diagnosing high school students’ mathematics misconceptions

ISSN 2309-0081 Shoaib & Akhter (2020)

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www.irss.academyirmbr.com December 2020

International Review of Social Sciences Vol. 8 Issue.12

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Diagnosing High School Students’ Mathematics

Misconceptions

ALMAS SHOAIB Ph.D. Scholar, Institute of Education and Research, University of the Punjab, Lahore.

Email: [email protected]

Tel: +923484819148

Dr. MUMTAZ AKHTER Prof., Dean School of Social Sciences and Humanities,

University of Management and Technology, Lahore.

Abstract

The study was designed to diagnose high school students‟ Mathematics misconceptions through a

descriptive survey. The 1500 students from 50 high schools of Lahore were selected by using a two-stage

cluster random sampling technique. The misconceptions two-tiered test was developed after reviewing the

literature and textbooks which cover various topics of Mathematics i.e. real and complex numbers, linear

equations and inequalities, algebraic expressions and formula (surds), quadratic equations, algebraic

manipulation, variation (ratio), sets and functions. The data were analyzed through item wise content

analysis technique. It is concluded from the results that high school students have numerous

misconceptions in the content of Mathematics. Students were memorized properties and shortcuts to work

with Mathematics problems. Correspondingly, they were unable to write the reasons. It is recommended

from the results that textbook of Mathematics at the secondary level may be revised by experts keeping in

mind that the content like LCM, HCF, sets and functions, inequality, radicals (surds), factorizations and

complex numbers, etc. given in textbooks should help students to remove their misconception instead of

leading to misconception.

Keywords: Complex Numbers, HCF, LCM, Mathematics Misconceptions, High School Students.

Introduction

Students come into the classrooms with their beliefs, perceptions, and thoughts about the world. When they

learn something new about any concepts or topic, they build new knowledge of their prior knowledge.

Correspondingly, they have an extraordinary reasoning framework that is utilized in sense-making and

expressing the world. These reasoning frameworks are flawed or inadequate as they establish the bases for

misconceptions (Docktor et al., 2015; Sarwadi & Shahrill, 2014). Misconceptions emerge when students

neglect to connect new information to past information for which the brain has established cognitive

networks. Students depend on strategies created through their involvement in comparative material when

new information isn't tied down to existing systems to tackle new issues (Hiebert & Grouws, 2007).

Students can't distinguish between terms of misconceptions and error. Luneta and Makonye (2010)

expressed that if the student comprehends an idea as essentially not quite the same as its logical importance,

at that point he/she most presumably will build misinterpretation. Misconceptions are the subset of

mistakes, which implies one can characterize all misconceptions as errors, however, all errors may not be a

misconception. At the end of the day, the error is the aftereffect of misconception, or misconception is a

sort of recognition that systematically produces an error (Aydogan & Gelbal, 2017). Though, an error is

viewed as an arbitrary mistake whereas misconceptions emerge when students erroneously apply

previously learned techniques to take care of new issues (Hiebert & Grouws, 2007).

mailto:[email protected]


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Mathematics has been the subject of a considerable body of research conducted over recent decades. It is a

progressively moderate difficult subject to learn at the school level. In Pakistan, different researchers have

indicated that Mathematics is a difficult subject, as far as scoring (for example students‟ academic

achievement) and a grip of the content matter (Ali, 2011; Akhter et al., 2015; Halai, 2007; Jameel & Ali,

2016). The misconceptions that students create during primary classes or carry with them to the school

from the community can make restraints in the progressing learning of mathematical conceptions,

subsequently delivering poor achievement in Mathematics (Mohyuddin & Khalil, 2016). According to

Shoaib and Saeed (2016), students try to learn Mathematics by heart like a theory. Therefore, they develop

misconceptions about various concepts of Mathematics i.e. Division, Multiplication, Addition, and

Subtraction (DMAS) rules, sets, functions, factorization, algebra, geometry, drawing the geometrical

shapes, algorithms and manipulate expressions (Kaput, 2017; Karakus & Karatas, 2014; Shoaib & Saeed,

2016). Experts believe that errors are the signal of deeper misunderstandings about algebraic concepts that

are not haphazard but are systematic. It derives from their own created theories and experiences (Russell et

al., 2009). The literature showed that in Pakistan fewer researches were conducted on students‟

misconceptions in Mathematics. But these researches were limited to Science subjects and Mathematics at

various levels (Bakar & Ali, 2018; Mohyuddin & Khalil, 2016; Naz, 2008; Shoaib & Saeed, 2016).

Therefore, it was needed to diagnose students‟ misconceptions about the Mathematics curriculum at the

secondary level in Pakistan.

Research Objective

The objective of the study was:

1. To diagnose students‟ misconceptions about the Mathematics curriculum at the secondary level in

Pakistan.

Literature Review

Studies on misconceptions in Mathematics have uncovered that students‟ misconceptions vary class-wise

and concepts wise as well. Students have conceptual misconceptions, such as the meaning of the square

root sign, the relationship between the square root and the square root expression (İşleyen & Mercan,

2013). A few investigations have been led on students‟ misconceptions and their relationship with academic

achievement (Alghazo & Alghazo, 2017; Egodawatte, 2011; Kaplan et al., 2011; Nassir et al., 2017).

Students' errors are causally decided, and frequently deliberate. Systematic errors are generally an outcome

of students‟ misconceptions. These can incorporate the inability to make associations with what they know.

Students‟ errors are extraordinary, and they mirror their comprehension of an idea, issue, or method

(Sarwadi & Shahrill, 2014). Karakus and Karatas (2014) made an investigation to decide secondary school

students‟ misconceptions about fractals. They used a test based on three dimensions of students‟

misconceptions i.e. misconceptions on the definition, recognition, and errors in drawing fractal shapes.

Results showed that students hold misunderstandings about the formal definition of a fractal and drawing

fractals, even though students could perceive instinctively the given shape as fractal or not.

