19750 23791 1 PB
PROFILE OF JUNIOR HIGH SCHOOL STUDENTS’ CRITICAL THINKING ON
MATHEMATICAL PROVING BASED ON SEX DIFFERENCES
Tiania Mewanda
Mathematics Education, Faculty of Mathematics and Natural Sciences, State University of Surabaya,
e-mail : [email protected]
Abdul Haris Rosyidi
Mathematics Education, Faculty of Mathematics and Natural Sciences, State University of Surabaya,
e-mail : [email protected]
Abstract
Developing critical thinking in solving a problem is one of the aims in learning mathematics.
Critical thinking is defined as a mental activity involving organized ways in processing information
which consists of looking for clarity, analyzing, evaluating and determining decisions. Critical thinking is
needed in mathematical proving and it can be seen through the process of solving mathematical proving
done by the students. Sex differences have probability to influence the students’ critical thinking.
This study is included into qualitative research by using interview method which was based on
the test aiming to describe the critical thinking profile of junior high school students in mathematical
proving of number based on sex differences. The subject of this study consisted of one male and one
female student with the same level of high competence chosen among five male and five female students
that closely identify critical thinking in proving based on the most variation in students’ answer. The data
were analyzed based on the indicators of critical thinking which consist of clarification, assessment,
inference and strategy.
Based on the result of the study, the steps and strategy done by male and female subject in doing
proving were relatively showing similarity. Both subjects used picture representation in proving. In
clarification category, both subjects identified information which was known and precisely proven by
interpreting word by word. Both subject also identified the relation of parts which was needed in proving
by using their prior knowledge. However, the male subject had more details than the female one. In the
assessment category, both subjects were able to evaluate the steps done in proving. In inference category,
both subjects were able to describe and explain the steps used as well as the reasons in every step. The
male student was able to do the proving with more than one method, even he was able to describe and
explain every step as well as the correct reasons so that he fulfilled the strategy category. On the other
hand, the female subject was unable to do the proving in more than one method so that she did not fulfill
the strategy category.
Keywords: Critical Thinking, Mathematical Proving, Sex
INTRODUCTION
1
Rapid
development
of
information
and
communication
technology requires skills
of
selecting
and
processing
information
properly. One of the most
important skills is critical
thinking. Furthermore, in
facing the complexity of
our life, critical thinking is
paramount in solving
many problems. Since
critical thinking is urgent
in daily basis, the teaching
of it in school, therefore,
is essential.
Curriculum
always
puts
critical
thinking as a requirement
that should be mastered
by students. One of the
subject
in
school
requiring critical thinking
skill is mathematic. It is
shown by the fact that
mathematic
standard
competence ruled out by
the decree of culture and
education
minister
number 21 year 2016
which requires students
to act logically, critically,
analytically,
precisely,
responsibly, responsively
and toughly in solving
problems.
One of the ways
to develop the critical
thinking skill of students
is through the process of
mathematical
proving.
Proving is one of the
categories of problem
according to Polya (1973).
In doing mathematical
proving, a comprehension
of mathematics concepts
is a must. Due to the fact
that mathematical proving
is not a simple process to
do, many students are
constraint to master it and
their ability to use it is
still below average. A
research by Knuth and Ko
(2009) finds that many
students had difficulties in
doing
mathematical
proving. Another research
done by Stavros Georgios
(2014) shows that there
were many mistakes made
by students while doing
the mathematical proving,
even the students let their
answer sheets blank and
only wrote down short
notes saying that they
were confused on how to
start
doing
the
mathematical proving.
One
of
the
problems causing the
inability of students to do
mathematical proving is
that students are not
accustomed
with
questions
requiring
mathematical proving. By
this fact, mathematical
proving
should
be
implemented
in
any
mathematics
teaching
learning
process
in
secondary school or even
primary
school
(Stylianides,
2016).
According to National
Council of Teachers of
Mathematics
(NCTM)
(2000)
(recited
by
Stylianides, 2016) and
English
National
Curriculum
for
mathematics (Department
of Education, 2013),
mathematical
proving
should be given to
students since in a
primary school to make
them accustomed with
guessing, connecting prior
knowledge,
elaborating
arguments
and
determining truth of a
statement. In this case,
proving becomes the
concern
of
the
international curriculum,
but the fact says that there
are still common errors in
mathematical
proving
done by students (Stavrou,
2014). While in Indonesia,
mathematical proving in
primary and secondary
schools has less attention.
Thus, the researcher is
going to capture the
process of students on
how to do mathematical
proving.
Number is one of
the materials that is going
to be taught to Junior
High students. However,
according to Cahyadi
(2016) says that the ability
of students in number of
Junior High is still low.
Therefore, emphasizing
on the sensitivity of
number is related to not
only how to calculate but
also how to use the
features of numeral.
To sum up, the
research questions for this
research is how the profile
of Junior High students
critical
thinking
in
mathematical
proving
based on the students’ sex
differences is. Moreover,
the objective of the study
is to describe the profile
of Junior High students
critical
thinking
in
mathematical
proving
based
on
the
sex
differences
of
the
students.
Critical Thinking
Critical thinking
is a daily activity done by
human. Thinking is a
mental
activity
experienced by someone
while he is facing a
problem needed to be
solved (Siswono, 2008).
Krulik
and
Rudnick
(1995) categorize critical
thinking into memorizing,
basic thinking, critical
thinking and creative
thinking. To reach the
highest level which is
creative
thinking,
somebody must think
critically.
Chukwuyenum
(2013)
explains
that
critical
thinking
is
thinking involving logical
reasoning, enhancing an
ability to assess something
before accepting and
refusing
ideas,
constructing, evaluating
and decision making. In
accordance with it, critical
thinking is also defined as
thinking which involves
trying, connecting and
evaluating all aspects of a
situation or problem,
including collecting and
organizing
problems
(Siswono, 2008).
Desmita (2009:
153)
defines
critical
thinking as a skill of
thinking
logically,
reflectively,
and
productively to make a
precise consideration and
decision. Critical thinking
means
reflecting
the
problems in depth and
selecting the information.
