An Analysis of Students’ Error in Solving Critical Thinking Problems in

Integral Calculus Course based on Newman Error Analysis Theory

Viewed from Gender Differences and Habits of Mind

Rezi Ariawan, Zetriuslita

Mathematics Education Study Program, Universitas Islam Riau, Pekanbaru, Indonesia

Keywords:

Mathematical Critical Thinking Problem, Newman Error Analysis, Gender differences, Habits of mind

Abstract:

This study aims to describe the types of students’ error in solving mathematical critical thinking problems in

Integral Calculus course based on Newman Error Analysis theory and viewed from gender differences and

habits of mind. The method of the research is a descriptive quantitative research. The research subjects were

47 students from the third semester who have taken Integral Calculus course. The instrument of data collection

was a test using the indicators of mathematical critical thinking skills and a questionnaire on the habit of mind.

The data collection techniques are test and non-test techniques. The data was analysed by using quantitative

descriptive technique. The result of this study indicated that: (1) The most dominant type of error is encoding

(92.44%) and the least is decoding (6.08%); (2) The most dominant type of error by male and female students

is encoding (96.43%) and (86.24%) and the least is decoding (24.28%) and (6.76%) ; (3) The most dominant

type of error in terms of habits of mind with very good, good, and fair criteria is encoding (88.28%), (94.43%),

and (77.7%) and the least one is decoding (10.56%), (19.76%), and (54.28%).

1 INTRODUCTION

Calculus is a course that exists in almost every

university that offers mathematics and science majors.

In addition, calculus must be taken and completed by

students in exact sciences, especially Mathematics.

In Mathematics Education Study Program, calculus

is a compulsory course divided into three subjects,

such as Differential Calculus, Integral Calculus and

Multivariable Calculus. Calculus 1 (Differential

Calculus) is provided in the ﬁrst semester, Calculus

2 (Integral Calculus) in the second semester and

Advanced Calculus (Multivariable Calculus) in the

third semester.

Given the importance of Calculus 2 course,

students are expected to have a good mastery and

understanding of the course. For this purpose,

lecturers can conduct an investigation to ﬁnd out how

the students’ ability can be improved.

One of the ways to investigate is to ﬁnd out

their error in solving Calculus 2 problems. This is

in line with SofriRizka Amalia’s statement (2017)

students’ mistakes in problem solving need to be

analyzed to determine their error and why these errors

occur. Furthermore, students’ mistakes need further

analysis, in order to get a clear and detailed picture

of students’ weakness in solving story problems

(Nurussafa’at et al., 2016).

Based on the experts’ opinions above,the ﬁnal

result of this investigation can provide an overview

of the types of error, so that lecturers can take

more appropriate steps to solve because the focus of

improvement is narrower and more apparent. The

students’ problem solving answers will reveal the

error, especially in the form of essay questions that

are included in High Order Thinking Skill (HOTS)

category. One of the HOT skills is critical thinking.

Subsequently, the researchers would like to create

mathematical critical thinking problems in the form

of essay to examine students’ ability in Calculus 2

course.

Errors are deviations from the right things which

are systematic, consistent, and incidental in certain

areas. Consistent and systematic errors are caused

by students’ competence, while the incidental ones

are not a result of their low mastery of the subject

(Abidin, 2012).

Learning error can be caused by: (a) low

intellectual ability; (b) emotional disorder (c) lack

of learning motivation; (d) students’ immaturity

in learning; (e) too young; (f) supporting social

background that does not support; (tid) (g) poor

336

Ariawan, R. and Zetriuslita, .

An Analysis of Students’ Error in Solving Critical Thinking Problems in Integral Calculus Course based on Newman Error Analysis Theory Viewed from Gender Differences and Habits of

Mind.

DOI: 10.5220/0009145003360342

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 336-342

ISBN: 978-989-758-464-0

study habits; (h) low memorization; (i) disruption of

the sensory devices for the development of puberty

(Rahimah, 2012).