Mohyuddin and Khalil (2016) analyzed that primary school students had misconceptions in eight

conceptual areas i.e. numbers, operations on numbers, fractions, operations on fractions, decimals,

measurement, information handling, and geometry. They concluded that students‟ misconceptions fall into

four categories i.e. the meaning of brackets; basic Mathematics rules; notation; and properties of

trigonometry (Nassir et al., 2017). Furthermore, Çelen (2018) conducted a study on students‟

misconceptions regarding the ratio of proportion that contributes to their elimination. The researcher found

that students had various misconceptions about the proportion and ratio. However, Kaplan et al. (2011)

made a study to determine the mistake and misconceptions of students about the proportion and ratio

concept. They had misconceptions in constructing ratio, proportion concepts, and proportional reasoning.

Similarly, Kucuk and Demir (2009) encountered misconceptions in teaching mathematics in the classroom


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and found that the students have many misconceptions about geometric shapes and their representations,

rational numbers, and operations with rational numbers.

However, Doğan and Çetin (2009) examines the conceptual misconceptions of right and inverse

proportions of 7th and 9th-grade students and found that students have various misconceptions. Almog and

Ilany (2012) indicated that the main types of mistakes that students consistently made when solving

absolute value inequalities. Gür and Barak (2007) explore high school students‟ mistakes and errors in

derivatives and determine the areas in which students have probable misconceptions. They concluded that

students could not understand derivative definition that depends on the limit, make mistakes in composite

functions and trigonometric functions, and establish wrong relations between slopes of tangent and normal.

Keceli and Turanli (2013) conducted a study to identify the students' misconceptions and common mistakes

about complex numbers. The authors also concluded that eleventh-grade students have similar

misconceptions about complex numbers as undergraduate students. Nonetheless, Erbas et al. (2009) said

that students faced difficulties to solve linear equations of mathematics and have misconceptions about

linear equations. Egodawatte (2011) made a study on students‟ errors and misconceptions in algebra at the

secondary school level. He said that the major difficulty in word problems was translating them from

natural language to algebraic language and students used guessing methods in their classrooms while

learning Mathematics. The literature showed that students have various misconceptions in Mathematics.

Theoretical Framework

Misconceptions are tough and difficult to snuff out through teaching. Educational psychologists were

inclined towards Piaget's constructivist epistemology and accepted that cognitive development is dependent

on the scaffold of explicit concepts and assimilations between these ideas gained during the dynamic life

expectancy of an individual (Agarkar & Brock, 2017; Arias et al., 2016; Furtak et al., 2016). Jiménez-

Aleixandre and Brocos (2017) portrayed equivalence between Piaget's ideas of assimilation and

accommodation. Students interpret tasks and instructional exercises including new ideas as far as their

earlier information. Misconceptions are the attributes of beginning periods of learning since students

existing information is lacking that supports only fractional understandings (Calais, 2018). Students learn

by changing and refining that earlier information into progressively sophisticated structures, as their current

information is perceived to be lacking to clarify phenomena and solve problems (Calais, 2018; Treagust et

al., 2014). The constructivist theory proposes that students can be fruitful in tackling an issue if they select

and apply the correct solving schema. Students likely had both the right and wrong schemas in their long-

term memory, however, they recovered inappropriate data. Despite the presence of the right data, there are

a few circumstances where students apply incorrect schemas while having the right ones in their minds.

Theory expressed that the purpose behind reviewing inappropriate data was that the right data secured or

restrained (Brod et al., 2013).

Research Methodology

In this study, the researchers adopted a descriptive survey to explore the secondary school students‟

misconceptions regarding the Mathematics curriculum through a test. The population of the present study

was comprised of 46146 (10th

grade) students (i.e. 22706 boys and 23440 girls) that are enrolled in

academic session 2017-19 in the 334 public sector school of Lahore District (Census of School Education

Department, 2018). According to Gay et al. (2009), the 1513 sample is required for the population of

46146, at a 95% confidence interval. The students were selected using a two-stage cluster random sampling

technique. For the first stage, 15% (23 boys and 27 girls) public sector schools were selected through a

proportionate stratified cluster random sampling technique. In the second stage, 1500 high school students

(30 from each 10th

-grade science section) were selected randomly to diagnose students‟ misconceptions.


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Research Instrumentation

Most of the studies related to mathematics misconceptions used traditional one-tier achievement tests

(Erbas et al., 2009; Gür & Barak, 2007: Keceli & Turanli, 2013; Kucuk & Demir, 2009, Tunç et al., 2011).

However, Peşman and Eryılmaz (2010) stated that the wrong answers given in these tests can be considered

as misconceptions although they are not. Students can give the wrong answer due to lack of knowledge,

wrong information in the question or faulty thinking during the test, but these do not prove that student has

misconceptions on that concept. Therefore, tests that included more than two tiers are stated to be more

appropriate to be able to detect the misconceptions.

Therefore, keeping in view the investigators' remarks, the misconceptions two-tiered test was developed

after reviewing the literature and textbooks to examine students‟ misconceptions about the Mathematics

curriculum. The test was cover various topics of Mathematics i.e. real and complex numbers, linear

equations and inequalities, algebraic expressions and formula (surds), quadratic equations, algebraic

manipulation, variation (ratio), sets, and functions. It consisted of 30 multiple-choice questions (MCQs).

Each item consisted of one correct option and three distractors. In the test, students had to write reasons to

select the option along with the selection of options. All questions were selected from the Mathematics

textbooks of the science section. The content validity of the test was determined by three subject specialists

and two assessment experts. The test was conducted on 240 students to confirm reliability. In the light of

validity and reliability, nine inappropriate MCQs were excluded from the final test. The final test

encompassed 21 items. The reliability of the test was α = 0.720. As stated by Cortina (1993) and Taber

(2018) the value of α > 0.70 is acceptable.