Fisher (2009:13) defines
critical thinking as an
activity fulfilling various
types
of
intellectual
standard, such as clarity,
relevancy,
sufficiency,
coherency and many
more. Besides, Ennis
(1996) claims that critical
thinking is a process
aimed to make a decision
about what should and
should not be done.
From all those
experts, the definition of
critical thinking in this
research is a mental
activity which involves
organization in processing
information in order to
look for clarity, analyze,
evaluate and make a
decision.
There are several
experts’ indicators used to
find out the critical
thinking
profile
of
students. According to
Ennis (1995), critical
thinking indicators are
part of critical thinking
indicators of Jacob and
Sam (2008). Moreover,
Fisher (2009) indicators
are just fulfilled by the
indicators of Jacob and
Sam (2008). In the critical
thinking indicators listed
by Jacob and Sam (2008),
there
are
several
strategies, such as open
minded
in
solving
problems
which
can
reveal the ability of
students to find other
alternatives in solving
problems. Therefore, this
research uses critical
thinking
indicators
adapted from Jacob and
Sam
(2008)
which
consists of clarification,
assessment, inference and
strategy.
The
following
table is critical thinking
indicators from Jacob and
Sam (2008).
category. It can
assess how the
students
understand about
the problem by
giving
the
meaning words
by words and
also to identify
the scope of the
problem. Besides
that, it can assess
how the students
identify
the
relation of parts
of the problem.
Menawhile, here
is the table of critical
thinking indicators in
proving adapted from
Jacob and Sam (2008).
4.
5.
2.
To describe all the
categories above, there are
several
aspects
used
according
to
Ramos
(2012). They are as
follows:
1. Giving meaning
to words and
statements.
This
aspect is used to
describe
the
clarification
3.
Justifying claim
Justifyin
g claim is the
other ways to
describe
clarification
category. By this
aspect,
the
students should
give
their
justification of
the truth.
Illustrating
by
using examples
This
aspect is used to
assess
the
inference
category. It will
show about how
deep
understanding
the
students
about
the
problem given.
6.
Logical
statement
and
proving
framework
This
aspect is the
other ways to
describe
inference
category. In this
aspect,
the
students should
give their logical
statement
and
proving
framework
to
evaluate
their
steps in proving.
Describing
the
process
of
determining steps
which will be
used and its
reasons in each
step.
This
aspect is used to
assess inference
category.
The
students should
give
their
explanation
about
their
proving with its
reasons in each
step.
Describing
the
process
of
determining in
finding
other
alternatives
in
proving with its
reasons.
This
aspect is used to
assess strategy
category.
The
students should
give
their
explanation
about
finding
other alternatives
in proving with
its reasons.
Mathematical Proving
Proving is an
activity which is strongly
related to mathematics
because
in
general,
mathematics products are
in a form of theorem that
the truth should be
verified. According to
Frasier (2014), proving is
a fundamental element of
mathematics and cannot
be separated in the sense
of mathematics without
proving
is
not
mathematics
at
all.
Learning
mathematics
will be like science
without experiments or
learning language without
writing.
Mathematical
proving is a set of logical
arguments explaining the
truth of a statement. These
arguments can be from
either the premise of the
statement itself, theorem,
definition, or postulate
(Hernadi,
2008).
According to Frasier
(2014), proving is a way
to
validate,
justify,
communicate,
and
systematize
the
knowledge
of
mathematics.
Proving
strategies
which
are
possible to be used by
students is formal proving
consisting
of
direct
proving,
contradictory
proving
and
contraposition
proving.
Moreover, the indirect
proving that may be used
is proving using picture
representation.
The relation of Critical
Thinking
and
Sex
Differences
Every
student
has different ability in
solving a problem, either
male or female students.
Arends (2008) explains
that there is a difference in
cognitive level between
male and female. Male
tends to be rational and
better in thinking logically
and critically. On the other
hand, female has stronger
memory than male and
more interested in verbal
skill.
Male tends to use
spatial strategy to solve
mathematics
problems,
while female uses verbal
strategy. Male is more
excellent than female.
However, some female
reach an achievement in
mathematics.
The
difference lies on how
those two opposite sex
differences
use
the
strategy to solve problems
and the difference of way
of thinking (Amir, 2013).
It goes the same
as the research of Subaidi
(2015) which states that
male critical thinking is
better than female. Male
passes the steps of critical
thinking precisely and
correctly, but female
cannot give any reason
while giving conclusion in
the result. Besides, the
research of Zu (2007)
concludes that male and
female have different
preference for the strategy
of problem solving.
Based on the
experts above, male and
female
may
create
differences in the critical
thinking skill. Thus, there
is a relation between
critical thinking and the
types of sex differences.
research
was
the
researcher
herself
supported
by
complementary
instrument such as test of
mathematics proving and
the interview guidelines.
The
data
collection
technique was interview
which was based on test.
The data analysis of
interview was through
data
reduction,
data
presentation,
and
conclusion drawing.
METHOD
This
research
belonged to qualitative
using interview method
which was based on test
aiming to describe the
critical thinking profile of
Junior High students in
mathematics
proving
based
on
the
sex
differences. This research
was conducted in class
RESULT
AND
VII-A SCI SMP Negeri 1
DISCUSSION
Krian. The subject of this
The test was
research consisted of one
given to 10 students of
male student and one
class VII-a SCI SMP
female student with the
Negeri 1 Krian. After that,
same high competence
one male and one female
chosen among five male
students were selected as
students and five female
the subject of the research.
students
who
are
The selection of subject
identified by the result of
was based on the result of
fulfilling
the
critical
students
who
were
thinking indicators the
identified to fulfill more
most while doing many
critical
thinking
ways in mathematics
indicators. Next, interview
proving.
These
high
was conducted to the
competence students were
selected subject regarding
the students who got 90 or
the process of doing
more in the accumulation
mathematics
proving.
of the final test and the
Followings are the test
assignments. Furthermore,
used in this research.
another
criterion
in
selecting the subject If
ofn is odd, then 3n+7 is even, n
this research is that
Make a conjecture, is it true or false?
Proof your conjecture!
students who had better
Are
there
any
alternative solutions that can support
communication skill in
your
answer? Explain please!
both spoken and written.
