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Profile of Metacognition of Mathematics and Mathematics Education Students in Understanding the Concept of Integral Calculus

This study describes the metacognition profile of mathematics and mathematics education students in understanding the concept of integral calculus. The metacognition profile is a natural and intact description of a person’s cognition that involves his own thinking in terms of using his knowledge, planning and monitoring his thinking process, and evaluating his thinking results when understanding a concept. The purpose of this study was to produce the metacognition profile of mathematics and mathematics education students in understanding the concept of integral calculus. This research method is explorative method with the qualitative approach. The subjects of this study are mathematics and mathematics education students who have studied integral calculus. The results of this study are as follows: (1) the summarizing category, the mathematics and mathematics education students can use metacognition knowledge and metacognition skills in understanding the concept of indefinite integrals. While the definite integrals, only mathematics education students use metacognition skills; and (2) the explaining category, mathematics students can use knowledge and metacognition skills in understanding the concept of indefinite integrals, while the definite integrals only use metacognition skills. In addition, mathematics education students can use knowledge and metacognition skills in understanding the concept of both indefinite and definite integrals.


Introduction
Integral calculus is one of the compulsory courses programmed by students of Mathematics Department and Mathematics Education Department. When viewed from the graduates of the Department of Mathematics Education is to produce a bachelor candidate for high school mathematics teacher, so it is expected to be able to comprehend integral calculus material, in addition to apply it to everyday life, and other relevant courses, also can teach it to students in high school. While the graduates of Department of Mathematics are producing pure mathematics scholars, so it is expected to be able to comprehend the material of integral calculus, and can apply it to everyday life, as well as other relevant subjects [4].
Based on the difference of interest of students in the selection of Mathematics Department and Mathematics Education Department, it is possible to differentiate students' metacognition in understand mathematical concepts. This can be seen from the difference of mathematics learning outcomes of students in high school and in the first semester of Mathematics Department or Mathematics Education Department. When viewed from the average value of the national exam of high school mathematics, for students better mathematics education than math students (63.58> 59.0), while the average achievement index first semester, mathematics students better than mathematics education students (3.17> 3.07).
The metacognition profile is a natural and intact description of a person's cognition that involves his own thinking in terms of his or her knowledge, and the ability to plan and monitor his thinking process, and evaluate the process and outcome of one's thinking when understanding a concept. According to Koriat, in Goh [8], Metacognition refers to what people know about cognition in general and about their own cognitive processes, in particular, as well as how they use this knowledge to adjust their informational processes and behavior. While cognitive understanding is a process that occurs internally within the central nervous system at the time of human thinking (Gagne [7]). Then Neisser [14] says that cognition is the acquisition, arrangement, and use of knowledge. The concept of integral calculus that is examined in this paper is the concept of integral indeterminate and integral of course. To understand the concept of integral calculus using the Bloom theory developed by Anderson et al.
[1], and Mayer [12], and can be seen in Table 1. This paper discusses only two categories of conceptual understanding, summarizing and explaining. The metacognition profile in understanding the concept of integral calculus especially the summarizing and explaining category can be seen from 2 components of metacognition, ie cognitive knowledge consists of declarative knowledge, procedural knowledge, and conditional knowledge (Flavell [6]) and metacognition skills consisting of planning, monitoring and evaluating (Dawson [5], Livingston [10], and CCD Malaysia [3]). The students' metacognition profile review was distinguished between mathematics students and mathematics education students. Thus, the purpose of this study was to produce the metacognition profile of mathematics and mathematics education students in understanding the concept of integral calculus.

Method
This research method is the explorative method with a qualitative approach. This explorative research is intended to explore the metacognition of mathematics students and mathematics education in understanding the concept of integral calculus so as to obtain a large profile of students' metacognition. The subjects of study (RS) are students of the mathematics department and Mathematics Education department who are studying integral calculus, consisting of 1 mathematics student and 1 student of mathematics education. Selection of the subject of this study based on the highest score of the Mathematics Ability Test (score >70), and consider the value of student achievement. The main data collection of this research was obtained by using interview technique. In addition, there is supporting data which is the result of written work of the research subject in understand integral calculus task. The data analysis process follows analysis model of Miles and Huberman [13], consisting of: (1) data reduction, data presentation, and (3) conclusion. Interpretation of data carried out simultaneously with data presentation activities. In this activity, researchers do an interpretation of research data (answers RS at the time of interview). Making conclusions and recommendations based on the results of data processing to answer research questions, namely finding the profile of students' metacognition in understanding the concept of integral calculus.

