Students’ Semantic-Proof Production in Proving a Mathematical Proposition

Journal of Education and Learning (EduLearn)
Vol.12, No.3, August 2018, pp. 560~567
ISSN: 2089-9823 DOI: 10.11591/edulearn.v12i3.5578  560
Journal homepage: http://journal.uad.ac.id/index.php/EduLearn
Students’ Semantic-Proof Production in Proving a
Mathematical Proposition
Syamsuri1
, Purwanto2
, Subanji3
, Santi Irawati4
1
Department of Mathematics Education, Universitas Sultan Ageng Tirtayasa, Indonesia
2,3,4
Department of Mathematics, Universitas Negeri Malang, Indonesia
Article Info ABSTRACT
Article history:
Received Sep 2, 2017
Revised Apr 27, 2018
Accepted May 11, 2018
Proving a proposition is emphasized in undergraduate mathematics learning.
There are three strategies in proving or proof-production, i.e.: procedural-
proof, syntactic-proof, and semantic-proof production. Students‟ difficulties
in proving can occur in constructing a proof. In this article, we focused on
students‟ thinking when proving using semantic-proof production. This
research is qualitative research that conducted on students majored in
mathematics education in public university in Banten province, Indonesia.
Data was obtained through asking students to solve proving-task using think-
aloud and then following by interview based task. Results show that
characterization of students‟ thinking using semantic-proof production can be
classified into three categories, i.e.: (1) false-semantic, (2) proof-semantic for
clarification of proposition, (3) proof-semantic for remembering concept.
Both category (1) and (2) occurred before students proven formally in
Representation System Proof (RSP). Nevertheless, category (3) occurred
when students have proven the task in RSP then step out from RSP while
proving. Based on the results, some suitable learning activities should be
designed to support the construction of these mental categories.
Keywords:
APOS theory
False-semantic proof
Mathematical proposition
Proving
Semantic-proof
Copyright © 2018 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Syamsuri,
Department of Mathematics Education,
Universitas Sultan Ageng Tirtayasa,
Jalan Raya Jakarta KM. 4 Pakupatan Kota Serang, Provinsi Banten, Indonesia.
Email: syamsuri@untirta.ac.id
1. INTRODUCTION
Mathematical proof is important component in mathematics learning. As well as knowledge,
mathematics develops through mathematical proof. Besides, a mathematical proposition can accepted as
knowledge if it has a correct proof. Therefore, mathematical proof is like a pillar in mathematics building [1].
The purpose of activity in constructing a proof is to prove and to explain a proposition [2]. As well
as Hanna suggested that proof as bearer a knowledge in context of mathematics learning [3]. Role of
mathematical proof is bearer of mathematical knowledge, because a proof contains a new knowledge that
constructed using knowledge before. Moreover, mathematical proof can generate problem-solving process to
think on constructing a mathematical knowledge [4].
Proving a proposition emphasize in undergraduate mathematics learning. Therefore, undergraduate
mathematics learning is different from school mathematics learning. In undergraduate learning put emphasis
on logics and rigorous mathematical thinking. For instance, at mathematical school, vector as mathematical
object that have magnitude and direction, also vector can be displayed as arrow or others symbol.
Nevertheless, at undergraduate learning, vector was defined axiomatically and constructed through formal
deductive. A proposition is valid if there is proof that showed from axioms and some definitions using logics
rule. Proving process use formal approach and can behave a condition and appropriate to proving context.
Therefore, mathematics learning in university involves mental confusion as connection among perception