Overall, the steps
It was important to do
and strategies used by the
since this research used
male student and female
interview to the subject so
student was relatively
that the ability of the
similar. In the clarification
students
in
category, there were the
communication can be
way of male student and
considered
and
female student identifying
accommodated.
information which was
The
main
known and things that
instrument
for
this
were proven by giving
meaning in words by
words and the statement
in the test. It is shown by
the interview between the
researcher (R) and the
male student (MS):
was 3n+7 is an even
number. On the other
hand, the female student
had a difficulty on
understanding 3n+7 is an
even number so that she
needed to read the
question carefully even
R Please mention what though
information
eventually
you get the
from the test?
female student understood
MS n is an odd number, about the test given.
asked to prove whether
Moreover, she did not
number
detail while mentioning
R Do you understand every
mentioned
orallyword the
known
in the test? Please, explain.
information in the test and
MS Insyaallah I understand.
what should be proven.
R What is the meaning of n, odd number,
3n+7, even number andShe only said that the
question was about an odd
MS Odd number is a number that cannot be
number and even number,
divided by 2 and remains 1.
and then students were
3n+7 is 3 multiplied by n then plus 7.
asked to prove whether
Even number is a number that can be
3n+7 was an even
divided by 2.
number. It was matched
n is the element of integer.
with
the
research
Male student and
conducted by Subaidi
female student were able
(2015) that male students
to understand the context
were more detail in
of the test precisely and
explaining the known
accurately. The male
information given in a
student
was
aware
test.
regarding the known
Both
subjects
information and what
used
their
prior
would
be
proven
knowledge
to
identify
the
accurately as well as
relation
between
parts
seeing the relation of each
needed in proving. The
part. Even he claimed that
prior knowledge was
the statement was true
about the knowledge of
based on his prior
even and odd numbers
knowledge. The male
including
definition,
student mentioned orally
meaning,
general
form
the known information
and the features of even
written in the test which
and
odd
numbers.
was n is an odd number, n
Although they were a bit
confused
in
explaining
whether or not
there were other
forms of even
number besides
2n, n element
Figure1. Part of mathematical proving by FS
of Z, but they
element of Z, and what
could
explain
their
should be proven which
knowledge about even
number correctly. The
male student explained
that there were other
forms of even number,
such as 2a, 2b, 2c until 2z
with a, b, c until z with
element of Z (by only
changing the variable) and
he presumed that 4n, n
element of Z was also
another form of even
number since 4 is the
multiple of 2 so that 4n, n
element of Z was also the
general form of even
number. However, after he
was asked to explain the
form 4n, n element of Z,
he
MS
R
MS
0.5 (thinking for a while) lho?
be the general form of even num
Why? Previously you said yes
No, it can’t. 0.5 is not intege
fulfills if the even number is 2. S
form of even number is only 2n
element of integer. Or it can be 2
2z, just change the variable.
It went the same
to the female students
while she presumed that
4n, n element of Z was the
general form of even
number under the same
reason as the male student
which was 4 is the
multiple of 2. However,
after she was asked to
realized
Part of mathematical proving by FS (continuing)
that 4n, n
element of Z was not a
explain the form 4n, n
general form of even
element of Z, she realized
number since there was no
that there was no value of
n fulfilling the even
n that could fulfill the
number of 2. It is shown
even number of 2 if the
by the interview between
general form of even
the researcher (R) and the
number is 4n, n element
male student (MS) which
of Z.
is as follows
Both
subjects
could also explain their
R
Okay, what is the general form of even
prior knowledge regarding
number?
integer. They were able to
MS
, with as the element of integer.
explain
that
integer
R
Why , with as theconsisted
element of of
integer
can
positive,
be the general form of negative,
even number?and
zero.
MS Even number is the multiple
of 2. they almost
Although
R
Is there any other form?forgot
Or that’s
that it?
zero was also
MS I guess yes
the part of integer.
R
If yes, please explain me the other form of
Both
subjects
even number besides 2n with n as the element
could justify the claim
of integer
by the number
presumption
MS 4n is possible since 4 isshown
the multiple
of
of
the
statement
truth. The
2.
male student said that the
R
Are you sure?
MS Yes, I’m quite sure. statement
was
right
R
If you said that 4n, with
n as he
thewas
element
because
tryingof
and
integer could be the explaining
general form
of even
the truth
of the
number, then what is the value of n?
statement by using the
R
What is your first step in order to solve this
question?
FS
The first step is I try to understand the question,
then I try to insert the odd number n to 3n+7
R
Please explain each step you have to do this
question
FS
It is similar to what I have said before. Firstly, I
draw this n (by pointing the answer) because n
is odd number so there must be 1 remaining or
there is one which does not have a pair. After
that I saw the question written 3n so that I draw
3 times for the n. Next, I draw 7 circles and
match them until there is only one left. So, I
match each 1 remaining circle so that all circles
R
Please
give meThus,
an example
theeven
othernumber
form
are matched.
3n+7 isor an
which
has
similar
meaning
to
the
test
with n as the element of integer is proven miss.
MS If z is an odd number, 5z+99 is an even
number, z is the element of integer.
features of even and odd
was way easier and easy
numbers. In line with that,
to understand. It showed
the female student was
that picture representation
also saying that the
could be one of the
statement
was
true
alternatives or a way that
because she had tried to
could be used to learn
change n with odd number
proving.
and the result was correct.
The male student
Both subjects showed that
explained the proving
they tended to use
precisely and accurately
inductive knowledge to
by using pictures that the
presume the truth of
general form of odd
statement given that was
number was 2n+1. Odd
through any odd number,
number n could be drawn
n was substituted into
with circles and in the end
3n+7.
would be added 1 circle
The process done
because odd number
by both subjects to
always remained 1 if
determine the steps in
divided by 2. Next,
doing the proving was
because the test was
giving meaning word by
written 3n, the pictures
word and relating all parts
representing it were made
in the text. The first step
3 times and later was
used by both subjects was
matched until it was only
trying to insert an odd
1 remaining. So, 3n was
number to 3n+7. After
an odd number. Based on
that, both subjects did it
the feature of odd number
by using pictures and
that odd number plus odd
applied the features of odd
number was equal to even
and even numbers. It
number, 3n+7 was an
proved that both subjects
even number under the
could describe the process
condition that 3n was the
in determining the steps
element of odd number.
used in proving.