Analysis

Item wise content analysis technique was used. The misconceptions two-tiered test consisted of 21 items

and administered to 1500 sampled students. The test papers were marked and categorized for

misconceptions. Content Analysis was used to analyze and interpret the presence of certain words, themes,

or concepts within some given qualitative data (i.e. text).

Misconception in Real and Complex Numbers

Item#1: What is the purest form of a complex number z?

The purest form of a complex number consisted of the real part (Dar & Haq, 2017). The responses of the

second item “the purest form of a complex number z is” showed that only 37.4% of students gave the right

answer i.e. bi. On the other hand, the majority of the students gave the wrong answer. The detail about the

selection of the wrong option showed that 18.5% of students „-bi‟. Similarly, 36.1% of students selected

„0i‟ and 5.7% of students selected „-0i‟. The rest of the 2.3% of students (35) left the item unanswered.

It was revealed from the review of students‟ test scripts that those students who selected the right answer

understood the difference between the real and imaginary numbers. They knew the purest form of a

complex number z is bi. Interestingly, one student transcribed in the test that “I have selected the option

because this is written in the Mathematics textbook”. Another student reported that “I have selected bi

because the complex numbers are always real and unequal number”.


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Contrary to this, those who selected „-bi‟ reflected that they have confusion in signs. It reflects that they

forgot that the general term of complex number has a positive sign i.e. „a + bi‟. One student stated that

complex value is a negative (-) sign. Whereas, most of the students who selected the „0i and -0i‟ forgot the

general form of complex number i.e. z = a + bi along with the rule that when we multiple zero „0‟ with any

number then the answer is also zero. One of them specified that “0 has no sign, therefore, the purest form

of a complex number is 0i”.

Item#2: What is the conjugate of 5+4i?

In conjugate of any complex number, the sign of imaginary part always varies i.e. negative into positive or

positive into negative (Dar & Haq, 2017). Analysis of test scripts revealed that students (82.6%) who

selected the right answer (5-4i) mentioned that conjugate always shows a negative sign with imaginary.

One of them stated, “It is a Mathematics rule”. On the other hand, 3% of students selected the (5+4i)

reported that they forgot the concept of the conjugate. One of them stated that “in Mathematics as „5‟ has

positive power” therefore he selected this option. 8.5% of students stated that in the conjugate we replace

the sign of real number i.e. „+‟ sign into „-‟ and vice versa therefore they selected „-5+4i‟. Moreover, 5.7%

of students specified that conjugate mean reverses the sign of the real and imaginary number in the opposite

sign, i.e. if there is a „+‟ sign we change it in „-‟ and vice versa so they chose „-5-4i‟.

Item#3: What is the relation between the following terms 2+2i & 4i?

The responses showed that the majority of students (41.8%) assumed that 2+2i is equal to 4i. Whereas,

7.1% of students reported that 2+2i is greater than 4i (>) and 13.9% of students described that there is less

than a sign (<) between the terms. On the other hand, 36.1% of students gave the correct answer which was

„≠‟ not equal sign. While 1.1% of students did not attempt the item. When students‟ test scripts were

reviewed to find the reasons to select the options then a few reasons were discovered. Some of the students

wrote on the test that imaginary value (i) cannot be added with simple value (real numbers), therefore 2 + 2i

is less than 4i. One of the students reported that “because i= 1 and there is no change when 2 is added in 2i

then answer is same (2+2i=4i)”. Another student stated that “the value of 2i is 2 and 2 = 2 therefore 2 + 2i

= 4i”. A boy student stated that “both sides are equal in my thought”.

One of the students who gave the correct answer stated that “If i is with 2 than 2i + 2i = 4i but there is no i

with 2 therefore 2 + 2i is not equal to 4i”. Another student knows that real and imaginary numbers cannot

be added but still he selected the wrong relation (i.e. > sign), as he stated in the test that he selected option a

because “2 + 2i cannot be sum(added) therefore 2+2i is greater than 4i”. Similarly, one more student

wrote the reason for selecting the option by following steps which are displayed in the given picture:


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Item#4: What is the reciprocal of i?

It was revealed from the collected data that the majority of the students (48%) gave the right answer i.e. the

reciprocal of i is = -i. On the other hand, 10.5% of students selected option a (1). Similarly, 30.6% of

students selected option b (-1) and 9.2% of students selected option c (i). And rest of the 1.7% of students

did not answer the item. Although the majority of the students selected the right option, when their

transcripts were explored to find out the reasons then it was found that they had misconceptions in terms of

the sign. They stated that i is positive, so it‟s reciprocal is negative i. Some of them reported that reciprocal

is like the conjugate the sign is reverse only. Conversely, one girl student reported that “when we try to

convert the value from denominator to numerator for equivalence of value this is called reciprocal”.

One of the students stated that as i=1 and then the reciprocal of -1 is equal to 1.

Item#5: If z= a + ib then what will be the ?

In response of „If z = a + ib then =___________‟, the data disclosed that one third (32.9%) of the students

gave the right answer i.e. option ( ). Conversely, 27.5% of students selected the first option (a – ib).

Similarly, 12.8% of students selected option ( ) and 24.1% of students selected option ( ).

Furthermore, 2.7% of students did not attempt the item. When students‟ test scripts were reviewed to find

the reasons to select the options then a few reasons were discovered. Some of the students wrote on the test

that if z = a + ib then = a – ib because reciprocal positive value change into negative therefore they

selected option (a – ib). According to one of the students' tests “in the sign changes the position (up and

down)”. A girl stated that “I selected option a because in reciprocal form positive change in negative and

vice versa”. Another student stated that “I selected the option d because the value a + ib is divided by 1

then the sign change (a – ib) and it come in the numerator and the denominator, we write a + ib”.


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It was exposed from the one of test that student has less practice. He skips the denominator in the last step

and chooses the wrong option.

Misconception in linear equations and inequalities

Item#6: If -2<x<3/2 then what is the value of x?