EduLearn ISSN: 2089-9823 
Students’ Semantic-Proof Production in Proving a Mathematical Proposition (Syamsuri)
561
and action, and then reorganize formally deductive, so that can build a new experience through
formal situation [5].
In students‟ thinking process in constructing proof, Pinto and Tall used term 'natural' to describe the
process of extract-meaning and term 'formal' to describe process of giving-meaning that work formally [6]. In
addition, Weber added idea of 'procedural' learning for students who are just trying to cope with formal
definitions and proving with rote learning [7]. Alcock and Weber customized student response to 'semantic'
and 'syntactic', based on language which refers to the meaning of language (semantic contents) and grammar
(syntax) [8]. Alcock and Weber described the syntactic approach as one strategy in proving that involved
mathematical definition formally, and semantic approach as a strategy in utilizing intuitive understanding of
a concept. It is actually in line with the term extract-meaning and giving meaning in the category of 'formal'
and 'natural'.
In this regard, according to [4] stated that there are three strategies are usually done to prove. A first
strategy is strategy of procedural proof production. In a procedural strategy of proof production, students
prove by looking for proof of similar examples. Furthermore, students modify it according to the statement to
be proved. Topics that are usually done with this strategy is the limit of a sequence and limit of function. A
second strategy is syntactic proof production. Syntactic proof production strategy is a strategy that began
with some definitions and assumptions are appropriate to the problem, and then draw conclusions based on
the definitions and assumptions by utilizing existing theorems and rules of logic. A third strategy is the
semantic proof production. Semantic proof production strategy is the strategy of mathematical proofs that use
different representations of mathematical concepts informally to guide writing of strict and formal proof. The
illustrations are, i.e.: graphs, drawings, or examples of cases. With hint of illustrations, the idea of proving is
expected to appear. Alcock and Inglis [9] defined proof production using Representations System of Proof
(RSP). RSP is a series of statement set based on logics and deductive to show that the proposition is true.
According that, they stated that if all step to produce statements in proving on RSP, so then it is called
syntactic-proof production. Moreover, it is called semantic-proof production when step out from
RSP while proving.
Many studies have revealed that students have difficulties in constructing a mathematical proof [10-
14]. The difficulties in proving can occurred in constructing a proof, so then made an incorrect proof. One
strategy to refine a constructing a proof is how to encourage students to make sense through example,
drawing a picture, sketch or other informal mathematical objects. We was suggested that semantic strategy
can help students to aware about their error in constructing a proof [15]. This strategy encourages students to
do reflective thinking so that they can be aware of their fault when constructing a proof.
In revealing a thinking process, this study is based APOS theory [16]. APOS theory proposes that an
individual has to have appropriate mental structures to make sense of a given mathematical concept. The
mental structures refer to actions, processes, objects and schema which required to learn the concept.
Research based on this theory requires that for a given concept the likely mental structures need to be
detected, and then suitable learning activities should be designed to support the construction of these mental
structures. Base on explain above, thus article will describe characterization of students‟ thinking using
semantic-proof production in proving a mathematical proposition.
2. RESEARCH METHOD
2.1. Participants
This research is descriptive exploratory study to reveal characteristics of semantic proof production
in proving. Therefore, research approach used is qualitative. The research conducted on students majored in
mathematics education in public university in Banten province, Indonesia. The consideration of that was
students were able to think a formal proof in mathematics. Students who constructed semantic proof
production was selected as research subject.
2.2. Instrument
The main instrument in qualitative research was researcher itself. The support instruments are
proving-task and interview guides. These instruments were evaluated and validated from two lecturers in
order to guarantee quality of instruments. The interview is open and it‟s needed to reveal students‟ response
about error in constructing a proof. Procedure to obtain data are 1) subject is given the task-proving and
asked him/her to accomplish the task by think-aloud. And then 2) subject is interviewed base on the-task.
Therefore, scratch of proving-task and transcript of interview is obtained. The proving-task is in the
following.
Prove: For any positive integers m & n, if m2
and n2
are divisible by 3, then m+ n is divisible by 3.