Results and Discussion
The results of this study reveal the metacognition profile of mathematics and mathematics education students in understanding the concept of integral calculus, especially the categories of summarizing and explaining. To reveal the metacognition profile of the students, first, look at the understanding of mathematics and mathematics education students about the concept of integral calculus based on the students' answers in completing the task of understanding the concept of integral. Second, through interviews can reveal the metacognition of students in understanding the concept of integral calculus.
3.1. Understanding of mathematics and mathematics education students on the concept of integral calculus on the Summarizing Category The answers of mathematics and mathematics education students in understanding the concept of integral calculus in the summarizing category can be seen in Table 2.
Based on the results of the answers in Table 2, mathematics and mathematics education students can summarize the main points given in the form of indefinite integrals so that the answer is correct. Meanwhile, the main points given in the form of definite integrals cannot be summarized so the answer is wrong.

Results of mathematics student interviews about understanding the concept of integral calculus
on the summarizing category Results of mathematics student interviews about understanding the concept of integral calculus on the category of summarizes consists of the concept of indefinite integrals and definite integrals:

The Indefinite Integral Concepts
The results of mathematics student interviews in understanding the concept of indefinite integrals in the Summarizing category as follows: (1) Students of mathematics can summarize the main points given in the form of improper integrals correctly ie is wrong (2) The mathematics student can not give the reason that the area of the function |x + 2| on the interval [-2,2] is above the x-axis, (3) Math students are less able to use the procedure to obtain the area of function |x + 2| at the interval [-2,2], (4) Math students have not been able to review their understanding that the area of function |x + 2| on the interval [-2,2] is above the x-axis, (5) The mathematics student has not been able to explain the steps of calculating the area of the function |x + 2| at the interval [-2 , and (6) Students of mathematics have not mastered the procedure to obtain results from 8 Thus, the Results of mathematics student interviews in understanding the concept of integral certainly in the Summarizing category has not shown metacognition knowledge and metacognition skills.

Results of mathematics education student interviews about understanding the concept of
integral calculus on the summarizing category Results of mathematics education student interviews about understanding the concept of integral calculus on the category of summarizes consists of the concept of indefinite integrals and definite integrals:

The Indefinite Integral Concepts
The results of mathematics education students interviews in understanding the concept of indefinite integrals in the summarizing category as follows: (1) Students of mathematics education can summarize the main points given in the form of improper integrals correctly ie The mathematics education students can give the reason that Thus, the result of the mathematics education student's interview in understanding the indefinite integrals concepts indicate that it has used metacognition knowledge and metacognition skills in understanding the concept of indefinite integrals in the summarizing category.

The Definite Integral Concepts
The results of interviews of mathematics students in understanding the concept of definite integrals on summarizing category as follows: (1) Students of mathematics education cannot summarize the main points given in form the definite integrals ie The mathematics education students cannot give the reason that the area of the function |x + 2| on So, the result of student interviews mathematics education in understanding the concept of definite integrals on summarizing category does not show knowledge of metacognition. After being reminded again of the concepts of definite integrals, students can use their metacognitive skills to understanding the concepts of definite integrals that.