In response to an item about the value of x if -2<x<3/2, the data showed that one-third (38.7%) of the

students gave the right answer i.e. x=0. On the other hand, 11.5% of students selected x= -5. Similarly,

26.6% students selected x=3. 20.3% students selected x=3/2. Whereas, 2.9% of students (63) did not

attempt the item. When students‟ test scripts were reviewed to find the reasons to select the options then a

few reasons were discovered. Few students stated that „-5‟ is the real number that‟s why we have chosen it.

One of the students selected the „3‟ due to the wrong solution.

Moreover, some of the students wrote on the test that we selected the 3/2 because -2 is smaller than x.

Item#7: If the capacity c of an elevator is at most 1600 then c will be?

In response to an item about the capacity of the elevator, the data described that one-half (53.9%) of the

students gave the right answer i.e. c ≤ 1600. Which was mostly based on their guessing as they did not

provide any reason for selecting the correct answer. Whereas, 44(2.9%) students did not attempt the item.

On the other hand, 8.8% of students selected the c >1600. Similarly, 20.1% students selected c <1600 and

14.3% students selected c ≥ 1600.


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Item#8: If a>b and c> 0 then what is the relation between ac & bc?

Multiplication of a positive number with any inequality does not change its sign (Dar & Haq, 2017). In

response to the item about the relationship between ac & bc if a>b and c> 0 then the results explained that

only 28.9% of students gave the right answer i.e. ac is greater than bc if a>b and c> 0. Conversely, 24% of

students selected „<‟. Similarly, 35.5% of students were selected ≥ and 8.6% of students were selected ≤.

Whereas, 45(3.0%) students did not attempt the item. It shows that students had a misconception about

inequality signs. Few students selected less than signs and stated that it is a mathematical rule. One of the

students definite that “I selected ≤ because a and c are greater than 0”. Four students stated that the „c‟ is

presented on both sides that why they selected „≥‟.

Item#9: If x, y, z are real numbers and z<0 then x<y will be equal to?

The expected answer was a greater sign. As the multiplication of inequality with any negative number

always reverses the sign of inequality in reverse i.e. less (<) into greater (>) and greater into less (Habib,

Ali, Khan, & Moeen, 2016). Consequently, the item response i.e. „If x, y, z are real numbers and z<0 then

x<y =?‟ showed that one-third (32.1%) of the students gave the right answer i.e. xz > yz. On the other hand,

21.9% of students selected xz<yz. Similarly, 26.1% of students selected xz = yz, and 15.7% of students

selected xz ≠yz. Whereas, 4.2% of students (63) did not attempt the item. Moreover, analyzing transcripts

different misconceptions were diagnosed about inequality sign/ relationship. Some of the students wrote on

the test that the x, y, and z are real numbers, therefore, they put the equal sign. One of them stated that “the

value is „y‟ is greater than „x‟ therefore xz = yz”. On the other hand, one of the students stated that “the „x‟

is equal to „0‟ and „y‟ is equal to „0‟thefere xz ≠ yz”. Similarly, few students reported that as „x‟ is smaller

than „y‟ and „z‟ are smaller than „0‟ so „xz‟ is smaller than „yz‟. A boy reported that “0 is small and when

this will multiple with anyone then its value will not change and as x<y so, therefore „xz‟ is also smaller

than yz”.

Misconception in Algebraic Expressions and Formula (Surds)

Item#10: What is the relationship between these surds ___________ ?

In response to an item ___________ , only 7.4% of students gave the right answer i.e.

option d (<). On the other hand, 2.2% of students selected the option (>). The data depicted that the

mainstream of the students (64.4%) selected option (=) as they assume that is equal to

. One of the students specified that “both are equal, and it is clear from the question”.


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Whereas, 25.3% of students selected option (≠) which was the correct answer, and only 10 (0.7%) students

did not attempt the item. Some of them specified that in radicals (surds) we can write two terms

like this separately. One student stated that “two radicals (surds) are only equal in

multiplication ( = ) not besides so ”. Similarly, a student tried to

give an example to prove that .

Misconception in Quadratic Equations

Item#11: What is the purest form of a quadratic equation?

The responses of item about the purest form of a quadratic equation describes that majority of the students

(61.9%) selected the option b (ax2+ bx + c = 0, a ≠ 0), which is the general form of a quadratic equation.

Similarly, 5.8% students selected the first option (bx + c = 0, b ≠ 0). 9.7% students selected option (ax2 +

bx =0, a≠0) and 2.7% students did not attempt the item. Conversely, only 19.9% students picked the correct

option (ax2 + c=0, a ≠ 0). When students‟ test scripts were reviewed to find the reasons to select the options

it was found that students either just randomly selected the options or some of the respondents were

confused in a pure and general form as no one wrote the reason for their answers.

Misconceptions in Algebraic Manipulation

Item#12: What is the Highest Common Factor (HCF) two algebraic expressions, i.e. x2-5x+6 and x

2-x-

6?

If two or more algebraic expressions are given, then their common factor of the highest power is called the

HCF of the expressions (Dar & Haq, 2017). The answer to an item about the Highest Common Factor

(HCF) showed that only 37.4% know that if x2-5x+6 and x

2-x-6 are two algebraic expressions, then HCF is

x-3. Contrary to this rest of the students' selected other options. The data disclosed that 27.1% of students

had chosen the option (x+2). 18.8% students selected the option (x-2) and 14.1% students selected option

(x+3). The rest of the 38(2.5%) students did not attempt the item. When students‟ test scripts were reviewed

to find the reasons to select the options then a few reasons were discovered. One of the students stated that

“because x2 and x-6 are only divisible by x-2 so HCF is x-2”. Furthermore, it was observed that most of the

students have a problem with factorization. The sample solution is shown in the following picture:


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Item#13: What is the Least Common Multiple (LCM) of two algebraic expressions a2+b

2 and a

4-b

4?