Furthermore, male student
Both
subjects
also explained that after
decided to use picture
drawing 3n, it could also
representation because it
be possible to draw 7
circles until 1 more
remaining. After that, the
remaining circle of 3n was
matched
with
the
remaining circle of 7 so
that all circles had their
pair and nothing left.
Thus, 3n+7 was even
number. It was shown by
the male subject that he
was able to explain the
steps used as well as the
reasons in each step.
It went the same
to the female student that
could be proven through
the figure below and
interview between the
researcher (R) and the
female student (FS).
given. It was if z was odd
number, then 5z+99 was
even number, with z as the
element of integer and the
subject was really sure
about the truth given in
his example. He could
explain the truth of other
examples by using the
features of odd and even
numbers. It was supported
by the interview between
the researcher and the
male student.
The
female
student was also able to
illustrate the example
which
had
similar
meaning to the test given.
It was that n was an odd
number, 11n+3 was an
even number and she was
On the orher
really sure about the
hand, it was supported by
answer. She could also
interview between the
explain the truth of the
reseacher (R) and the
other example by using
female
student
(FS)
the features of odd and
below:
even number.
Both
subjects
Proving
using
could also identify the aim
picture
representation
of the statement existed in
could be the stepping
the proving process. The
stone to lead in the formal
male student explained
proving. After the students
that 3n is element of odd
understood about the
number needed to be
concept of odd and even
written in order to
nubers
using
picture
highlight that 3n was an
representation, the next
odd number. Besides, the
step is to focus on the
female student added an
number of pairs formed in
example
and
some
the odd numbers. In this
features needed to do the
case, if odd number of 23
proving of a statement.
Figure.
Direft proof
male student
was taken, the number of
The(formal) by
basic
pairs formed in the picture
difference between male
and female students was
representation
was
male student was able to
find the alternatives in
.
proving. The process of
The male student
finding other alternatives
could also illustrate the
in proving was through
use of example which
the general form of odd
meant the same as the test
numbers
done
in
deductive way. It was
supported by the proving
of male student in figure
below:
It was supported
by the male student’s
was in line with the
research
done
by
Gallagher, Ann. M, et. al.
(2000) which was saying
that male students were
better
than
female
proving
below.
students in using and
finding the strategy for
problem solving. Besides,
the result of this research
was also matched with
Arends (2008) which
explained that there was a
cognitive level difference
between male and female.
Male tended to be rational
and better in thinking
logically and critically.
in
the
figure
On the other
hand,
the
interview
between the researcher
and the male student
which was as follows:
Based on the
figure 3, the female
subject was actually trying
to use alternatives to
prove the truth of a
statement, even she had a
deductive knowledge to
do proving. Before she did
the proving by using
picture representation, she
explained that she tried to
do it by using direct
proving (formal proving)
in order to verify. Yet, she
was confused and not sure
with the answer so that the
answer was marked. She
was unable to change the
form
in
a
general form of even
number so that she did not
get the result and decided
to leave the process of
proving. Thus, she was
unable to find the
alternatives in proving. It
CONCLUSION
Based on the result
of this research, we have
conclusions as follows:
1. In the clarification
category,
both
subjects
identify
known information
that
is
proven
precisely
and
accurately by giving
meaning in word by
word. Both subjects
are also able to
identify the relation
between
parts
needed
in
the
proving
through
prior knowledge they
have.
However,
explanations of male
student are more
detail
than
the
female one.
Figure 3. Part of proving by female student
In the
assessment category,
that he fulfills the
both subjects are
criteria of strategy
able to evaluate the
category. On the
processes done in
other
hand,
the
proving.
female student is
unable to complete
3. In
inference
or finish the proving
category,
both
in more than one
subjects are able to
way so that she does
describe and explain
not fulfill the criteria
the steps used as
of strategy category.
well as the reasons
given in each step.
Suggestion
4. Male student is able
Based on the
to do the proving in
result of this research,
more than one ways,
teachers are expected to
even he can describe
familiarize the Junior
and explain every
High students to do the
step as well as the
proving test. By this,
reasons correctly so
teachers are also expected
R
Explain me the way you take in solving the
problems given!
MS First, I use the general form of odd number and
algorithm.
The general form of odd number is
atau
2.
,
so
that
the
R
MS
result is
Supposedly
is b, so
2b becomes the general form of even number. It
is proven that 3n+7 is an even number.
Why is
?
It is from the general form of odd number
. n was the odd number so that n can be
R
MS
R
MS
changed into
.
Then what is the next step? How do you say
?
I have learnt it in algorithm that it is multiplied
one by one with the numbers inside the bracket.
So, 3 is multiplied by 2a and then plus 3
multiplied by 1. So the answer is
The next is why do you say
6a is still the same. It directly proceeds and
=10, so
R
What about this
MS
I divide
with two.
Why do you divide it with two?
To have the general form, so get the result of
R
MS
R
MS
?
Then I assume
is , so 2b is
the general form of even number with b as the
element of integer.
How do you justify that b is the element of
integer
It is obvious because n is also integer.
to develop the critical
thinking of the students.
Moreover,
in
doing
mathematics
proving,
teachers are expected to
use multi representation,
and one of which is
picture representation. It
is shown by the result of
this research that both
subjects tend to use
picture representation to
do proving. It is possible
for teachers that picture
representation can be the
alternative of mathematics
proving although not all
proving can be done by
using
picture
representation
(only
particular cases).
Mathematics
among
Senior
Secondary
School Students
in Lagos State.
IOSR Journal of
Research
&
Method
in
Education,
eISSN:
23207388,p-ISSN:
2320-737X
Volume 3, Issue
5, 18-25.
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Perkembangan
Peserta Didik.
Bandung:
PT
Remaja
Rosdakarya.
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R. H. (1996).
Critical
Thinking.
University
of
Illnois.
Fisher,
A.
(2009).
Berpikir Kritis.
Jakarta:
Erlangga.