Furthermore, there are several errors in learning

mathematics, namely; (a) lack of understanding of

symbols; (b) concerning place value; (c) use of

the wrong process; (d) error in calculations; (e)

writing error. Students’ error in solving Mathematical

problems can be analyzed with several methods. One

of the methods is Newman Error Analysis theory

(NEA). Newman Error Analysis was ﬁrst introduced

by Allan Leslie White as a simple diagnostic

procedure to observe students’ behaviour in solving

story problems. In line with that, researchers also

plan to use the procedure of the Newman theory in

diagnosing the students’ error in solving the problems

of critical thinking skill in Calculus 2 course.

Furthermore, the error in mathematical problem

solving can be inﬂuenced by several factors, one of

which is the presence of gender differences. Gender

differences in creativity were areas of controversy

(Indrawati and Tasni, 2016) .Abra and Valentine

French in Nenny Indrawati and Nurfaidah Tasni

(2016) stated that some experts suggest that men

are more creative than women, but other experts

reveal that women are more creative than men.

Moreover, Nenny Indrawati and Nurfaidah Tasni

(2016) stated that some researchers believe that

the inﬂuence of gender in mathematics is due to

biological differences in the child’s brain that are

known through observation.

Another factor that can affect students’ error in

Integral Calculus problem solving is called habits of

mind. Habit of mind is a characteristic of intelligent

people when faced with problems whose solutions

cannot be identiﬁed easily (Costa and Kallick, 2008).

Habit of mind is a group of skills, attitudes, and

values that allow people to bring up performance

or behavioral intelligence based on the stimulus to

guide students to face or resolve existing issues

(Marita, 2014). Based on these explanations, it can

be concluded that habit of mind is a very important

aspect to be explored, especially on students’ problem

solving error in Integral Calculus course. Therefore,

the researcher was interested in analyzing students’

error based on Newman’s theory viewed from gender

differences and habits of mind.

2 RESEARCH METHODS

The research used in this study is descriptive.

According to Nana Syaodih Sukmadinata (2010).

descriptive research is the most basic research,

intended to describe the existing phenomena, both

natural and man-made. Meanwhile, Descriptive

research is a research that intends to describe

situations or events (Suryabrata, 2014). If the study

wants to describe the size, number or frequency,

then the research is more appropriately named

as quantitative descriptive research (Sukmadinata,

2010).

Based on several opinions above and referring

to the research objective, this type of research is

quantitative descriptive research. It generates the

number, size or frequency of students who commit

errors in solving mathematical critical thinking

problems in Integral Calculus course by using

Newman Error Analysis theory based on gender

differences and habits of mind.

To obtain the data about the subject’s ability to

express opinions and ideas, the researchers needed

an auxiliary instrument in the form of a test of

mathematical critical thinking problems, which was

referred as the ﬁrst auxiliary instrument. Meanwhile,

to get the information about the students’ habit of

mind, the researcher used a questionnaire sheet as

the second auxiliary instrument. The data collection

techniques were test and non-test techniques. The

test technique used the ﬁrst instrument and non-test

technique used the second one.

The data analysis technique is a systematic

process of searching and compiling data obtained

from the result of written test. The data analysis

process in this study was carried out with the

following steps: (1) Analyzing written data by

examining the answer error given by students

to the test of mathematical critical thinking skill

based on Newman Error Analysis theory ; (2)

Analyzing students’ error based on Newman Error

Analysis theory in terms of gender differences;

(3) Checking the answers to the students’ Habits

of Mind questionnaire to collect the scores; (4)

Classifying students into three Habits of Mind

categories (self-regulation , critical thinking, and

creative thinking) based on the results of each

student’s questionnaire, as seen in Table 1 below.

Table 1: Classiﬁcation of Students’ Habits of Mind

NO Score CRITERIA

1 0 – 20 Habits of Mind Very Bad

2 21 – 40 Habits of Mind Bad

3 41 – 60 Habits of Mind Fair

4 61 – 80 Habits of Mind Good

5 81 – 100 Habits of Mind Very Good

Source: Adapted from (Riduwan and Sunarto, 2013)

The procedure of this research was carried out

with the following steps, such as:

An Analysis of Students’ Error in Solving Critical Thinking Problems in Integral Calculus Course based on Newman Error Analysis Theory

Viewed from Gender Differences and Habits of Mind

337

• Prepare the instruments, such as test instruments

with table of speciﬁcation and alternative

answers, and habits of mind questionnaire sheets,

• Provide questions that have been prepared for

students to complete.