If an algebraic expression p(x) is exactly divisible by two or more expressions, then p(x) is called the

common multiple of the given expression. The Least Common Multiple (LCM) is the product of common

factors together with not common factors of the algebraic expressions (Dar & Haq, 2017). The answer of

item about Least Common Multiple (LCM) exposed that 37.7% knew that if a2+b

2 and a

4-b

4 are two

algebraic expressions, then LCM is a4 – b

4. Contrary to this rest of the students chosen other options. It was

revealed from the data that 11.7% of students had chosen the option (LCM= a2+b

2). Similarly, 43.5%

students selected the option (LCM= a2-b

2) and 5.3% students selected option (LCM= a

4+b

4). The rest of the

28 (1.9%) students did not attempt the item. It was explored from the script analysis that students had less

understanding of the concept of LCM. Some of the students reported that in LCM we take the highest

power therefore answer is a4+b

4. Correspondingly, few students claimed that LCM is the highest most

common value or number.

Two students stated that we selected a2-b

2 because it divides both expressions. A girl student specified that

“I selected the a2-b

2 by doing midterm break of these two algebraic expressions”. Two students selected the

same option but with the wrong solution as shown in the given pictures:

Misconception in variation (Ratio)

Item#14: In ratio, what is the relationship between 4:5 & 5:4?

The order of the element in a ratio is important (Habib et al., 2016). The result about the relationship

between 4:5 & 5:4 in ratio showed that the majority 59.9% of students selected the option (::) as they

assumed that this was the question about proportion. One of them specified that “because the relation

between two ratios is expressed by proportion (::)”.


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Similarly, 13.9% students selected option (=) and 7% students selected option (/). Conversely, only 17.8%

of students selected the option (≠). They knew that in ratio if two elements or numbers are the same then

they are not equal i.e. 4:5 ≠5:4. The rest of the 20(1.3%) students did not attempt the item.

Misconceptions in Sets and Functions

Item#15: If A= {1, 2, 3, 4, 4, 5}, then what will be A called?

According to Habib et al. (2016), a set is a well-defined and distinct collection of objects/numbers. If any

object or number is repeated in a set, then this is not called a set. It was discovered from responses that only

one-third of the students (33.4%) understand that A= {1, 2, 3, 4, 4, 5} is not a set. Conversely, most of the

students (37.5%) stated it is a finite set. Some of them expressed that this is a finite set because it is a single

set, limited numbers are presented in the set and we can calculate this. One of them stated that “this is a

finite set as one value is repeated”. A girl specified that “this set consists of 6 digits and enclose in

brackets, therefore, this is a finite set”. Another girl student stated that “this is not the subset and not null

set so it may be infinite set”. 24.7% students defined A= {1, 2, 3, 4, 4, 5} as subset. Few students reported

that this is a subset because it has curly brackets. One of them stated that “this is subset because there are

few elements in the set”. A boy student expressed that “I selected subset as 4 is repeated two times”.

Likewise, 2.2% of students selected the null set and the rest of 2.2% of students left the item unsolved.

Item#16: In pairs, what is the relationship between (x, y) & (y, x)?

According to Habib et al. (2016), the order of the numbers in a pair is important. The result about the

relationship between (x, y) & (y, x) in pairs shows that most of the students (41.3%) reported that (x, y) is

not equal to (y, x) in pairs as they know that the order is important. Contrariwise, 3.5% students selected

option (>) and 14.1% students selected option (<). Comparatively, many students (38.8%) selected the

option (=). The rest of the 35 students did not attempt the item.

Item#17: If A ⊆ B, then A- B is equal to what?

If A is a subset of B so all the elements of A belong to B then A-B equal to the empty set (Habib et al.,

2016). In response to item If A⊆B (A is the subset of B), then A- B (difference) is equal to ________, the

result confirms that most of the students (43.5%) knew that if A is a subset of B then A-B is equal to an

empty set consequently they selected option ∅. However, 37(2.5%) students did not attempt the item.

Contrariwise, 31.5% students selected option (B-A). Some of them reported that A is a smaller subset of B,


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so A- B = B-A. One of them stated that “because set B is a subset and greater than set A so, therefore, I

selected B-A”. Similarly, 9.1% of students chosen option “A-B = A”. One of them said that “Because set B

is smaller than set A, therefore I carefully have chosen A-B = A”. Moreover, 13.5% of students selected the

option “A- B =B” and stated that because B is a subset of A, so they selected „B‟. One of the students draw

Venn diagrams but she looks confused between difference and union.

Item#18: If A and B are disjoint sets, then A∩B will be equal what?

The result regarding disjoint sets exposed that the majority of students (59.8%) stated that “If A and B are

disjoint sets, then A∩B is equal to B∩A”. Few students reported that according to associative law, A∩B is

equal to B∩A. A girl student stated that “I selected B∩A due to assertive law i.e. A∩B= B∩A” it seems she

also did not know the property name of sets. Likewise, 4.1% of students chosen option (A∩B =A). One of

them reported that “I choose option a because if you consider a = (2, 3), b = (1, 2, 3) than A∩B = (2, 3) =

A”. It seems that she did not consider the condition of the item that both sets are disjoint.

Moreover, 8.7% of students option (A∩B =B). Some of the students reported that both sets are equal

therefore they selected „B‟. Comparatively, few students (25.4%) selected the right option (∅). The rest of

the 31(2.1%) students did not attempt the item.

Item#19: If A= {a, b, c} and B= {e, f} then what is the relationship between A* B & B*A?

The result of the relationship between A*B & B*A expressed that most of the students (40.9%) selected

option „≠‟ and stated that if A={a, b, c}, and B={e, f} then A*B is not equal B*A. Interestingly almost

same number of students (47.3%) selected option „=‟ and reported that if A={a, b, c} and B={e, f} then A*

B is equal to B*A as A * B = 3 * 2 = 6 and B* A = 6. Some of them stated various reasons like both sides

have the same element, therefore, option equal sign; in multiplication A * B and B* A are always equal.