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(2013). Impact
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on
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Frasier, B. J. (2014). Cara
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dkk.
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Commmonwealt
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Knuth, E., & Ko, Y.-Y.
(2009).
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mathematics
majors’ writing
performance
producing proof
and
counterexamples
about
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Behavior, 68-77.
Jacob, & Sam. (2008).
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Critical
Thinking
in
Problem Solving
through Online
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Year University
Mathematics.
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the International
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Ramos, J. P., & dkk.
(2011).
An
Assessment
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Comprehension
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Mathematics.
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10.1007/s10649011-9349-7, 318.
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Proving in the
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International
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8(2), 187-203.
ISSN
14431475, Shannon
Research Press
MATHEMATICAL PROVING BASED ON SEX DIFFERENCES
Tiania Mewanda
Mathematics Education, Faculty of Mathematics and Natural Sciences, State University of Surabaya,
e-mail : [email protected]
Abdul Haris Rosyidi
Mathematics Education, Faculty of Mathematics and Natural Sciences, State University of Surabaya,
e-mail : [email protected]
Abstract
Developing critical thinking in solving a problem is one of the aims in learning mathematics.
Critical thinking is defined as a mental activity involving organized ways in processing information
which consists of looking for clarity, analyzing, evaluating and determining decisions. Critical thinking is
needed in mathematical proving and it can be seen through the process of solving mathematical proving
done by the students. Sex differences have probability to influence the students’ critical thinking.
This study is included into qualitative research by using interview method which was based on
the test aiming to describe the critical thinking profile of junior high school students in mathematical
proving of number based on sex differences. The subject of this study consisted of one male and one
female student with the same level of high competence chosen among five male and five female students
that closely identify critical thinking in proving based on the most variation in students’ answer. The data
were analyzed based on the indicators of critical thinking which consist of clarification, assessment,
inference and strategy.
Based on the result of the study, the steps and strategy done by male and female subject in doing
proving were relatively showing similarity. Both subjects used picture representation in proving. In
clarification category, both subjects identified information which was known and precisely proven by
interpreting word by word. Both subject also identified the relation of parts which was needed in proving
by using their prior knowledge. However, the male subject had more details than the female one. In the
assessment category, both subjects were able to evaluate the steps done in proving. In inference category,
both subjects were able to describe and explain the steps used as well as the reasons in every step. The
male student was able to do the proving with more than one method, even he was able to describe and
explain every step as well as the correct reasons so that he fulfilled the strategy category. On the other
hand, the female subject was unable to do the proving in more than one method so that she did not fulfill
the strategy category.
Keywords: Critical Thinking, Mathematical Proving, Sex
INTRODUCTION
1
Rapid
development
of
information
and
communication
technology requires skills
of
selecting
and
processing
information
properly. One of the most
important skills is critical
thinking. Furthermore, in
facing the complexity of
our life, critical thinking is
paramount in solving
many problems. Since
critical thinking is urgent
in daily basis, the teaching
of it in school, therefore,
is essential.
Curriculum
always
puts
critical
thinking as a requirement
that should be mastered
by students. One of the
subject
in
school
requiring critical thinking
skill is mathematic. It is
shown by the fact that
mathematic
standard
competence ruled out by
the decree of culture and
education
minister
number 21 year 2016
which requires students
to act logically, critically,
analytically,
precisely,
responsibly, responsively
and toughly in solving
problems.
One of the ways
to develop the critical
thinking skill of students
is through the process of
mathematical
proving.
Proving is one of the
categories of problem
according to Polya (1973).
In doing mathematical
proving, a comprehension
of mathematics concepts
is a must. Due to the fact
that mathematical proving
is not a simple process to
do, many students are
constraint to master it and
their ability to use it is
still below average. A
research by Knuth and Ko
(2009) finds that many
students had difficulties in
doing
mathematical
proving. Another research
done by Stavros Georgios
(2014) shows that there
were many mistakes made
by students while doing
the mathematical proving,
even the students let their
answer sheets blank and
only wrote down short
notes saying that they
were confused on how to
start
doing
the
mathematical proving.
One
of
the
problems causing the
inability of students to do
mathematical proving is
that students are not
accustomed
with
questions
requiring
mathematical proving. By
this fact, mathematical
proving
should
be
implemented
in
any
mathematics
teaching
learning
process
in
secondary school or even
primary
school
(Stylianides,
2016).
According to National
Council of Teachers of
Mathematics
(NCTM)
(2000)
(recited
by
Stylianides, 2016) and
English
National
Curriculum
for
mathematics (Department
of Education, 2013),
mathematical
proving
should be given to
students since in a
primary school to make
them accustomed with
guessing, connecting prior
knowledge,
elaborating
arguments
and
determining truth of a
statement. In this case,
proving becomes the
concern
of
the
international curriculum,
but the fact says that there
are still common errors in
mathematical
proving
done by students (Stavrou,
2014). While in Indonesia,
mathematical proving in
primary and secondary
schools has less attention.
Thus, the researcher is
going to capture the
process of students on
how to do mathematical
proving.
Number is one of
the materials that is going
to be taught to Junior
High students. However,
according to Cahyadi
(2016) says that the ability
of students in number of
Junior High is still low.
Therefore, emphasizing
on the sensitivity of
number is related to not
only how to calculate but
also how to use the
features of numeral.
To sum up, the
research questions for this
research is how the profile
of Junior High students
critical
thinking
in
mathematical
proving
based on the students’ sex
differences is. Moreover,
the objective of the study
is to describe the profile
of Junior High students
critical
thinking
in
mathematical
proving
based
on
the
sex
differences
of
the
students.
Critical Thinking
Critical thinking
is a daily activity done by
human. Thinking is a
mental
activity
experienced by someone
while he is facing a
problem needed to be
solved (Siswono, 2008).
Krulik
and
Rudnick
(1995) categorize critical
thinking into memorizing,
basic thinking, critical
thinking and creative
thinking. To reach the
highest level which is
creative
thinking,
somebody must think
critically.