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EduLearn Vol. 12, No. 3, August 2018 : 560 – 567
562
We used this task because some methods can use for solving, i.e.: direct proof, contradiction, and
contra-positive. Besides, we would like to test students‟ comprehension about mathematical induction
method, because some students have an opinion that using mathematical induction to prove a number which
“divisible by 3”. This research is descriptive exploratory study to reveal characteristics of semantic proof
production in proving. Therefore, research approach used is qualitative.
2.3. Data Analyse
Data obtained are recording of think-aloud, scratch of proving-task-sheet, and recording of
interview. These data was collected to describe students‟ thinking process in constructing a semantic proof
production. Therefore, reliability of this research is source of data.
To analyse students‟ thinking process, we use APOS theory. In APOS theory, the main mental
mechanisms for building the mental structures of action, process, object, and schema are called
interiorization and encapsulation [17]. The mental structures of action, process, object, and schema constitute
the acronym APOS. APOS theory postulates that a mathematical concept develops as one tries to transform
existing physical or mental objects. The descriptions of action, process, object and schema in this research
are given below:
a. Action: A transformation is first conceived as an action, when it is a reaction to stimuli which an
individual perceives as external. Action‟s indicator is student has given some correct-example for the
proposition.
b. Process: As an individual repeats and reflects on an action, it may be interiorized into a mental process.
A process is a mental structure that performs the same operation as the action, but wholly in the mind of
the individual. Process‟ indicator is students could create mathematical equation model into variable
form.
c. Object: If one becomes aware of a process as a totality, realizes that transformations can act on that
totality and can actually construct such transformations (explicitly or in one‟s imagination), then we say
the individual has encapsulated the process into a cognitive object. For example, students could create
other representation from mathematical equation model in order to connect to other information in the
proposition.
d. Schema: A mathematical topic often involves many actions, processes, and objects that need to be
organized and linked into a coherent framework, called a schema. It is coherent in that it provides an
individual with a way of deciding, when presented with a particular mathematical situation, whether the
schema applies. For example, there are good connection among integer concept, divisibility concept,
logic of implication in the proposition.
3. RESULTS AND ANALYSIS
Of the 25 students who were given the proving-task. There were 12 students who constructed proof
using semantic-proof production. The remaining 13 students are constructed using syntactic proof
production. The semantic-proof production is occurred before students prove formally in Representation
System Proof (RSP) or when students prove in RSP. Of the 12 students who constructed proof using
semantic-proof production, their thinking process can be classified into 3 categories, i.e.: false-semantic,
proof-semantic for clarification of proposition, and proof-semantic for remembering concept. False-semantic
and semantic-proof for clarification of proposition occurred before students prove formally in Representation
System Proof (RSP). And semantic-proof for remembering concept occurred when students prove in RSP.
3.1. False-semantic Proof Production
There are 4 students who constructed false-semantic proof. Based on our observation, founded that
some students tried to make sense of proposition by incorrect illustration. Proving using incorrect
illustrations called “false-semantic”[18]. Figure 1 shows an example of false-semantic proof production. For
instance, S1 given illustration for the proving-task by choosing m=1 and n=1, so that m2
=12
=1 is not divisible
by 3. So then, S1 concluded that the proving-task is unproved. Other student, namely S2, she given an
example in form even number, m=2 and n=2 are not divisible by 3, whereas the example is given by S1/S2
are incorrect.

563
1a. A false-semantic proof is written by S1
1b. A false-semantic proof is written by S2
Figure 1. Example of False-Semantic Proof Production
According to Balacheff [19], a proof was constructed by either S1 or S2 is naïve-empirism, but
incorrect illustration. S1/S2 didn‟t have a good comprehension about the proposition. Understanding about
content of proposition is important to do next proving step. Andrew stated two main components in
evaluation students‟ proof, that are conceptual understanding and proof-structure [20]. If conceptual
understanding was not appeared, so then students will be difficult to construct a proof. It is in line with
Moore [10] who stated that students have difficult how to begin a proof.
Corresponding to scratch of student proof of S1/S2, we can see that both S1 and S2 have done
„Action‟ using illustration in order to understand the proving-task proposition. Yet S1/S2 was gave incorrect
illustration. Corresponding to transcript of think-aloud, both S1 and S2 was thought “For any positive
integers m & n, if m2
and n2
are divisible by 3, then …” by only emphasized on “For any positive integers m
& n”, neither S1 and S2 was thought on “if m2
and n2
are divisible by 3”. Its showed that both S1 and S2
thought imperfect on proof-structure, so that they didn‟t use all information in the task-proving proposition.
Therefore, they done imperfection “Action‟, so then they didn‟t appear interiorization “Action‟ into
“Process”.
Students‟ characteristics in false-semantic have no proof-structure, and also have a little conceptual
understanding. They done imperfection “Action‟, so then they didn‟t perform interiorization “Action‟ into
“Process”. Therefore, the students need assistance to refine a proof-structure and conceptual understanding
about proposition. One of method is learning using worked-example [7]. Worked-example is example how to
proving a proposition, and involved arguments in every step. Worked-example can refine students‟
knowledge, 1) how to begin a proof, 2) how to understand about end of proof, 3) how to give argumentation
for each step, and 4) how to select mathematical concept is needed. Point 1) and 2) related to refine proof-
structure, in addition point 3) and 4) related to refine conceptual-understanding.
3.2. Semantic-proof for Clarification of Proposition
There are 5 students who constructed proof-semantic for clarification of proposition. Based on our
observation, founded that some students tried to make sense of proposition by correct illustration. Figure 2
shows an example of semantic-proof for clarification of proposition. For instance, S3 given illustration for
the proving-task by choosing m=6 and n=3, so that m2
=62
=36 and n2
=32
=9 are divisible by 3. So then, S3
concluded that the proving-task is proved. Other student, namely S4, she given an example in form even
number, m=6 and n=6 are divisible by 3.

