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Mathematical proof analysis using mathematical induction of grade XI
students
To cite this article: G Emeira et al 2020 J. Phys.: Conf. Ser. 1480 012044

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National Conference on Mathematics Education (NaCoME)
IOP Conf. Series: Journal of Physics: Conf. Series 1480 (2020) 012044

IOP Publishing
doi:10.1088/1742-6596/1480/1/012044

Mathematical proof analysis using mathematical induction of
grade XI students
G Emeira1, Hapizah1*, and Scristia1
1

Mathematics Education Sriwijaya University, Palembang, Indonesia
Corresponding author’s email: hapizah@fkip.unsri.ac.id

*

Abstract. The purpose of this study is to provide an overview on how students do
mathematical proof using mathematical induction through the activity of compiling and
validating proof. It is known that mathematical induction is a proof used to prove the truth of a
mathematical statement related to natural numbers. The subjects of this study were 34 students
of grade XI SMA Negeri 13 Palembang. The results showed that (1) students were able to
prove on the base of the induction step by showing the truth when 𝑛 is assumed by one
member of a natural number, (2) students were able to make a mathematical induction
hypothesis by assuming the statement n = 𝑘 is true, (3) students still make mistakes in
performing mathematical induction steps precisely on the part when doing algebra
manipulation to prove 𝑛 = 𝑘 + 1 is true from the statement 𝑛 = 𝑘 that has been considered
true, and (4) Students still do not understand the concept of mathematical induction as
indicated by the large number of students unable to do the activities of compiling proof by
mathematical induction and there are still some students who write conclusions on the activity
of compiling proof that the statement is proven true without showing the truth of the induction
step
1. Introduction
In learning mathematics one of the activities that students must master is the activity of conducting
proof. Proof has an important role in mathematics learning [1-4]. Proof is an absolute and fundamental
part of learning mathematics or an inseparable part of learning mathematics [5-7]. Proof is a series of
logical arguments where the argument here comes from theorems, premises, definitions and postulates
which later from this argument will explain the truth of a statement [8]. Several benefits can be
obtained from conducting proving activities including being able to improve students' skills in solving
a problem, persuasive, creativity, reasoning, and mathematical thinking [8,9]. Also, students who are
accustomed to carrying out proof activities can verify a truth [10]. From the statement above it is
known that proof is an ability that students should have in learning mathematics
In carrying out proof, several methods can be used, one of them is by mathematical induction.
Mathematical induction is a standard proof technique in mathematics [11]. It is known that induction
is used to prove the truth of a mathematical statement related to natural numbers [11-14]. A statement
can be proven true if it has shown the principle of mathematical induction. It is known that there are 3
principles of mathematical induction, namely the principle of standard induction, the principle of
strong induction and the principle of induction which is confiscated [11]. The principle of
mathematical induction is generally formalized as follows: suppose 𝑃 (𝑛) a statement relating to
natural numbers. To prove that 𝑃 (𝑛) is true it must show that if (1) P (1) is true, (2) 𝑃 (𝑘) is true →
𝑃 (𝑘 + 1) is true for every 𝑘 ∈ ℕ, then 𝑃 (𝑛) is proven true for every 𝑛 ∈ ℕ [12,14]. Mathematical
induction is the third goal of the school curriculum where the teacher is expected to give more

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National Conference on Mathematics Education (NaCoME)
IOP Conf. Series: Journal of Physics: Conf. Series 1480 (2020) 012044

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doi:10.1088/1742-6596/1480/1/012044

