Assessing Students’ Conviction in Writing Mathematical Proofs
I. Maryono
1,2
, R. Kariadinata
1
, T. K. Rachmawati
1
and I. Nuraida
1
1
Mathematics Program, UIN Sunan Gunung Djati Bandung, Jl. A.H. Nasution No. 105, Bandung, Indonesia
2
School of Postragraduate Studies, Universitas Pendidikan Indonesia, Jl. Setiabudi No. 225, Bandung, Indonesia
Keywords: Assessment, Self-confidence, writing mathematical proofs.
Abstract: Assessing self-conviction in writing mathematical proofs is important to provide feedback for students. The
purpose of this study is to determine the correlation between the levels of self-conviction with the score of
writing evidence and to provide feedback for students who are still wrong in writing the proofs. The method
used in this study is a correlation study between the levels of self-conviction to write the proofs with the score
of writing the proofs. The subjects of this study were forty third semester students of mathematics education.
These participants were tested writing proofs along with the claim, then the correlation between the claim
with the score of the write proofs was be examined. Furthermore, students who believe that their proofs are
wrong were interviewed. The results of this study indicated that the correlation between the level of
confidence with the score of the ability to write mathematical proofs was weak. Despite their weak correlation,
assessment of conviction levels is important for providing feedback to students who believe in the proofs they
have written even though the proofs they have written were still wrong. The results of this study imply that
mathematics learning that focuses on the ability to provide mathematical proofs must provide an assessment
on the aspects of student self-conviction.
1 INTRODUCTION
Assessing students' self-conviction in writing
mathematical evidence is important. The results of
this assessment can be used as feedback for students
regarding their understanding. In writing the
evidence, convincing oneself is the first level before
convincing others. Mason, Burton and Stacey (2010)
states that there are three levels in the convincing
process: (1) convince yourself; (2) convince a friend;
(3) convince an enemy.
One of the functions of mathematical proofs and
the action of making proofs is to verify and justify a
proposition (Bell, 1976; Renz, 1981; Villiers, 1990).
The activity of verification and justification cannot be
separated from the self-conviction level of the claim.
Therefore, the degree of self-conviction in claims is a
factor to be considered in providing an assessment of
the claim.
The facts show that some students have the view
that convincing arguments are different from
mathematical proofs. They are convinced of his
opinion even though that opinion is not a
mathematical proof (Weber, 2010). This shows that
self-conviction in the claim of proof does not
guarantee that the claim is true. In some cases
students sometimes do not believe in the truth of
mathematical proofs. They are more convinced in the
truth of their inductively obtained claims. Therefore,
the lecturer needs to provide an assessment
(feedback) on the student's claim to differ which
claim is a mathematical proof, and where is not a
mathematical proof.
The main purpose of this study is to investigate
the correlation of students' self-conviction in writing
the proofs with the truth of the proofs they write and
to trace the causes in the case of students who have
high self-conviction but low in writing proofs. The
researcher gave the test of writing mathematical
proofs to 40 students after attending a lecture for
seven meetings with self-explanation technique. At
each step of proof-writing, the student must provide
his or her claims in two choices that are "certain" and
"less certain" with the assumption that the "less
certain" option is indicated by not providing an
answer.
The results of the study provide a description on
how the relationship of student self-conviction to his
claim. Furthermore, feedback to students may be
given in three possibilities, namely: ‘sure and true’ is
a category of students who have high confidence and
score. ‘not sure but true’ is a category of students who
have low self-conviction but having high scores. ‘sure
Maryono, I., Kariadinata, R., Rachmawati, T. and Nuraida, I.
Assessing Students’ Conviction in Writing Mathematical Proofs.
In Proceedings of the 2nd International Conference on Sociology Education (ICSE 2017) - Volume 1, pages 341-344
ISBN: 978-989-758-316-2
Copyright © 2018 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
341

but not true’ is a category of students who have high
confidence but low scores.
2 LITERATURE REVIEW
The essence of the assessment is to look for a link
between what should be learned in the curriculum and
what students have learned. So the main problem in
the assessment is how to recognize the existing
learning and data about what has been learned by
students (Cumming and Wyatt-Smith, 2009).
Therefore, the assessment should pay attention to the
learning process and learning achievement so as to
obtain accurate conclusions about the condition of the
students.
Students' self-conviction in the claim of writing
mathematical proofs needs to be revealed so that
feedback can be provided for students who are very
convinced that their claims are true, but the claims are
actually wrong. This is where the importance of
assessment, which serves as a means to obtain
information about students' knowledge, motivation
and potential and to provide feedback (Latta, 2007
and Ginsburg, 2009).
In mathematics, proving is the method used to
derive a clear conclusion. The importance of the
ability for mathematics teacher candidates to give
proofs can be described in the function of proofs and
proving acts, such as : (1) verification or justification
on a proposition; (2) explanation to the truth; (3)
systematisation; (4) discovery of new findings; (5)
communication (Bell, 1976 and Villiers, 1990). The
importance of the proofs can also be reviewed based
on the purpose. Renz (1981) describes seven
objectives of evidence in mathematics, namely to: (1)
Clarify the relationships between traits; (2) Giving us
pleasure in constructing arguments and finding out
the proof; (3) Helps remember important and useful
results; (4) Guiding us along the right path formally
where our intuition may be weak or misleading; (5)
Guiding calculations; (6) Exploring the nature of the
formal system; (7) Offering a different perspective.