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564
Figure 2. Example of semantic proof for clarification of proposition is written by s3
According to Balacheff [19], a proof was constructed by S3/S4 is naïve empirism. Both S3 and S4
only recalled an example of proposition. Students in this group could not use high order thinking for
analysing in order to connect to other concepts. Conceptual understanding about content of proposition is
partial, it‟s showed that proof schema was not built. Neither S3 nor S4 have connected concept well. Besides,
they didn‟t appear proof-structure, because they didn‟t definite a first step in proving. The first step is
important for next step [10].
Corresponding to scratch of student proof of S3/S4, we can see that both S3 and S4 have done
„Action‟ well using illustration in order to understand the proving-task proposition. Both S3 and S4 was gave
correct illustration. Corresponding to transcript of think-aloud, both S3 and S4 were thought “For any
positive integers m & n, if m2
and n2
are divisible by 3, then …” well. Both S3 and S4 used information in
hypothesis of task-proving. Nevertheless, S3/S4 couldn‟t bring up a connection among divisibility concept of
m and n, so then interiorization wasn‟t appeared. Therefore, they done imperfection “Action‟, so then they
didn‟t appear interiorization “Action‟ into “Process”.
Students‟ characteristics in this group have no proof-structure. Therefore, the students need
assistance to refine a proof-structure. One method for refining proof-structure is learning using structural
method. This method is adopted from Leron [21]. The method triggered by recent ideas from computer
science, is intended to increase the comprehensibility of mathematical presentations while retaining their
rigor. The basic idea underlying the structural method is to arrange the proof in levels, proceeding from the
top down; the levels themselves consist of short autonomous "modules," each embodying one major idea of
the proof. The top level gives in very general (but precise) terms the main line of the proof. The second level
elaborates on the generalities of the top level, supplying proofs for unsubstantiated statements, details for
general descriptions, specific constructions for objects whose existence has been merely asserted, and so on.
If some such sub procedure is itself complicated, we may choose to give it in the second level only a "top-
level description," pushing the details further down to lower levels.
3.3. Semantic-proof for Remembering of Concept
There are three students who constructed proof-semantic for clarification of concept. The students in
proof-semantic for remembering of concept have proven the task in RSP, then step out from RSP while
proving. The aim of this strategy is to remember or clarify concept by example. Figure 3 shows an example
of semantic proof for remembering of concept. For instance, S5 gave illustration whether if (m+n)(m-n) is
divisible by 3 then (m+n) is divisible by 3. S5 wrote (5+4)(5-1)|3 and clarified whether its statement correct
or incorrect. But actually the case is 3|(5+4) (5-1). Another students, S6 gave illustration for getting meaning
of a≡b(mod m) by a=2m+b and 4≡7(mod 3). While S7 gave illustration for getting meaning of a|b by
b=am+k.
According to Balacheff [19], a proof was constructed by them is thought-experiment. They used
high order thinking for analysing in order to connect to other concepts. Conceptual understanding about
content of proposition is integrated, it‟s showed that proof schema was built. Nevertheless, they needed some
aid for reinforcement a claim using semantic strategy. They have a good comprehension about mathematical
concept and ability to connect them. Besides, they appeared proof-structure, because they defined a first step
in proving well.
Corresponding to transcript of think-aloud, S5 was thought “For any positive integers m & n, if m2
and n2
are divisible by 3, then …” well. S5 used integrated information in hypothesis of task-proving. S5
done “Action‟ well, so then S5 appeared interiorization “Action‟ into “Process”. And then, encapsulation

565
„Process‟ into „Object‟ is successful, it showed by correct claim that m2
+n2
is divisible by 3. Nevertheless,
when S5 claimed that m+n is divisible by 3, S5 only paid attention to m+n without m-n, but actually the case
is m-n is divisible by 3 too. Besides, notation is written by S5 was incorrect, get accidently exchanged among
divided and divisor. It‟s indicated that Object is not formed, so that Schema is uncompleted too.
Figure 3. Example of semantic proof for remembering of concept is written by s5 (left) and s6 (right)
Corresponding to transcript of think-aloud, Both S6 and S7 were thought “For any positive integers
m & n, if m2
and n2
are divisible by 3, then …” well. Either S6 or S7 used integrated information in
hypothesis of task-proving. They done “Action‟ well, so then they appeared interiorization “Action‟ into
“Process”. And then, encapsulation „Process‟ into „Object‟ is unsuccessful, it showed by incorrect claim that
m+n is not divisible by 3. S6 argued that 3k
m n
m n
 