attention to learning proof by mathematical induction [15]. Mathematical induction is one of the most
powerful tools in proving statements related to discrete mathematics [16]. From the statement above it
can be seen that the activities of proof by mathematical induction are important for secondary school
students.
According to Selden and Selden [17] the evidentiary activity consists of the activity of compiling
proof and validating proof. In carrying out the activities of compiling proof students must have the
ability to use methods of proof, entries, definitions, theorems to show the truth of a mathematical
statement. In validating proof students must have the ability to criticize proof related to the types of
proof that often appear in mathematics. There are several activities that can be carried out during the
validation of proof, including reading mathematical proof to determine the truth and error of the proof
by paying attention to the suitability of the axiom system, the premise, existing mathematical results
(theorems or entries) with deductive reasoning flow, completing the proof if mistakes are found and
compare the effectiveness of the proof with each other [17,18].
In the previous research, it was known that mathematical proofing activities usually only looked at
how the students did the activity of compiling proof. As conducted Netti researchers [19] who saw the
stages of mathematical thinking of students from the activities of proof construction on the material
function. There is no previous research that discusses how students conduct proof through the
activities of compile and validation proof on mathematical induction material which is an important
material for secondary school students. For the reasons above, the researcher is interested in taking the
title of the study " Mathematical Proof Analysis Using Mathematical Induction of Grade XI Students".
The purpose of this study is to describe how students do mathematical proof using mathematical
induction through the activities of compiling and validating proof.
2. Method
This research is a descriptive study that aims to provide an overview of the results of mathematical
proofing analysis using mathematical induction. This research has been carried out in SMA Negeri 13
Palembang in 2019/2020 academic year recorded from July 2019 to September 2019. The research
subjects used were students of class XI MIPA 1 of SMA Negeri 13 Palembang. The procedure of this
study consisted of three stages, namely the preparation phase, the implementation stage and the data
analysis stage. In the preparation stage, the thing to do is to observe the school of SMA Negeri 13
Palembang to determine the class that will be used as the subject of research and prepare a research
instrument consisting of RPP, LKPD and test questions consisting of 2 questions with different types.
Problem number 1 asks about how students validation proof related to series and problem number 2
asks how students compile proof related to division. At the stage of implementing things done is
teaching mathematics induction material in class according to the RPP that has been designed, giving
LKPD during the learning process and at the end of student learning will be given test questions.
While at the data analysis stage, the test results sheets that have been done by students are then
grouped based on the similarity of errors which are then analyzed to find out how students validate the
proof and arrange proof by looking at the principles of mathematical induction. The principle of
induction is divided into several steps including (1) the induction base step, (2) the induction
hypothesis, (3) the induction step and (4) the conclusion [12]. From the validation questions, the
students can do the induction base step if the students can show the truth in statement 1, for the
induction hypothesis of truth in statement 2, for the induction statement on 3, 4, and 5 as well as the
conclusion in statement 6.
3. Results and Discussion
The test results sheets that students have done first are grouped based on similarities and wrong
answers which are then analyzed to find out how students can validate the proof and arrange the proof
by looking at the principles of mathematical induction. From the validation of the proof activities
students can perform induction basis steps if students can show the truth in statement 1, for the
induction hypothesis of truth in statement 2, for the induction step in statements 3, 4, and 5 as well as
the conclusions in statement 6. For the activity of compiling the results of the answers students who

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IOP Conf. Series: Journal of Physics: Conf. Series 1480 (2020) 012044

IOP Publishing
doi:10.1088/1742-6596/1480/1/012044

have been classified based on similarity and error answers obtained several categories. The categories
are described in Table 1 below.
Table 1. Categories of activities for compiling student proof.
Category

Answer type

A1

Compile proof by fulfilling the principle of induction
correctly and completely.
Compile proof by fulfilling the principle of induction but
make mistakes.
Compiling proof only fulfills the principle of mathematical
induction on the basis of the induction step and the
induction hypothesis.
Compiling proof does not fulfill the principle of induction
but keeps writing conclusions.
Unable to conduct compile proof.

A2
A3

A4
A5

The results of the data analysis validated the proof and compiled the proof of class XI students of
MIPA 1 with mathematical induction as follows.
3.1. Validation of Mathematical Induction Proof
The results of data analysis from the student test answer sheets validating the proof of question
number 1 related to the series material is presented in Table 2 below.
Table 2. Distribution of student proof validation .
Principle of Induction

Validation of
Proof

Correct

Incorrect

Induction Basis Step

Statement 1

28

6

Induction Hypothesis

Statement 2

22

12

Statement 3

8

26

Statement 4

20

14

Statement 5

5

29

Statement 5

30

4

Induction step
Conclusion

Total students

34 People

From the results of data analysis validating the proof in Table 2 it is known that the average
student already knows that statement 1 is an induction basis step which is shown from the test answers
of students who write information that statement 1 is an induction basis step. Then students also can
do the induction basis step by showing the truth in statement 1 that contains 𝑃2 : 2𝑎 + 𝑏 = 2𝑎 + 𝑏,
when suppose with n = 2. Students write the reason behind the statement by adding up the first term
and the second term to show the truth of the left side so that 𝑎 + 𝑎 + 𝑏 = 2𝑎 + 𝑏. In the right
𝑛
side, students substitute the value of 𝑛 = 2 to the statement [2𝑎 + (𝑛 − 1)𝑏] so that it is evident
2