• Analyze students’ error in solving critical

thinking problems in Integral Calculus course

based on Newman Error Analysis theory .

• Divide students in terms of gender differences.

• Analyze students’ error in solving critical

thinking problems in Integral Calculus course

based on Newman Error Analysis theory viewed

from gender differences.

• Distribute habits of mind questionnaire sheets to

students.

• Calculate the scores of habits of mind

questionnaires that have been distributed.

• Classify students based on habits of mind criteria.

• Analyze students’ error in solving critical

thinking problems in Integral Calculus course

based on Newman Error Analysis theory viewed

from habits of mind.

3 RESULTS AND DISCUSSIONS

3.1 Result

The subjects in this study were 2nd semester students

of Mathematics Education Study Program who were

taking Calculus 2. The researchers only taught

one class, class A, with a total of 47 students.

Therefore, the subjects were 47 students from the

second semester in the academic year of 2017/2018.

3.1.1 Students’ Error based on the Newman

Error Analysis Theory in Terms of the Test

Items.

The questions were prepared using the indicators of

mathematical critical thinking skills and referred to

the teaching material in Integral Calculus course.

Table 2 shows the distribution of teaching materials

and indicators used in the development of the research

instruments.

The test was carried out in two stages. Test I was

conducted after the teaching material in test 1 was

completed, while Test II was held after ﬁnishing the

teaching material in Test II. The types of error that

students committed in Test 1 and Test II can be seen

in Table 3.

3.1.2 Student’s Error based on the Newman

Error Analysis Theory Viewed from

Gender Differences

The following table 4 is presents the data regarding

the percentage of students in terms of gender

differences.

Table 4: Percentage of Students based on Gender

Differences

Gender Total Percentage (%)

Male 6 12.80%

Female 41 87.20%

Total 47 100%

Table 4 indicates that the number of male and

female subjects has a very signiﬁcant difference,

which is equal to 74.4%. However, according to the

researcher, it can still be used as a source of data to

be used as a reference in looking at the types of error

made by the subjects of the study. The following table

5 shows the data related to the types of error viewed

from gender differences. ’

In addition to Test I, the researchers also

conducted a second test. The following table 6

presents the data regarding the percentage of subjects

in terms of gender differences in Test II.

3.1.3 Students’ Error based on the Newman

Error Analysis Theory Viewed from

Habits of Mind

To obtain the data about the students’ habits of mind,

researchers distributed questionnaire sheets. The

following table 7 presents the data related to the

students’ habits of mind.

Table 7: Number and Percentage of Students in terms of

Habits of Mind

Habits of Mind Criteria Number of Students

Percentage

( % )

Very Good 18 37.5

Good 26 54.2

Fair 4 8.3

Bad 0 0

Very Bad 0 0

Total 48 100

Source: Processed Data

Table 8 reveals the students’ habits of mind

viewed from gender differences (male and female).

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Table 2: Details of Teaching Materials and Indicators of Mathematical Critical Thinking Ability Used in the Development of

Test Items in Integral Calculus Course

Teaching Materials Indicators of Mathematical Critical Thinking Ability Test Items

IndeﬁniteIntegral The ability to identify and justify concepts, namely to provide

reasons for mastering concepts.

No. 1 (Test 1)

Integration Technique by partial integral The ability to identify and justify concepts, namely to provide

reasons for mastering concepts.

No. 2 (Test 1)

Integration technique by trigonometry substitution The ability to analyze algorithm, namely to evaluate or examine an

algorithm.

No. 3 (Test 1)

Integration technique by trigonometry function The ability to generalize, namely to complete the data or

supporting information.

No. 4 (Test 1)

Integration technique by trigonometry substitution The ability to generalize, namely to complete the data or

supporting information.