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On the other hand, 3.3% of students selected option (>) and 5.6% of students selected option (<). The

remaining, 2.9% of students did not attempt the item.

Item#20: In which quadrant point (-1, 4) lies?

In response to another item, the position of a point in quadrant showed that the majority of the students

(67.5%) gave the right answer i.e. quadrant II. On the other hand, 4.1% of students selected the first

quadrant I. Similarly, 19.4% of students selected quadrant III, and 8.1% of students selected quadrant IV.

One of the students selected the fourth quadrant by incorrect multiplication.

Whereas, 0.9% of students did not attempt the item. It was discovered from the test that students had a

misconception about the Cartesian coordinate xy-plane. They reverse the xy-plane horizontal into vertical

and vice versa. The sample pictures are given below:

Item#21: What is the relation in these elements {(1, 2), (2, 3), (3, 3), (3, 4)}?

In this item misconception about the definition of function was examined. According to the condition of

function definition that each element in the domain is matched with only one element in the range (Habib et

al., 2016), the result of the {(1, 2), (2, 3), (3, 3), (3, 4)} discloses that one-half students (50.6%) understand

that this is not function. On the other side, 17.2% students stated onto function; 16.1% students claimed into

function and 13.2% students selected one-one function. Whereas, 2.9% of students (43) did not attempt the

item. When students‟ test scripts were reviewed to find the reasons for the selection of the answers then a

few reasons were discovered. Some of the students wrote on the test that one-one function because all the

distinct elements of „B‟ has a distinct image in A. The following sampled pictures exposed that students

had misconceptions about functions.


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Some of them have chosen „Into function‟ as one element of B, which is not an image of A. Those who

selected the „onto function‟ stated that as the relationship has two functions therefore, they chose onto

function. One of them stated that “in range, the number is repeated i.e. 3”. It was revealed from the

response that she has confusion in the domain and range of a function. Students‟ concept image of

functions plays an important role in their understanding (Goldenberg & Mason, 2008; Habre & Abboud,

2006).

Discussion

The study explored that students have misconceptions between HCF and LCM. They have misconceptions

in general and pure forms of complex numbers and quadratic equations. Some of them have misconceptions

in other concepts i.e. sets and functions, inequality, radicals (surds), and factorizations of the Mathematics

curriculum. The results of various researches are in line with the findings of the misconceptions of various

concepts of Mathematics curriculum (Ali, 2011; Almog & Ilany, 2012; Ashlock, 2002; Baidoo, 2019;

Barcellos, 2005; Budak & Kapusuz, 2004; Cansoz et al., 2011; Crisan, 2008, 2012; Dubinsky & Wilson,

2013; Egodawatte, 2011; Erlandson, 2013; Gomez & Buhlea, 2009; İşleyen & Mercan, 2013; Kusuma et

al., 2018; Li, 2006; Maguire, 2012; Mohyuddin & Khalil, 2016; Peşman & Eryılmaz, 2010; Roneau et al.,

2014; Shoaib & Saeed, 2016).

Conclusion

It is concluded from the results that high school students have numerous misconceptions in the content of

Mathematics. Students are frequently taught to memorize properties and shortcuts to work with

Mathematics problems. Learning Mathematics in this manner can hinder a students‟ sense of numbers and

aid the development of misconceptions. At high school, students have misconceptions in the conjugate of

complex numbers, general and pure forms of complex numbers, and quadratic equations. They have

misconceptions between reciprocal and conjugate of complex numbers, HCF and LCM. Similarly, it is also

concluded that students have misconceptions about the relationship between inequality and radicals (surds)

of algebraic expressions.

Moreover, students have confused between ratio and proportion. It is also concluded from the results that

concept of sets and function, nature of subsets and disjoint sets, the importance of position in pairs of sets,

the position of a point (x, y) Cartesian coordinate xy-plane. The researchers also conclude from the results

there that are multiple reasons for high school students‟ misconceptions about Mathematics: confusion; test

anxiety; fewer practices; forgot formulas; use of guessing method; surprise test; lack of concept clarity; less

focus. Therefore, they are unable to give justification and reasons for the concepts. It seemed that they

memorized (crammed) Mathematical formulas, procedures, and concepts instead of understanding them in

their classrooms which leads to misconceptions in Mathematics at the secondary level.


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Recommendations

Based on the findings and conclusions of this study, the following recommendations have been formulated:

1. The students‟ misconceptions may be communicated with them and their parents through

parent-teacher conferences. Subsequently, parents may correct their children's misconceptions

about Mathematics at home.

2. Textbook of Mathematics at the secondary level may be revised by experts keeping in mind

that the content like LCM, HCF, sets and functions, inequality, radicals (surds), factorizations

and complex numbers, etc. given in textbooks should help students to remove their

misconception instead of leading to misconception.

3. All available textbooks of Mathematics being used at different levels of schools may be

reviewed to identify the content leading to misconceptions of students.

Implications of the study

The study has the following implications: the investigations of the misconceptions of students may guide

teachers, researchers, and educationists who want to identify students‟ misconceptions at the secondary

level. This study may be a source of inspiration for researchers to work upon misconceptions of students

and conduct researches on some teaching strategies to test practicability and feasibility at the secondary

level. This research may provide guidelines for present and future Mathematics teachers to effectively

address the misconceptions of students. The study may be supportive for curriculum developers and

textbook authors to consider students‟ misconceptions while developing, Mathematics curriculum and

subject matter.

References

Agarkar, S., & Brock, R. (2017). Learning theories in science education. Rotterdam, Netherlands: Sense

Publishers.

Akhter, N., Akhtar, M., & Abaidullah, M. (2015). The perceptions of high school Mathematics problem-

solving teaching methods in Mathematics education. Bulletin of Education and Research, 37(1), 1-23.

Alghazo, Y. M., & Alghazo, R. (2017). Exploring common misconceptions and errors about fractions

among college students in Saudi Arabia. International Education Studies, 10(4), 133-140.