𝑛

[2𝑎 + (𝑛 − 1)𝑏] when
the number of series of the left side is the same as the result of statement
2
𝑛 = 2. In the statement 2, the average student has been able to validate by giving reasons from dari
𝑘
𝑃𝑘 ∶ 𝑎 + 𝑎 + 𝑏 + 𝑎 + 2𝑏 + 𝑎 + 3𝑏 + 𝑎 + 4𝑏 + ⋯ + 𝑎 + (𝑘 − 1)𝑏 = [2𝑎 + (𝑘 − 1)𝑏]
is
an
2
induction hypothesis.
In the statement 3, only a small proportion of students can validate correctly. This is known from
the results of the student evidence validation activity where only 8 students answered correctly and
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IOP Conf. Series: Journal of Physics: Conf. Series 1480 (2020) 012044

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doi:10.1088/1742-6596/1480/1/012044

correctly in validating the statement 3. Students who validate correctly can give reasons 𝑃𝑘+1 : 𝑎 + 𝑎 +
𝑘+1
[2𝑎 + 𝑏𝑘] is an induction
𝑏 + 𝑎 + 2𝑏 + 𝑎 + 3𝑏 + 𝑎 + 4𝑏 + ⋯ + 𝑎 + (𝑘 − 1)𝑏 + 𝑎 + 𝑏𝑘 =
2
step. Students also explain that the statement 3 is a step used to show the truth of 𝑛 = 𝑘 + 1 by
using the statement 𝑛 = 𝑘 which has been assumed to be true. Students also have shown the
statement 𝑎 + 𝑎 + 𝑏 + 𝑎 + 2𝑏 + 𝑎 + 3𝑏 + 𝑎 + 4𝑏 + ⋯ + 𝑎 + (𝑘 − 1)𝑏 because it adds 1 term from
the statement n = k which we have assumed to be true in the 3rd statement by substituting k + 1 into
𝑘+1
the statement 𝑎 + (𝑘 − 1) 𝑏 to obtain 𝑎 + 𝑏𝑘. For statements
( 2𝑎 + 𝑏𝑘) obtained by
2

𝑘

substituting 𝑘 + 1 into the statement [2𝑎 + (𝑘 − 1)𝑏]. While students who are not quite right in
2
validating statement 3 are only giving reasons that statement 3 is an induction step used to show the
truth of 𝑛 = 𝑘 + 1 using the statement 𝑛 = 𝑘 which we already consider to be true in the
statement.

Figure 1. The test answers validate student 1 proof.

Figure 2. The test answers validate student 2.
For the statement 4 students make a lot of mistakes from giving reasons because 𝑎 + 𝑎 + 𝑏 + 𝑎 +
𝑘
2𝑏 + 𝑎 + 3𝑏 + 𝑎 + 4𝑏 + ⋯ + 𝑎 + (𝑘 − 1)𝑏 + 𝑎 + 𝑏𝑘 = [2𝑎 + (𝑘 − 1)𝑏], then the statement on
the left side can be changed to

𝑘+1
2

2

[2𝑎 + 𝑏𝑘] even though students should write on the left side it can
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IOP Conf. Series: Journal of Physics: Conf. Series 1480 (2020) 012044

be changed to

𝑘
2

IOP Publishing
doi:10.1088/1742-6596/1480/1/012044

[2𝑎 + (𝑘 − 1)𝑏] + 𝑎 + 𝑏𝑘. Furthermore, in the statement 5 it is known that only 5
𝑘

𝑘+1

[2𝑎 + (𝑘 − 1)𝑏] + 𝑎 + 𝑏𝑘 is equal to
[2𝑎 + 𝑏𝑘] by
students can validate the statement
2
2
showing the truth of the statement (figure 1). Almost all students cannot show the truth of the
statement. It is known that students often make mistakes in doing algebraic manipulation shown in
Figure 2. From Figure 2 it is known that student 2 cannot apply the distributive nature of algebra. This
𝑘
𝑘
[3𝑎 + 𝑏𝑘 2 − 𝑏] where
is shown by student 2 who writes the result of [2𝑎 + 𝑏𝑘 − 𝑏] + 𝑎 + 𝑏𝑘 =
the result should be

