia-
tives proposed by the government is the Massive Online Open Course (MOOC)
is web-based learning that can be accessed anywhere and anytime. Integrating
the technology into the learning process can help improve understanding of the
subject matter such as mathematics (Abdul Wahab et al., 2018; Nordin et al.,
2016).

Funding
This work was supported by UKM [Grant PP-FPEND-2019] and [Grant GG-
2019-018].

Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this pa-
per.

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