Chukwuyenum
(2013)
explains
that
critical
thinking
is
thinking involving logical
reasoning, enhancing an
ability to assess something
before accepting and
refusing
ideas,
constructing, evaluating
and decision making. In
accordance with it, critical
thinking is also defined as
thinking which involves
trying, connecting and
evaluating all aspects of a
situation or problem,
including collecting and
organizing
problems
(Siswono, 2008).
Desmita (2009:
153)
defines
critical
thinking as a skill of
thinking
logically,
reflectively,
and
productively to make a
precise consideration and
decision. Critical thinking
means
reflecting
the
problems in depth and
selecting the information.
Fisher (2009:13) defines
critical thinking as an
activity fulfilling various
types
of
intellectual
standard, such as clarity,
relevancy,
sufficiency,
coherency and many
more. Besides, Ennis
(1996) claims that critical
thinking is a process
aimed to make a decision
about what should and
should not be done.
From all those
experts, the definition of
critical thinking in this
research is a mental
activity which involves
organization in processing
information in order to
look for clarity, analyze,
evaluate and make a
decision.
There are several
experts’ indicators used to
find out the critical
thinking
profile
of
students. According to
Ennis (1995), critical
thinking indicators are
part of critical thinking
indicators of Jacob and
Sam (2008). Moreover,
Fisher (2009) indicators
are just fulfilled by the
indicators of Jacob and
Sam (2008). In the critical
thinking indicators listed
by Jacob and Sam (2008),
there
are
several
strategies, such as open
minded
in
solving
problems
which
can
reveal the ability of
students to find other
alternatives in solving
problems. Therefore, this
research uses critical
thinking
indicators
adapted from Jacob and
Sam
(2008)
which
consists of clarification,
assessment, inference and
strategy.
The
following
table is critical thinking
indicators from Jacob and
Sam (2008).
category. It can
assess how the
students
understand about
the problem by
giving
the
meaning words
by words and
also to identify
the scope of the
problem. Besides
that, it can assess
how the students
identify
the
relation of parts
of the problem.
Menawhile, here
is the table of critical
thinking indicators in
proving adapted from
Jacob and Sam (2008).
4.
5.
2.
To describe all the
categories above, there are
several
aspects
used
according
to
Ramos
(2012). They are as
follows:
1. Giving meaning
to words and
statements.
This
aspect is used to
describe
the
clarification
3.
Justifying claim
Justifyin
g claim is the
other ways to
describe
clarification
category. By this
aspect,
the
students should
give
their
justification of
the truth.
Illustrating
by
using examples
This
aspect is used to
assess
the
inference
category. It will
show about how
deep
understanding
the
students
about
the
problem given.
6.
Logical
statement
and
proving
framework
This
aspect is the
other ways to
describe
inference
category. In this
aspect,
the
students should
give their logical
statement
and
proving
framework
to
evaluate
their
steps in proving.
Describing
the
process
of
determining steps
which will be
used and its
reasons in each
step.
This
aspect is used to
assess inference
category.
The
students should
give
their
explanation
about
their
proving with its
reasons in each
step.
Describing
the
process
of
determining in
finding
other
alternatives
in
proving with its
reasons.
This
aspect is used to
assess strategy
category.
The
students should
give
their
explanation
about
finding
other alternatives
in proving with
its reasons.
Mathematical Proving
Proving is an
activity which is strongly
related to mathematics
because
in
general,
mathematics products are
in a form of theorem that
the truth should be
verified. According to
Frasier (2014), proving is
a fundamental element of
mathematics and cannot
be separated in the sense
of mathematics without
proving
is
not
mathematics
at
all.
Learning
mathematics
will be like science
without experiments or
learning language without
writing.
Mathematical
proving is a set of logical
arguments explaining the
truth of a statement. These
arguments can be from
either the premise of the
statement itself, theorem,
definition, or postulate
(Hernadi,
2008).
According to Frasier
(2014), proving is a way
to
validate,
justify,
communicate,
and
systematize
the
knowledge
of
mathematics.
Proving
strategies
which
are
possible to be used by
students is formal proving
consisting
of
direct
proving,
contradictory
proving
and
contraposition
proving.
Moreover, the indirect
proving that may be used
is proving using picture
representation.
The relation of Critical
Thinking
and
Sex
Differences
Every
student
has different ability in
solving a problem, either
male or female students.
Arends (2008) explains
that there is a difference in
cognitive level between
male and female. Male
tends to be rational and
better in thinking logically
and critically. On the other
hand, female has stronger
memory than male and
more interested in verbal
skill.
Male tends to use
spatial strategy to solve
mathematics
problems,
while female uses verbal
strategy. Male is more
excellent than female.
However, some female
reach an achievement in
mathematics.
The
difference lies on how
those two opposite sex
differences
use
the
strategy to solve problems
and the difference of way
of thinking (Amir, 2013).
It goes the same
as the research of Subaidi
(2015) which states that
male critical thinking is
better than female. Male
passes the steps of critical
thinking precisely and
correctly, but female
cannot give any reason
while giving conclusion in
the result. Besides, the
research of Zu (2007)
concludes that male and
female have different
preference for the strategy
of problem solving.
Based on the
experts above, male and
female
may
create
differences in the critical
thinking skill. Thus, there
is a relation between
critical thinking and the
types of sex differences.
research
was
the
researcher
herself
supported
by
complementary
instrument such as test of
mathematics proving and
the interview guidelines.
The
data
collection
technique was interview
which was based on test.
The data analysis of
interview was through
data
reduction,
data
presentation,
and
conclusion drawing.
METHOD
This
research
belonged to qualitative
using interview method
which was based on test
aiming to describe the
critical thinking profile of
Junior High students in
mathematics
proving
based
on
the
sex
differences. This research
was conducted in class
RESULT
AND
VII-A SCI SMP Negeri 1
DISCUSSION
Krian. The subject of this
The test was
research consisted of one
given to 10 students of
male student and one
class VII-a SCI SMP
female student with the
Negeri 1 Krian. After that,
same high competence
one male and one female
chosen among five male
students were selected as
students and five female
the subject of the research.
students
who
are
The selection of subject
identified by the result of
was based on the result of
fulfilling
the
critical
students
who
were
thinking indicators the
identified to fulfill more
most while doing many
critical
thinking
ways in mathematics
indicators. Next, interview
proving.