2𝑎𝑘+𝑏𝑘 2 −𝑏𝑘+2𝑎+𝑏𝑘
2

2

2

. As a result of mistakes made by students in doing algebraic
𝑘

𝑘+1

[2𝑎 +
manipulation, students cannot show that [2𝑎 + (𝑘 − 1)𝑏] + 𝑎 + 𝑏𝑘 is the same value as
2
2
𝑏𝑘]. Because students cannot show the truth of the statement, students cannot validate the proof of the
statement 5 which is an induction step.
In the statement 6 it is known that students have been able to give reasons from the conclusions
given why the statement 𝑃 (𝑛) can be proven to be true because it has shown the induction basis step
and the induction step is proven to be true. But in the concept of mathematical induction the activity of
validating evidence in the statement 6 made by students is wrong. Because most students make
mistakes and mistakes in validating statement 5 which is an induction step in mathematics. So it can
be said that students cannot validate correctly in statement 5.
From the above explanation it can be concluded that students have not been able to validate the
evidence in statements 3, 4 and 5 because they still make mistakes in showing the truth of the
statement. In addition, it is known that students' understanding of algebraic concepts is still weak
which is indicated by students who cannot validate the truth of the 5 th statement. This is in line with
research conducted by Kong [20] which explains that the lack of knowledge students have on certain
topics in metaphorics affects the ability of students to complete proof with mathematical induction.
3.2. Compile Proof by Mathematical Induction
The results of data analysis from the answer sheets of student tests compiling proof with mathematical
induction on item number 2 related to the subject matter of division are presented in Table 3 below.
Table 3. Distribution compile student proof
Category

Frequency
( People)

A1
A2
A3
A4
A5
Total students

5
2
11
6
10
34

From Table 3 which is the distribution of activities to compile student proof it is known that only 5
people fall into the A1 category. Prove the truth of the statement 𝑎2𝑛 − 𝑏 2𝑛 divide by (𝑎 + 𝑏) by
fulfilling the principles of mathematical induction correctly and completely, which shows the truth of
the induction basis step, makes the induction hypothesis, shows the truth of the induction step and
makes conclusion. In the A2 category, students have proven the truth of the statements 𝑎2𝑛 −
𝑏 2𝑛 divided (𝑎 + 𝑏)by fulfilling the principle of induction but making mistakes and mistakes.
Students who make mistakes can be seen in Figure 3 and students who make mistakes can be seen in
Figure 4 below.

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IOP Conf. Series: Journal of Physics: Conf. Series 1480 (2020) 012044

IOP Publishing
doi:10.1088/1742-6596/1480/1/012044

Figure 4. Student error 4 in compiling
proof at the basis step.

Figure 3. Student error 3 in compiling
proof at the induction step.

It is known from Figure 3, student 3 made a mistake in the induction step to show the truth of 𝑛 =
𝑘 + 1. Student 3 made a mistake in changing the form 𝑎2𝑘 . 𝑎2 − 𝑏 2𝑘 . 𝑏 2 to (𝑎2𝑘 − 𝑏 2𝑘 )( 𝑎2 + 𝑏 2 ).
It is known that the result of the multiplication factor (𝑎2𝑘 − 𝑏 2𝑘 )( 𝑎2 + 𝑏 2 ) = 𝑎2 𝑎2𝑘 + 𝑎2𝑘 𝑏 2 −
𝑎2 𝑏 2𝑘 − 𝑏 2 𝑏 2𝑘 not 𝑎2𝑘 . 𝑎2 − 𝑏 2𝑘 . 𝑏 2 . From Figure 4 it is known that student 4 made a mistake at the
basis induction step. To show the truth at the basis step of student induction 4, suppose 𝑛 = −1.
While the problem is known that n is a member of a natural number where the member of a natural
number starts from number 1. Negative numbers are not included in the original number.
In category A3 students can only carry out activities to compile proof up to the step of the
induction basis and the induction hypothesis. Students have not been able to show the truth at the
induction step, leading statements 𝑎2(𝑘+1) − 𝑏 2(𝑘+1) can be completely divided (𝑎 + 𝑏). In addition,
there are some students who can only do induction steps limited to writing 𝑎2(𝑘+1) − 𝑏 2(𝑘+1) . The
student cannot continue, herding statements 𝑎2(𝑘+1) − so that they have been divided (𝑎 + 𝑏)and
there are also some answers of category A3 students in showing the truth in the induction step of
making a mistake in applying the distributive nature of algebra. The tendency of students to make
mistakes in applying distributive algebra is supported by Kurniati research [21]. In Kurniati research
explained that many students have difficulty in working on algebraic form problems is to apply the
distributive nature of algebra.
Students in the A4 category in carrying out the activity of compiling evidence tend to keep writing
the conclusion that the statements 𝑎2𝑛 − 𝑏 2𝑛 are completely divided (𝑎 + 𝑏) proven to be true,
without showing the truth in the induction step as shown in the following Figure 5.