Ali, T. (2011). Exploring students‟ learning difficulties in secondary mathematics classroom in Gilgit

Baltistan and teachers‟ effort to help students overcome these difficulties. Bulletin of Education and

Research, 33(1), 7-69.

Almog, N., & Ilany, B. S. (2012). Absolute value inequalities: High school students‟ solutions and

misconceptions. Educational Studies in Mathematics, 81(3), 347-364.

Arias, A. M., Davis, E. A., Marino, J. C., Kademian, S. M., & Palincsar, A. S. (2016). Teachers‟ use of

educative curriculum materials to engage students in science practices. International Journal of

Science Education, 38(9), 1504-1526.

Ashlock, R. B. (2002). Error patterns in computation: Using error patterns to improve instruction. Upper

Saddle River, NJ: Prentice Hall.

Aydogan, S., & Gelbal, S. (2017). Examination of the misconceptions of teacher candidates in the

measurement and evaluation in education classes misconceptions in measurement and evaluation

concepts. Journal of Measurement and Evaluation in Education and Psychology, 8(4), 404-420.

Baidoo, J. (2019). Dealing with grade 10 learners‟ misconceptions and errors when simplifying algebraic

fractions. Journal of Emerging Trends in Educational Research and Policy Studies

(JETERAPS), 10(1), 47-55.

Bakar, Z. A., & Ali, R. (2018). Epistemological Beliefs in Classroom Learning Among Secondary School

Students in Pakistan. Advanced Science Letters, 24(5), 3507-3511.


383

I



R S S

Barcellos, A. (2005). Mathematical misconceptions of college-age algebra students (doctoral dissertation).

Available from ProQuest Dissertations & Theses Global. (UMI No 305031112).

Brod, G., Werkle-Bergner, M., & Shing, Y. L. (2013). The influence of prior knowledge on memory: A

developmental cognitive neuroscience perspective. Frontiers in Behavioral Neuroscience, 7(1), 139-

148.

Budak, I., & Kapusuz, A. (2004). Elementary and secondary school students‟ misconceptions of

mathematical set concept. In D. E. McDougall, & J. A. Ross (Eds.), Proceedings of the twenty-sixth

annual meeting of the North American Chapter of the International Group for the Psychology of

Mathematics Education. Toronto, Canada.

Calais, G. J. (2018). The multidimensional measure of conceptual complexity: Elucidating misconceptions

research in Science. National Forum of Applied Educational Research Journal, 31(1), 1-11.

Cansoz, S., Kucuk, B., & Isleyen, T. (2011). Identifying the secondary school students‟ misconceptions

about functions. Procedia-Social and Behavioral Sciences, 15(1), 3837-3842.

Çelen, Y. (2018). Misconceptions about the Ratio of Proportion of 7th Grade Students. Journal of Modern

Education Review, 8(1), 126-135.

Cortina, J. M. (1993). What is coefficient alpha? An examination of theory and applications. Journal of

Applied Psychology, 78(1), 98-104.

Crisan, C. (2008). Square roots. Mathematics Teaching, 209(1), 44-45.

Crisan, C. (2012). What is the square root of sixteen? Is this the question?. Mathematics Teaching, 230(1),

21-22.

Dar, K. H., & Haq, I. (2017). Mathematics (Science group) 9. Lahore: Caravan book house.

Docktor, J. L., Strand, N. E., Mestre, J. P., & Ross, B. H. (2015). Conceptual problem-solving in high

school physics. Physical Review Special Topics-Physics Education Research, 11(2), 02010601-

02010613

Doğan, A. & Çetin, I. (2009). On the direct and inverse proportion: The concept of students 7th and 9th

misconceptions. University Journal of Social Sciences, 2(2), 118–128.

Dubinsky, E., & Wilson, R. T. (2013). High school students‟ understanding of the function concept. The

Journal of Mathematical Behavior, 32(1), 83-101.

Egodawatte, G. (2011). Secondary school students‟ misconceptions in algebra (Doctoral thesis). University

of Toronto, Canada.

Erbas, A. K., Cetinkaya, B., & Ersoy, Y. (2009). Student difficulties and misconceptions in solving simple

linear equations. Education and Science, 34(152), 44-59.

Erlandson, K. M. (2013). A study of college students' misconceptions of radical expressions (Master‟s

thesis). University of New York, New York.

Furtak, E. M., Kiemer, K., Circi, R. K., Swanson, R., de León, V., Morrison, D., & Heredia, S. C. (2016).

Teachers‟ formative assessment abilities and their relationship to student learning: findings from a

four-year intervention study. Instructional Science, 44(3), 267-291.

Gay, L. R., Mills, G., & Airasian, P. W. (2009). Educational research: Competencies for analysis and

interpretation. Upper Saddle Back, NJ: Merrill Prentice-Hall.

Goldenberg, P., & Mason, J. (2008). Shedding light on and with example misconceptions spaces.

Educational Studies in Mathematics, 69(2), 183-194.

Gomez, B., & Buhlea, C. (2009). The ambiguity of the sign√. In Proceedings of the Sixth Congress of the

European Society for Research in Mathematics Education (pp. 509-518).

Gür, H., & Barak, B. (2007). The erroneous derivative examples of eleventh grade students. Kuramve

Uygulamada Egitim Bilimleri, 7(1), 473-480.

Habib, M., Ali, A., Khan, A. R., & Moeen, M. (2016). Mathematics (Science group) 10. Lahore: Ilmi Kitab

khana.

Habre, S., & Abboud, M. (2006). Students' conceptual understanding of a function and its derivative in an

experimental calculus course. The Journal of Mathematical Behavior, 25(1), 57-72.

Halai, A. (2007). Learning Mathematics in English medium classrooms in Pakistan: Implications for policy

and practice. Bulletin of Education and Research, 29(1), 1-16.


384

I



R S S

Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students‟ learning.

