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 : tianiamewanda@mhs.unesa.ac.id
Abdul Haris Rosyidi
Mathematics Education, Faculty of Mathematics and Natural Sciences, State University of Surabaya, e-mail : abdulharis@unesa.ac.id
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
Rapid 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 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.
(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).
Menawhile, here is the table of critical thinking indicators in proving adapted from Jacob and Sam (2008).
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
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.
2. 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.
3. 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.
4. 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. 5. 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.
6. Describing the process of determining in finding other alternatives in proving with its reasons.
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). 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 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.
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 VII-A SCI SMP Negeri 1 Krian. The subject of this research consisted of one male student and one identified by the result of fulfilling the critical thinking indicators the most while doing many ways in mathematics proving. These high competence students were the students who got 90 or more in the accumulation of the final test and the interview to the subject so that the ability of the
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 Negeri 1 Krian. After that, one male and one female students were selected as the subject of the research. The selection of subject was based on the result of students who were mathematics proving. Followings are the test used in this research.
Overall, the steps and strategies used by the male student and female student was relatively similar. In the clarification category, there were the Make a conjecture, is it true or false?
Proof your conjecture!
Part of mathematical proving by FS (continuing)
Figure1. Part of mathematical proving by FS 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
R Do you understand every word mentioned in the test? Please, explain.
MS Insyaallah I understand.
R What is the meaning of n, odd number, 3n+7, even number and
MS Odd number is a number that cannot be divided by 2 and remains 1.
3n+7 is 3 multiplied by n then plus 7. Even number is a number that can be divided by 2.
n is the element of integer. Male student and
female student were able to understand the context of the test precisely and seeing the relation of each part. Even he claimed that the statement was true based on his prior knowledge. The male student mentioned orally the known information understanding 3n+7 is an even number so that she needed to read the question carefully even though eventually the female student understood about the test given. Moreover, she did not detail while mentioning orally the known information in the test and what should be proven. She only said that the question was about an odd number and even number, and then students were (2015) that male students were more detail in explaining the known information given in a test.
Both subjects used their prior knowledge to identify the relation between parts Although they were a bit confused in explaining whether or not there were other forms of even number besides 2n, n element 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 number since there was no n fulfilling the even number of 2. It is shown by the interview between the researcher (R) and the male student (MS) which is as follows
R Okay, what is the general form of even number?
MS , with as the element of integer.
R Why , with as the element of integer can be the general form of even number?
MS Even number is the multiple of 2. R Is there any other form? Or that’s it? MS I guess yes
R If yes, please explain me the other form of even number besides 2n with n as the element of integer
MS 4n is possible since 4 is the multiple number of 2.
R Are you sure? MS Yes, I’m quite sure.
R If you said that 4n, with n as the element of integer could be the general form of even number, then what is the value of n?
MS 0.5 (thinking for a while) lho? So, 4n cannot be the general form of even number.
R Why? Previously you said yes
MS No, it can’t. 0.5 is not integer. No integer fulfills if the even number is 2. So, the general form of even number is only 2n with n as the element of integer. Or it can be 2a, 2b, 2c until 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
explain the form 4n, n element of Z, she realized that there was no value of n that could fulfill the even number of 2 if the general form of even number is 4n, n element of Z.
Both subjects could also explain their prior knowledge regarding integer. They were able to explain that integer consisted of positive, negative, and zero. Although they almost forgot that zero was also the part of integer.
features of even and odd numbers. In line with that, the female student was also saying that the statement was true because she had tried to change n with odd number and the result was correct. Both subjects showed that they tended to use inductive knowledge to presume the truth of statement given that was through any odd number, n was substituted into word and relating all parts in the text. The first step used by both subjects was trying to insert an odd number to 3n+7. After that, both subjects did it by using pictures and applied the features of odd and even numbers. It proved that both subjects could describe the process in determining the steps used in proving.
Both subjects decided to use picture representation because it
was way easier and easy to understand. It showed that picture representation could be one of the alternatives or a way that could be used to learn proving.
The male student explained the proving precisely and accurately by using pictures that the general form of odd number was 2n+1. Odd number n could be drawn with circles and in the end would be added 1 circle representing it were made 3 times and later was matched until it was only 1 remaining. So, 3n was an odd number. Based on the feature of odd number that odd number plus odd number was equal to even number, 3n+7 was an even number under the condition that 3n was the element of odd number. Furthermore, male student also explained that after drawing 3n, it could also 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 hand, it was supported by interview between the reseacher (R) and the female student (FS) below:
Proving using picture representation could be the stepping stone to lead in the formal proving. After the students understood about the concept of odd and even nubers using picture representation, the next step is to focus on the number of pairs formed in the odd numbers. In this case, if odd number of 23 was taken, the number of pairs formed in the picture representation was
.
The male student could also illustrate the use of example which meant the same as the test
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 meaning to the test given. It was that n was an odd number, 11n+3 was an even number and she was really sure about the
Both subjects could also identify the aim of the statement existed in the proving process. The male student explained that 3n is element of odd number needed to be written in order to highlight that 3n was an odd number. Besides, the female student added an example and some features needed to do the proving of a statement.
The basic difference between male and female students was male student was able to find the alternatives in proving. The process of finding other alternatives in proving was through the general form of odd numbers done in 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 are matched. Thus, 3n+7 is an even number with n as the element of integer is proven miss. R Please give me an example or the other form
which has similar meaning to the test
MS If z is an odd number, 5z+99 is an even number, z is the element of integer.
Figure 3. Part of proving by female student deductive way. It was
supported by the proving of male student in figure below:
It was supported by the male student’s
proving in the figure below.
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
was in line with the research done by Gallagher, Ann. M, et. al. (2000) which was saying that male students were better than female
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.
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.
2. In the
assessment category, both subjects are able to evaluate the processes done in proving.
3. In inference category, both subjects are able to describe and explain the steps used as well as the reasons given in each step.
4. Male student is able to do the proving in more than one ways, even he can describe and explain every step as well as the reasons correctly so
that he fulfills the criteria of strategy category. On the other hand, the female student is unable to complete or finish the proving in more than one way so that she does not fulfill the criteria of strategy category.
Suggestion
Based on the result of this research, teachers are expected to familiarize the Junior High students to do the proving test. By this, 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 ,
so that
the result is Supposedly is b, so 2b becomes the general form of even number. It is proven that 3n+7 is an even number.
R Why is ?
MS It is from the general form of odd number . n was the odd number so that n can be changed into .
R Then what is the next step? How do you say ?
MS 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
R The next is why do you say
MS 6a is still the same. It directly proceeds and =10, so
R What about this ? MS I divide with two. R Why do you divide it with two?
MS To have the general form, so get the result of Then I assume is , so 2b is the general form of even number with b as the element of integer.
R How do you justify that b is the element of integer
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