Figure 5. Students' 5 test answers categorized as A4.
From Figure 5 above it is known that student 5 does not fulfill the principle of induction by not
showing the truth in the induction step. This is because student 5 did not complete the activity of
compiling proof at the induction step and also the student made a mistake in doing multiplication of
rank. Student 5 writes the result of the multiplication of 𝑎2(𝑘+1) − 𝑏 2(𝑘+1) = 22𝑘 . 𝑎1 − 𝑏 2𝑘 . 𝑏1 . The
result of the multiplication of the rank is 𝑎2𝑘 . 𝑎2 − 𝑏 2𝑘 . 𝑏 2 . In addition, on the answer sheet of student
test 5 it is known that the student wrote a conclusion because it is in accordance with the principle of
mathematical induction when it shows the truth of the initial step (induction basis step) and the
induction step then the statement of 𝑎2(1) − 𝑏 2(1)divisible by (𝑎 + 𝑏). The tendency of students in
writing conclusions without showing the truth of the basis and induction step is known to not

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IOP Conf. Series: Journal of Physics: Conf. Series 1480 (2020) 012044

IOP Publishing
doi:10.1088/1742-6596/1480/1/012044

understand the concept of mathematical induction where the truth of the induction basis step implies
the truth of the induction step then the truth of the induction step implies the statement can be proven
true. So that if the basis and induction steps are proven to be true then it can be concluded that the
statement is true but students tend to follow the rules of the mathematical induction principle which
must write the basis of induction steps, induction hypotheses, induction steps and conclusions. This is
in line with previous studies conducted by Harel [22], and Styliandes et al [23]. In that study explained
that students tend to follow the rules of the principle of induction without understanding the concept of
induction.
Students in the A5 category are known to have no understanding at all of how to carry out activities
to compile proof. This is known through the results of student answer sheets in which category A5
students cannot answer at all question number 2. The inability of students to carry out the activities of
compiling proof is because students are still unfamiliar with proof learning. Usually in the learning
process students rely more on memorizing existing evidence without understanding the evidence and
how to arrange the proof. This explanation is in line with what was said by Moore [24].
From the analysis of the student answer sheets described above it can be seen that the average
student cannot do the activity of compiling proof using mathematical induction. Students still
experience technical difficulties shown by students who still make mistakes in doing algebraic
manipulation in the induction step. This is in line with research conducted by Kong [20]. In addition
students still do not understand the concept of mathematical induction in compiling proof.
4. Conclusion
Based on the results of this study it can be concluded that through the activities of validation proof and
compile proof, students can already take an induction base step. Students also can make a hypothesis
of the induction step by assuming that the statement 𝑛 = 𝑘 is true. But apparently, most students still
do not understand the concept of proof of mathematical induction. It is known from many students still
write the conclusions that statements are proven true without showing the truth of the induction step.
Also, students still experience technical difficulties from the induction step shown from the answer
sheet of students who are still making mistakes in algebra manipulation when showing the truth of the
statement 𝑛 = 𝑘 + 1 by using the statement 𝑛 = 𝑘 that has been assumed is true.
Based on the conclusion above, the researcher suggests to the teacher to develop LKPD by the
principle of induction to make it easier for students to understand the concept of mathematical
induction. Also, it is expected that teachers will strengthen the concept of algebra in students so
students do not make mistakes in manipulating algebra in the induction step and emphasizing the
concept of mathematical induction that the steps in the induction principle are mutually implicating.
5. Acknowledgments
Researchers would like thank Mr. Yusuf Hartono for giving advice related to proof from mathematics.
The author also expresses her thanks to Mrs. Elika Kurniadi, M.Sc who validated the instrument in
this study and did not forget the researchers also expressed their thanks to the principal of SMA
Negeri 13 Palembang and Mrs. Dra. Badiah Asni as a mathematics teacher in class XI MIPA 1 so that
this article can be realized.
6. References
[1] NCTM 2003
Programs for Initial Preparation of Mathematics Teachers online:
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[2] Baiduri 2017 JPM 2 99
[3] Findel B R 2001 Learning and Understanding in Abstract Algebra Doctoral Disseration
(University of New Hampshire)
[4] Tarhadi S and Pujiastuti S L 2006 Jurnal Pendidikan Terbuka dan Jarak Jauh 7 121
[5] Dickerson 2008 High School Mathematics Teachers' Understandings of the Purposes of
Mathematical Proof Doctoral Dissertation (Syracuse University)

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IOP Conf. Series: Journal of Physics: Conf. Series 1480 (2020) 012044

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[23]
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doi:10.1088/1742-6596/1480/1/012044

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