No. 1 (Test II)

Integration technique by rational function The ability to identify and justify concepts, namely to provide

reasons for mastering concepts.

No. 2 (Test II)

Application of Deﬁnite Integral (Areas between Curves

and Volumes of Solids)

The ability to analyze algorithm, namely to evaluate or examine an

algorithm.

No. 3 (Test II)

Table 3: Percentage of Students’ Error Based on Test Items (Newman Error Analysis Theory)

Newman Error Analysis (NEA) Stages

Percentage of Students’ Error Based on Test Items (%)

Test I Test II

Q1 Q2 Q3 Q4 Q1 Q2 Q3

Decoding

9 3 11 5 0 0 0

22.50% 6.80% 2.40% 10.90% 0% 0% 0%

Comprehension

19 4 16 22 0 4 0

47.50% 9.10% 34.80% 47.80% 0% 8.30% 0%

Transformation

28 11 23 30 4 27 26

70% 25% 50% 65.20% 8.90% 56.30% 66.70%

Process Skill

39 37 36 38 11 37 37

97.50% 84.10% 78.30% 82.60% 24.40% 80.40% 94.90%

Encoding

39 41 42 42 17 48 39

97.50% 93.20% 91.30% 91.30% 73.80% 100% 100%

Source: Processed Data

Description :

1. In Test 1 Question No. 1, the amount of data analyzed was 40 out of 48 students

2. In Test I Question No. 2, the amount of data analyzed was 44 out of 48 students.

3. In Test I Question No. 3, the amount of data analyzed was 46 out of 48 students.

4. In Test I Question No. 4, the amount of data analyzed was 46 out of 48 students.

5. In Test II Question No. 1, the amount of data analyzed was 45 out of 48 students.

6. In Test II Question No. 2, the amount of data analyzed was 48 out of 48 students.

7. In Test II Question No. 3, the amount of data analyzed was 39 out of 48 students.

Table 8: Number and Percentage of Students’ Habits of

Mind Based on Gender Differences

Habits of Mind Criteria

Number and Percentage of

Student’s Habits of Mind based on

Gender Differences

Male Female

Very Good 2 (33.3%) 16 (38.1%)

Good 2 (33.3%) 24 (57.1%)

Fair 2 (33.3%) 2 (4.8%)

Bad 0 (0%) 0 (0%)

Very Bad 0 (0%) 0 (0%)

Total 6 (12.5%) 42 (87.5%)

Source: Processed Data

The following table shows the distribution of

types of students’ error according to Newman’s error

analysis theory based on habits of mind, see Table 9.

3.2 Discussion

Based on the results of this study, the most dominant

type of error according to Newman Error Analysis

theory is encoding and the least one is decoding.

In this study, encoding error occurs when students

cannot write correct answers in the form of numbers,

symbols or words even though they have gone

through the ”treatment” stage . Therefore, it can be

stated that most students cannot express the correct

answers to the questions, neither viewed from gender

differences nor habits of mind. From the analysis

of the answers, it was found that most of them

were unable to identify an appropriate solution to the

problem. Only a small number of students solved the

problems correctly. Furthermore, the least dominant

An Analysis of Students’ Error in Solving Critical Thinking Problems in Integral Calculus Course based on Newman Error Analysis Theory

Viewed from Gender Differences and Habits of Mind

339

Table 5: Students’ Error According to the Newman Error Analysis Theory Viewed from Gender Differences in Test 1.

Newman Error Analysis (NEA) Stages

Percentage of Students’ Error Viewed from Gender Differences

(%)

Test I

Q1 Q2 Q3 Q4

M F M F M F M F

Decoding

3 6 1 2 5 6 2 3

60% 20% 20% 5.10% 50% 15% 40% 7.20%

Comprehension

4 15 2 2 6 10 4 18

80% 40% 40% 5.10% 100% 25% 80% 43.90%

Transformation

5 23 2 9 6 17 5 25

100 70% 40% 23.10% 100% 42.50% 100% 61%

Process Skill

5 34 5 32 6 30 5 33

100 97.10% 100% 82.10% 100% 75% 100% 80.50%

Encoding

5 34 5 36 6 36 5 37

100 97.10% 100 92.30% 100% 90% 100% 90.20%

Source: Processed Data

Description:

1. In Test 1 Question No. 1, the amount of data analyzed was 40 (5 men and 35 women) out of 48 students.

2. In Test 1 Question No. 2, the amount of data analyzed was 44 (5 men and 39 women) out of 48 students.

3. In Test I Question No. 3, the amount of data analyzed was 46 (6 men and 40 women) out of 48 students.

4. In Test I Question No. 4, the amount of data analyzed was 46 (5 men and 41 women) out of 48 students.

Table 6: Students’ Error According to the Newman Error Analysis Theory Viewed from Gender Differences in Test II.

Newman Error Analysis (NEA) Stages

Percentage of Students’ Error Viewed from Gender Differences (%)

Test II

Q1 Q2 Q3

M F M F M F

Decoding

0 0 0 0 0 0

0% 0% 0% 0% 0% 0%

Comprehension

0 0 0 4 0 0

0% 0% 0% -9.50% 0% 0%

Transformation

2 2 5 22 2 24

50% -4.90% 83.3%) -52.40% -50% -68.60%

Process Skill

2 9 6 31 4 34

50% -22% -100% -73.80% -100% -97.10%

Encoding

14 (34.1%)

6 42 4 35

-75% -100% -100% -100% -100%

Source: Processed Data

Description:

1. In Test II Question No. 1, the amount of data analyzed was 45 (4 men and 41 women) out of 48 students.

2. In Test II Question No. 2, the amount of data analyzed was 48 (6 men and 42 women) out of 48 students.

3. In Test II Question No. 3, the amount of data analyzed was 39 (4 men and 35 women) out of 48 students.

type of error is decoding. It occurs because students

cannot recognize the terms in the problem, recognize

symbols nor comprehend the questions. This type

of error is the least dominant one which means that

most students are able to recognize symbols or to

understand the questions well, yet they can’t ﬁnish it

correctly.

Based on the explanation above, it can be

concluded that from all the questions presented, male

and female students did all types of error according to

Newman Error Analysis theory, but the percentage of

male subjects is higher than that of female students.

Gender differences do not separate the students from

making types of error.

According to Subanti (2014), women in general

are better at remembering, while men are better at

logical thinking. Generally, men and women are

the same, but male students have better abstraction

than female students, allowing male students to be

better than female students in the ﬁeld of mathematics

in terms of abstract understanding. Furthermore,

Abra and Valentine-French in Neni Indrawati and

Nurfaidah Tasni (2016) stated that men are more

creative than women, but many researchers expressed

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Table 9: Types of Students’ Error According to the Newman Error Analysis Theory Based on Habits of Mind in Test I

Newman Error Analysis (NEA) Stage

Percentage of Students’ Error in Each Test based on Habits of Mind(%)

Test I

Q1 Q2 Q3 Q4

VG G F VG G F VG G F VG G F

Decoding

2 4 2 2 1 0 1 8 2 2 6 2

13.30% 17.40% 100% 13.30% 4.20% 0% 5.60% 34.5 40 9.5 26.1 100

Comprehension

6 9 2 2 1 1 5 9 2 3 5 2

40% 39.10% 100% 13.30% 4.20% 20% 27.80% 39.1 40 14.3 21.7 100

Transformation

11 14 2 4 5 1 5 13 2 5 7 2

73.30% 60.90% 100% 26.70% 20.80% 20% 27.80% 56.5 40 23.8 30.4 100

Process Skill

14 23 2 11 21 3 15 15 3 5 15 2

93.30% 100% 100% 73.30% 87.50% 60% 83.3 62.5 60 23.8 65.2 100

Encoding

14 23 2 12 24 3 18 20 3 15 20 2

93.30% 100% 100% 80% 100% 60% 100% 87 60 71.4 87 100

Source: Processed Data

Description:

1. In Test 1 Question No. 1, the number of data analyzed were 40 (15 HOM VG, 23 HOM G people, 2 HOM F people) out of 48 students.