From a pedagogical point of view, proving is a
process of convincing the validity of a statement
through logical arguments. There are three levels in
the convincing process: (1) convince yourself; (2)
convince a friend; (3) convince an enemy (Mason,
Burton and Stacey; 2010). In the process of self-
conviction, one should be convinced to oneself.
However, self-conviction in the truth of the written
argument does not guarantee that the argument is
valid.
The process of learning to practice the ability to
prove at least consists of: (1) providing
counterexamples to claims that are false; (2)
evaluating a statement to know its truth by
justification; (3) analysing the work of another
student whether there is still a mistake in his
reasoning (Thompson, 2012) .Technique used in this
research is self-explanation technique. This technique
provides guidance to students in learning proof by
asking questions: (1) Do you understand the idea? (2)
Do you understand why the idea is used?, (3) How
can the idea be used/linked to other ideas (other
theorems, prior knowledge) in proof? (Hodds,
Alcock, and Inglis, 2014).
.
3 METHODS
The method used in this study was a correlation study
between the level of confidence with the truth of
writing proofs. The subjects of this research were 40
students of mathematics teacher candidates in third
semester. The instruments used in this study were
proving ability test and interview guidance.
This research tries to analyse the results of proof
writing skill test from 40 students of mathematics
teacher candidate. In the test instructions the students
were instructed to write their conviction on each
proving steps in two categories: Sure and Less Sure.
The first data obtained is the scores of writing proofs
and the level of self-conviction. These two data were
tested for their correlation resulting in several
categories of students, namely: ‘sure and true’; ‘Sure
but not true’; and ‘less sure but true’. Furthermore,
researchers interviewed students who categorized
‘sure but not true’. The level of truth consists of two
categories namely high and low. The level is high
(ranging from 70 to 100) and low (ranging from 0 to
60). Level of conviction is divided into two
categories, namely high and low. The high category
is in the range of 70% to 80% and the low category
ranges from 00% to 60% of the standard proof
measures performed.
4 RESULTS
The correlation analysis between the conviction
levels with the ability to write proofs is presented in
Table 1. This table shows there is a positive weak
correlation between the levels of conviction with the
score of proof writing ability at 0.361. This means
that the relationship between the level of confidence
with the score of writing ability was linear, indicating
that the higher the level of conviction, the higher the
acquired score of the ability to write proofs.
Similarly, the lower the level of conviction, the lower
the acquired score of the ability to write proofs.
ICSE 2017 - 2nd International Conference on Sociology Education
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Table 1: Correlation between the conviction levels with
the ability to write proofs
Score Level of conviction
Score
Pearson
Correlation
1 0.36
*
Si
g
.
(
2-tailed
)
0.02
N 40 40
*. Correlation is significant at the 0.05 level (2-tailed).
Furthermore, the hypothesis with the level of trust
α = 0.05 was tested and the conclusion is there is a
significant relationship between the level of
conviction to the scores of proof writing ability.
Based on the coefficient of determination, the level of
confidence to the score of proof writing ability was
13.0321% and 86.9679% was determined by other
variables..
Table 2 shows that five students have very high
confidence when the truth score of writing evidence
is very low. This usually causes students to be
disappointed because their expectations are far from
reality.
Table 2: The scores of students
Name Level of conviction (%) Scores (%)
A 76.8 25
B 86 25
C 100 45
D 100 50
E 100 50
Based on the interviews of these five students, it
is found that (1) they do not understand the axiom
system; (2) they cannot connect between concepts in
an axiomatic system; (3) they were in rush to make
conclusions, with lack of reasoning to be declared
right.
5 DISCUSSION
In the context of writing mathematical proof, the
factor of self-conviction is not enough to guarantee
the truth of proofs. What constitutes a convincing
argument for one person may not at all convince
others (Harel and Sowder, 1998). This is consistent
with the findings in this study that the self-confidence
level of self-confidence score was only 13.0321%.
However, since the standard of evidence in
mathematics is clear, the role of the assessment of
mathematical proof claims is crucial to the success of
the student.
There are two components in the assessment of
mathematical proofs that are self-understanding of
the principles of proof and the ability to write the
proofs (McCrone and Martin, 2004). In this study the
assessment of mathematical proofs was done on the
ability to write proofs and conviction against self-
claims. Based on the assessment results of the claim
to write proofs, the lecturer can provide feedback for
students who believe the proof is true and still wrong.
The findings in this study as feedback for students are
(1) understanding the axiom system so that it caught
what became the relationship between concepts; (2)
the student must re-examine whether the causal link
of the chain of statements made logical or not,
whether there is still a disconnected or not.
6 CONCLUSION
There is a weak correlation between the level sof
conviction with the score of the ability to write
mathematical proofs. Although there was a weak
influence of self-confidence on the truth of this
mathematical proof, the assessment of the level of
self-conviction is useful to provide feedback for
students who believe in their claims, but found false
in proving.
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