These
high
was conducted to the
competence students were
selected subject regarding
the students who got 90 or
the process of doing
more in the accumulation
mathematics
proving.
of the final test and the
Followings are the test
assignments. Furthermore,
used in this research.
another
criterion
in
selecting the subject If
ofn is odd, then 3n+7 is even, n
this research is that
Make a conjecture, is it true or false?
Proof your conjecture!
students who had better
Are
there
any
alternative solutions that can support
communication skill in
your
answer? Explain please!
both spoken and written.
Overall, the steps
It was important to do
and strategies used by the
since this research used
male student and female
interview to the subject so
student was relatively
that the ability of the
similar. In the clarification
students
in
category, there were the
communication can be
way of male student and
considered
and
female student identifying
accommodated.
information which was
The
main
known and things that
instrument
for
this
were proven by giving
meaning in words by
words and the statement
in the test. It is shown by
the interview between the
researcher (R) and the
male student (MS):
was 3n+7 is an even
number. On the other
hand, the female student
had a difficulty on
understanding 3n+7 is an
even number so that she
needed to read the
question carefully even
R Please mention what though
information
eventually
you get the
from the test?
female student understood
MS n is an odd number, about the test given.
asked to prove whether
Moreover, she did not
number
detail while mentioning
R Do you understand every
mentioned
orallyword the
known
in the test? Please, explain.
information in the test and
MS Insyaallah I understand.
what should be proven.
R What is the meaning of n, odd number,
3n+7, even number andShe only said that the
question was about an odd
MS Odd number is a number that cannot be
number and even number,
divided by 2 and remains 1.
and then students were
3n+7 is 3 multiplied by n then plus 7.
asked to prove whether
Even number is a number that can be
3n+7 was an even
divided by 2.
number. It was matched
n is the element of integer.
with
the
research
Male student and
conducted by Subaidi
female student were able
(2015) that male students
to understand the context
were more detail in
of the test precisely and
explaining the known
accurately. The male
information given in a
student
was
aware
test.
regarding the known
Both
subjects
information and what
used
their
prior
would
be
proven
knowledge
to
identify
the
accurately as well as
relation
between
parts
seeing the relation of each
needed in proving. The
part. Even he claimed that
prior knowledge was
the statement was true
about the knowledge of
based on his prior
even and odd numbers
knowledge. The male
including
definition,
student mentioned orally
meaning,
general
form
the known information
and the features of even
written in the test which
and
odd
numbers.
was n is an odd number, n
Although they were a bit
confused
in
explaining
whether or not
there were other
forms of even
number besides
2n, n element
Figure1. Part of mathematical proving by FS
of Z, but they
element of Z, and what
could
explain
their
should be proven which
knowledge about even
number correctly. The
male student explained
that there were other
forms of even number,
such as 2a, 2b, 2c until 2z
with a, b, c until z with
element of Z (by only
changing the variable) and
he presumed that 4n, n
element of Z was also
another form of even
number since 4 is the
multiple of 2 so that 4n, n
element of Z was also the
general form of even
number. However, after he
was asked to explain the
form 4n, n element of Z,
he
MS
R
MS
0.5 (thinking for a while) lho?
be the general form of even num
Why? Previously you said yes
No, it can’t. 0.5 is not intege
fulfills if the even number is 2. S
form of even number is only 2n
element of integer. Or it can be 2
2z, just change the variable.
It went the same
to the female students
while she presumed that
4n, n element of Z was the
general form of even
number under the same
reason as the male student
which was 4 is the
multiple of 2. However,
after she was asked to
realized
Part of mathematical proving by FS (continuing)
that 4n, n
element of Z was not a
explain the form 4n, n
general form of even
element of Z, she realized
number since there was no
that there was no value of
n fulfilling the even
n that could fulfill the
number of 2. It is shown
even number of 2 if the
by the interview between
general form of even
the researcher (R) and the
number is 4n, n element
male student (MS) which
of Z.
is as follows
Both
subjects
could also explain their
R
Okay, what is the general form of even
prior knowledge regarding
number?
integer. They were able to
MS
, with as the element of integer.
explain
that
integer
R
Why , with as theconsisted
element of of
integer
can
positive,
be the general form of negative,
even number?and
zero.
MS Even number is the multiple
of 2. they almost
Although
R
Is there any other form?forgot
Or that’s
that it?
zero was also
MS I guess yes
the part of integer.
R
If yes, please explain me the other form of
Both
subjects
even number besides 2n with n as the element
could justify the claim
of integer
by the number
presumption
MS 4n is possible since 4 isshown
the multiple
of
of
the
statement
truth. The
2.
male student said that the
R
Are you sure?
MS Yes, I’m quite sure. statement
was
right
R
If you said that 4n, with
n as he
thewas
element
because
tryingof
and
integer could be the explaining
general form
of even
the truth
of the
number, then what is the value of n?
statement by using the
R
What is your first step in order to solve this
question?
FS
The first step is I try to understand the question,
then I try to insert the odd number n to 3n+7
R
Please explain each step you have to do this
question
FS
It is similar to what I have said before. Firstly, I
draw this n (by pointing the answer) because n
is odd number so there must be 1 remaining or
there is one which does not have a pair. After
that I saw the question written 3n so that I draw
3 times for the n. Next, I draw 7 circles and
match them until there is only one left. So, I
match each 1 remaining circle so that all circles
R
Please
give meThus,
an example
theeven
othernumber
form
are matched.
3n+7 isor an
which
has
similar
meaning
to
the
test
with n as the element of integer is proven miss.
MS If z is an odd number, 5z+99 is an even
number, z is the element of integer.
features of even and odd
was way easier and easy
numbers. In line with that,
to understand. It showed
the female student was
that picture representation
also saying that the
could be one of the
statement
was
true
alternatives or a way that
because she had tried to
could be used to learn
change n with odd number
proving.
and the result was correct.