2. In Test I Question No. 2, the number of data analyzed were 44 (15 HOM VG people, 24 HOM G people, 5 HOM F people) out of 48 students.

3. In Test I Question No. 3, the number of data analyzed were 46 ( 18 HOM VG people, 23 HOM G people, 5 HOM F people) out of 48 students.

4. In Test I Question No. 4, the number of data analyzed was 46 ( 21 HOM VG, 23 HOM G, 2 HOM F) out of 48 students.

Table 10: Types of Students’ Error Based on Habits of Mind in Test II

Newman Error Analysis (NEA) Stages

Percentage of Students’ Error in Each Test based on Habits of Mind(%)

Test II

Q1 Q2 Q3

VG G F VG G F VG G F

Decoding

2 4 2 2 1 0 1 8 2

13.30% 17.40% 100% 13.30% 4.20% 0% 5.60% 34.5 40

Comprehension

6 9 2 2 1 1 5 9 2

40% 39.10% 100% 13.30% 4.20% 20% 27.80% 39.1 40

Transformation

11 13 2 4 5 1 5 13 2

73.30% 54.20% 100% 26.70% 20.80% 20% 27.80% 56.5 40

Process Skill

14 23 2 11 21 3 15 15 3

93.30% 100% 100% 73.30% 87.50% 60% 83.3 62.5 60

Encoding

14 23 2 12 24 3 18 20 3

93.30% 100% 100% 80% 100% 60% 100% 87 60

Source: Processed Data

Description:

1. In Test 1 Question No. 1, the number of data analyzed were 40 (15 HOM VG people, 23 HOM G people, 2 HOM F people) out of 48 students.

2. In Test I Question No. 2, the number of data analyzed were 44 (15 HOM VG people, 24 HOM G people, 5 HOM F people) out of 48 students.

3. In Test I Question No. 3, the number of data analyzed were 46 ( 18 HOM VG people, 23 HOM G people, 5 HOM F people) out of 48 students.

that women are more creative than men. Some

researchers believe that the gender inﬂuence in

mathematics is related to biological differences in

their brains. Through observation, women in general

are superior in language and writing, while men are

superior in mathematics because of their better spatial

ability. Both of these opinions contradict the ﬁndings

of this study. According to the analysis of researchers,

the level of ability of both male and female students

might be inﬂuential. Based on the ﬁnal test results

of Integral Calculus course,all male students had

lower scores than female students. Therefore, the

results from two previous studies contradict what the

researchers found.

Furthermore, the habits of mind of the students

do not affect the level or type of error. Students as

the research subjects have very good, good and fair

criteria with ﬁve types of error. Therefore, it can be

stated that the habits of mind cannot distinguish the

types of error made by the students as well.

4 CONCLUSIONS AND

SUGGESTIONS

Based on the research, it can be concluded that: (1)

The most dominant type of error based on Newman

Error Analysis theory is encoding (92.44%) and the

least dominant type of error is decoding (6.08%); (2)

Viewed from gender differences, the most dominant

type of error by male students is encoding (96.43%)

and the least type of error is decoding (24.28%). On

the other hand, the dominant type of error by female

students is encoding (86.24%) and the least dominant

one is decoding (6.76%); (3) In terms of habits of

mind, the type of error by students with Very Good

criteria is encoding (88.28%) and the least dominant

type of error is decoding (10.56%). Then, the type

of error by students with Good criteria is encoding

(94.43%) and the type of error is decoding (19.76%).

Lastly, the type of error by students with Fair criteria

is encoding (77.7%) and the least dominant type of

error is decoding (54.28%).

Furthermore, the researchers propose the

An Analysis of Students’ Error in Solving Critical Thinking Problems in Integral Calculus Course based on Newman Error Analysis Theory

Viewed from Gender Differences and Habits of Mind

341

following suggestions such as: (1) Further studies

should be conducted on the causes of the types of

error, both based on gender differences and habits

of mind; (2) An in-depth study can be conducted

through non-test interviews, either unstructured or

structured interviews.

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