In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 371-

404). USA: Information age publishing Inc.

İşleyen, T., & Mercan, E. (2013). Examining the difficulties experienced by 8th grade students on the

subject of square root numbers. Journal of Theory and Practice in Education, 9(4), 529-543.

Jameel, T., & Ali, H. H. (2016). Causes of poor performance in mathematics from the perspective of

students, teachers and parents. American Scientific Research Journal for Engineering, Technology, and

Sciences (ASRJETS), 15(1), 122-136.

Jiménez-Aleixandre, M. P., & Brocos, P. (2017). Shifts in epistemic status in argumentation and in

conceptual change. In T. G., Amin, & O. Levrini, (Eds.), Converging Perspectives on Conceptual

Change: Mapping an Emerging Paradigm in the Learning Sciences (pp.52-70). New York: Routledge.

Kaplan, A., İşleyen, T., & Öztürk, M. (2011). The misconceptions in ratio and proportion concept among

6th grade students. Kastamonu Education Journal, 19(3), 953-968.

Kaput, J. J. (2017). What is algebra? What is algebraic reasoning? Algebra in the early grades. New York:

Routledge

Karakus, F., & Karatas, I. (2014). Secondary school students‟ misconceptions about fractals. Journal of

Education and Human Development, 3(3), 241-250.

Keceli, V., & Turanli, N. (2013). Misconceptions and common errors in complex numbers. Hacettepe

University Journal of Education, 28(1), 223-234.

Kucuk, A., & Demir, B. (2009). Primary education 6–8. a study on some concept errors in mathematics

teaching in classrooms. Dicle University Journal of ZiyaGökalp Education Faculty, 42(13), 97-112.

Kusuma, N. F., Subanti, S., & Usodo, B. (2018). Students‟ misconception on equal sign. Journal of

Physics: Conference Series, 1008 (1), 1-6. UK: IOP Publishing.

Li, X. (2006). Cognitive analysis of student's errors and misconceptions in variables, equations and

functions (Doctoral dissertation, Texas A & M University, College Station, Texas). Retrieved

fromhttps://oaktrust.library.tamu.edu/bitstream/handle/1969.1/ETD-TAMU-1098/LI-

DISSERTATION.pdf?sequence=1

Luneta, K., & Makonye, P. J. (2010). Learners errors and misconceptions in elementary analysis: A case

study of a grade 12 class in South Africa. Acta Didactica Napocenia, 3(3), 36-45.

Maguire, T. J. (2012). Investigation of the misconceptions related to the concepts of equivalence and literal

symbols held by underprepared community college students (Doctoral dissertation, The University of

San Francisco, San Francisco). Retrieved

fromhttp://repository.usfca.edu/cgi/viewcontent.cgi?article=1039&context=diss

Mohyuddin, R. G. & Khalil, U. (2016). Misconceptions of students in learning mathematics at primary

level. Bulletin of Education and Research, 38(1), 133-162.

Nassir, A. A., Abdullah, N. H. M., Ahmad, S., Tarmuji, N. H., & Idris, A. S. (2017). Mathematical

misconception in calculus 1: Identification and gender difference. In AIP Conference Proceedings,

1870(1), 01-15.

Naz, A. (2008). An investigation of misconceptions of biology students at secondary level and effectiveness

of conceptual change instructional strategy (Doctoral dissertation, University of the Punjab, Lahore).

Retrieved from http://prr.hec.gov.pk/jspui/handle/123456789/1100

Peşman, H., & Eryılmaz, A. (2010). Development of a three-tier test to assess misconceptions about simple

electric circuits. The Journal of Educational Research, 103(3), 208-222.

Punjab, Government of (2018). Census of School Education Department Punjab 2018. Lahore: Ministry of

Education.

Roneau, R.N., Meyer, D., & Crites, T. (2014). Putting essential understanding of functions into practice in

grades 9-12. Restton, VA: NCTM

Russell, M., O‟Dwyer, L. M., & Miranda, H. (2009). Diagnosing students‟ misconceptions in algebra:

Results from an experimental pilot study. Behavior Research Methods, 41(2), 414-424.

Sarwadi, H. R. H., & Shahrill, M. (2014). Understanding students‟ mathematical errors and

misconceptions: The case of year 11 repeating students. Mathematics Education Trends and Research,

10(2), 1-10.

Sbaragli, S. (2011). The skills in mathematics. Difficulty in Mathematics, 7(2), 143-156.

https://oaktrust.library.tamu.edu/bitstream/handle/1969.1/ETD-TAMU-1098/LI-DISSERTATION.pdf?sequence=1

https://oaktrust.library.tamu.edu/bitstream/handle/1969.1/ETD-TAMU-1098/LI-DISSERTATION.pdf?sequence=1

http://repository.usfca.edu/cgi/viewcontent.cgi?article=1039&context=diss

http://prr.hec.gov.pk/jspui/handle/123456789/1100


385

I



R S S

Schnepper, L. C., & McCoy, L. P. (2013). Analysis of misconceptions in high school

mathematics. Networks: An Online Journal for Teacher Research, 15(1), 625-625.

Shoaib, A., & Saeed, M. (2016). Exploring factors promoting students‟ learning in mathematics at

secondary level. Journal of Educational Sciences & Research, 3(2), 11-20.

Taber, K. S. (2018). The use of Cronbach‟s alpha when developing and reporting research instruments in

science education. Research in Science Education, 48(6), 1273-1296.

Treagust, D., Won, M., & Duit, R. (2014). Paradigms in science education research. In N. G. Lederman, &

S. K. Abell (Eds.), Handbook of Research on Science Education, (pp.3-17). New York: Routledge.

Tunç, T., Akçam, H. K., & Dökme, İ. (2011). Primary school teacher candidates‟ misconceptions on some

fundamental science concepts with three annotated questions. Gazi University Journal of Gazi

Educational Faculty, 31(2), 817-842.

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