The male student
Both subjects showed that
explained the proving
they tended to use
precisely and accurately
inductive knowledge to
by using pictures that the
presume the truth of
general form of odd
statement given that was
number was 2n+1. Odd
through any odd number,
number n could be drawn
n was substituted into
with circles and in the end
3n+7.
would be added 1 circle
The process done
because odd number
by both subjects to
always remained 1 if
determine the steps in
divided by 2. Next,
doing the proving was
because the test was
giving meaning word by
written 3n, the pictures
word and relating all parts
representing it were made
in the text. The first step
3 times and later was
used by both subjects was
matched until it was only
trying to insert an odd
1 remaining. So, 3n was
number to 3n+7. After
an odd number. Based on
that, both subjects did it
the feature of odd number
by using pictures and
that odd number plus odd
applied the features of odd
number was equal to even
and even numbers. It
number, 3n+7 was an
proved that both subjects
even number under the
could describe the process
condition that 3n was the
in determining the steps
element of odd number.
used in proving.
Furthermore, male student
Both
subjects
also explained that after
decided to use picture
drawing 3n, it could also
representation because it
be possible to draw 7
circles until 1 more
remaining. After that, the
remaining circle of 3n was
matched
with
the
remaining circle of 7 so
that all circles had their
pair and nothing left.
Thus, 3n+7 was even
number. It was shown by
the male subject that he
was able to explain the
steps used as well as the
reasons in each step.
It went the same
to the female student that
could be proven through
the figure below and
interview between the
researcher (R) and the
female student (FS).
given. It was if z was odd
number, then 5z+99 was
even number, with z as the
element of integer and the
subject was really sure
about the truth given in
his example. He could
explain the truth of other
examples by using the
features of odd and even
numbers. It was supported
by the interview between
the researcher and the
male student.
The
female
student was also able to
illustrate the example
which
had
similar
meaning to the test given.
It was that n was an odd
number, 11n+3 was an
even number and she was
On the orher
really sure about the
hand, it was supported by
answer. She could also
interview between the
explain the truth of the
reseacher (R) and the
other example by using
female
student
(FS)
the features of odd and
below:
even number.
Both
subjects
Proving
using
could also identify the aim
picture
representation
of the statement existed in
could be the stepping
the proving process. The
stone to lead in the formal
male student explained
proving. After the students
that 3n is element of odd
understood about the
number needed to be
concept of odd and even
written in order to
nubers
using
picture
highlight that 3n was an
representation, the next
odd number. Besides, the
step is to focus on the
female student added an
number of pairs formed in
example
and
some
the odd numbers. In this
features needed to do the
case, if odd number of 23
proving of a statement.
Figure.
Direft proof
male student
was taken, the number of
The(formal) by
basic
pairs formed in the picture
difference between male
and female students was
representation
was
male student was able to
find the alternatives in
.
proving. The process of
The male student
finding other alternatives
could also illustrate the
in proving was through
use of example which
the general form of odd
meant the same as the test
numbers
done
in
deductive way. It was
supported by the proving
of male student in figure
below:
It was supported
by the male student’s
was in line with the
research
done
by
Gallagher, Ann. M, et. al.
(2000) which was saying
that male students were
better
than
female
proving
below.
students in using and
finding the strategy for
problem solving. Besides,
the result of this research
was also matched with
Arends (2008) which
explained that there was a
cognitive level difference
between male and female.
Male tended to be rational
and better in thinking
logically and critically.
in
the
figure
On the other
hand,
the
interview
between the researcher
and the male student
which was as follows:
Based on the
figure 3, the female
subject was actually trying
to use alternatives to
prove the truth of a
statement, even she had a
deductive knowledge to
do proving. Before she did
the proving by using
picture representation, she
explained that she tried to
do it by using direct
proving (formal proving)
in order to verify. Yet, she
was confused and not sure
with the answer so that the
answer was marked. She
was unable to change the
form
in
a
general form of even
number so that she did not
get the result and decided
to leave the process of
proving. Thus, she was
unable to find the
alternatives in proving. It
CONCLUSION
Based on the result
of this research, we have
conclusions as follows:
1. In the clarification
category,
both
subjects
identify
known information
that
is
proven
precisely
and
accurately by giving
meaning in word by
word. Both subjects
are also able to
identify the relation
between
parts
needed
in
the
proving
through
prior knowledge they
have.
However,
explanations of male
student are more
detail
than
the
female one.
Figure 3. Part of proving by female student
In the
assessment category,
that he fulfills the
both subjects are
criteria of strategy
able to evaluate the
category. On the
processes done in
other
hand,
the
proving.
female student is
unable to complete
3. In
inference
or finish the proving
category,
both
in more than one
subjects are able to
way so that she does
describe and explain
not fulfill the criteria
the steps used as
of strategy category.
well as the reasons
given in each step.
Suggestion
4. Male student is able
Based on the
to do the proving in
result of this research,
more than one ways,
teachers are expected to
even he can describe
familiarize the Junior
and explain every
High students to do the
step as well as the
proving test. By this,
reasons correctly so
teachers are also expected
R
Explain me the way you take in solving the
problems given!
MS First, I use the general form of odd number and
algorithm.
The general form of odd number is
atau
2.
,
so
that
the
R
MS
result is
Supposedly
is b, so
2b becomes the general form of even number. It
is proven that 3n+7 is an even number.
Why is
?
It is from the general form of odd number
. n was the odd number so that n can be
R
MS
R
MS
changed into
.
Then what is the next step? How do you say
?
I have learnt it in algorithm that it is multiplied
one by one with the numbers inside the bracket.
So, 3 is multiplied by 2a and then plus 3
multiplied by 1. So the answer is
The next is why do you say
6a is still the same. It directly proceeds and
=10, so
R
What about this
MS
I divide
with two.
Why do you divide it with two?
To have the general form, so get the result of
R
MS
R
MS
?
Then I assume
is , so 2b is
the general form of even number with b as the
element of integer.
How do you justify that b is the element of
integer
It is obvious because n is also integer.
to develop the critical
thinking of the students.
Moreover,
in
doing
mathematics
proving,
teachers are expected to
use multi representation,
and one of which is
picture representation. It
is shown by the result of
this research that both
subjects tend to use
picture representation to
do proving. It is possible
for teachers that picture
representation can be the
alternative of mathematics
proving although not all
proving can be done by
using
picture
representation
(only
particular cases).
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