is not divisible by 3. It‟s indicated that encapsulation
is unsuccessful. Meanwhile S7 only showed 2 1
3 3
m n b k a k
     . It‟s indicated that S7 failed in
encapsulation „Process‟ into „Object‟. It‟s indicated that Object is not formed, so that Schema is
uncompleted too.
Students‟ characteristics in this group have a proof-structure, but a little of concept is forgotten.
Therefore, the students need assistance to refine by remembering a needed concept. One method for refining
conceptual-understanding is learning that emphasized meaning of concept, such as reading for meaning [22].
Accordingly, mathematical concept such as definition, notation or applied matter can be understood well.
4. CONCLUSION
Characterization of students‟ thinking using semantic-proof production can be classified into 3
categories, that are false-semantic, proof-semantic for clarification of proposition, and proof-semantic for
remembering concept. False-semantic and semantic-proof for clarification of proposition occurred before
students prove formally in Representation System Proof (RSP), but semantic-proof for remembering concept
occurred when students prove in RSP.
Students‟ characteristics in false-semantic have no proof-structure, and also have a little conceptual
understanding. They done imperfection “Action‟, so then they didn‟t perform interiorization “Action‟ into
“Process”. Therefore the students need assistance to refine a proof-structure and conceptual understanding
about proposition. One of method is learning using worked-example.
Students‟ characteristics in proof-semantic for clarification of proposition have no proof-structure.
They could not use high order thinking for analysing in order to connect to other concepts. Conceptual
understanding about content of proposition is partial, it‟s showed that proof schema was not built. They done
imperfection “Action‟, so then they didn‟t perform interiorization “Action‟ into “Process”. Therefore, the
students need assistance to refine a proof-structure. One method for refining proof-structure is learning using
structural method.
Students in proof-semantic for remembering of concept have proven the task in RSP, then step out
from RSP while proving. A proof was constructed by them is thought-experiment. They used high order
thinking for analysing in order to connect to other concepts. They appeared interiorization “Action‟ into
“Process”, but Schema is not formed well. Students‟ characteristics in these categories have conceptual-

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566
understanding and proof-structure, but some concept was forgotten. Therefore, to the students need assistance
to refine by remembering a needed concept. One method for refining conceptual-understanding is learning
that emphasized meaning of concept, such as reading for meaning.
ACKNOWLEDGEMENTS
The first author gives appreciation and thanks to The Ministry of Research, Technology and Higher
Education of Indonesia which provides the opportunity to continue his studies in mathematics education
study program of Universitas Negeri Malang.
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BIOGRAPHIES OF AUTHORS
Syamsuri, Dr., M.Si., S.Si. is currently an assistant professor in the Department of Mathematics
Education at the Universitas Sultan Ageng Tirtayasa. He holds a S.Si. in Mathematics from
Institut Pertanian Bogor (IPB) and a M.Si. in Applied Mathematics from IPB. He earned his
Doctor degree in Mathematics Education from Universitas Negeri Malang (2017), with a
concentration in thinking process reasearch on mathematics and its learning. His research spans
the field of mathematics education and applied mathematics.
Purwanto, Prof., PhD., Drs. is currently a professor in the Department of Mathematics at the
Universitas Negeri Malang. He holds a Drs. in Mathematics Education from IKIP Yogyakarta
and a Ph.D in Mathematics from Curtin Univ. of Tech., Australia (1992). His research spans the
field of combinatorics, graph theory and mathematics education.
Subanji, Dr., M.Si., S.Pd. is currently an associate professor in the Department of Mathematics at
the Universitas Negeri Malang. He holds a S.Pd. in Mathematics Education from IKIP Malang; a
M.Si. in Mathematics from Institut Teknologi Bandung (ITB) and a Dr. in Mathematics
Education from Universitas Negeri Surabaya (2007). His research spans the field of mathematics
education, especially in thinking process reasearch.
Santi Irawati, Ph.D., M.Si., Dra. is currently an associate professor in the Department of
Mathematics at the Universitas Negeri Malang. She holds a Dra. in Mathematics Education from
IKIP Malang ; a M.Si. in Mathematics from Institut Teknologi Bandung (ITB) and a Ph.D. in
Mathematics from Hyogo University of Teacher Education, Japan (2007). Her research spans the
field of mathematics education and mathematics, especially in algebra.

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