Understanding of mathematics students and mathematics education on the concept of integral
calculus on the explaining category The answers of mathematics and mathematics education students in understanding the concept of integral calculus in the explaining category can be seen in Table 3.  Let f(x) be a polynomial function with the form f(x) = x 2 + 2x + 3. To integrate the function of this polynomial can be done one by one ie: Let f(x) be a polynomial function with the form f(x) = a n x n + a n-1 x n-1 + … + a 1 x + a 0 . To integrate the function of this polynomial can be done one by one ie: dx a x a x a n n ) ...
Based on your knowledge of the definitions of definite integrals, and the sequence properties of the real numbers, try to explain whether this property applies also to certain intervals containing four consecutive points a, b, c, and d when f is integrated at that interval.
If f on the interval [a, d] contains four dots a, b, c, and d respectively, illustrated If f on the interval [a, d] contains three dots a, b, and c respectively, illustrated If f on the interval [a, d] contains four dots a, b, c, and d respectively, illustrated Based on the results of the answers in Table 3, the mathematics student can explain a model of the consequences of a given property of either indefinite integrals or definite integrals, but the explanation is very brief and not detailed. While of mathematics education students can explain a model of the consequences of a given property of either indefinite integrals or definite integrals, with a very detailed explanation.

The Indefinite Integral Concepts
The results of the mathematics student interviews in understanding the concept of indefinite integrals in the explaining category as follows: (1) Students of mathematics can explain a model of an indefinite integral with a polynomial function based on the linearly properties, ie (2) Students of mathematics can give the reason that the polynomial function applies the linearly properties so that Students of mathematics can use the procedure to explain dx , and (6) Students of mathematics can conclude that the function of polynomial is a linear function then apply the linearly properties, so that Thus, the results of mathematical student interviews in understanding the concepts of indefinite integrals indicate that it has used metacognition knowledge and metacognition skills in understand the concept of indefinite integrals in the explaining category.

The Definite Integral Concepts
The results of interviews of mathematics students in understanding the concept of definite Integrals in the Explaining category as follows: (1) The mathematics student can explain a model of the definite integrals of function f at intervals [a, d] based on the sequence properties of the real numbers with f Thus, the results of the mathematics student's interview in understanding the definite integral concepts of the explaining category indicate that they have used metacognition knowledge, but have not used metacognition skills especially in planning and monitoring.

The Definite Integral Concepts
Thus, the result of the mathematics education student's interview in understanding the definite integral concepts shows that it has used metacognition knowledge and metacognition skills in understand the concept of definite integrals in the explaining category.
The results above show that there are differences in metacognition profile between mathematics students and mathematics education students in understanding the concept of integral calculus, especially the integral concept of both the summarizing category and the explaining category. In the summarizing category, mathematics education students can review their understanding that the area of function |x + 2| at interval [-2.2] is above the x-axis, and can explain the steps of calculating the area of the function |x +2| at the interval [-2,2]. While the mathematics student has not been able to review his understanding of the area of function |x + 2| at interval [-2,2] and can not yet explain the steps to calculate the area of the function area. This is because math students are not yet accustomed to drawing graphs of functions of absolute values. Although the mathematics students are theoretically directed to think mathematics, because they are not familiar with the concept they are doing, the result is not good. This is in the opinion of Masami & Katagiri [11], there is no way a person could teach in such a way as to cultivate mathematical thinking without first understanding the kinds of mathematical thinking that exist. On the other hand, Wittrock [15], stated that some studies have found no effects of summarization and the conditions below are not well understood. Then, in the explaining category, Mathematics education students can explain and write the integral form that if f is integral to [a, d] then f can also be integrated into each of the intervals [a, b], [b, c], and [c, d]. While the mathematics student can write the integral form, but can not yet explain that if f is integral to [a, d] then f can also be integrated into each of the intervals [a, b], [b, c], and [c,d]. This is because math students are not accustomed to voicing their thoughts. He is only skilled at completing a concept, but it is very difficult to reveal the results of his work to others.
Therefore, mathematics students need to learn about how to express their thoughts to others. This is as stated by Hattie et al, [9], teaching metacognitive strategies to the student can lead to marked improvement in their achievement. Then, Butler & Winn [2], suggests that Students can be learning to think their own thinking processes and apply specific learning strategies to think themselves through difficult task.

Conclusion
Based on the results of research and discussion above, it can be concluded that: The summarizing category, mathematics students, and mathematics education students can use knowledge and metacognition skills in understanding the concept of indefinite integrals. While the definite integral concepts, mathematics students have not been able to use his/her knowledge and skills of metacognition. And mathematics education students only use skills metacognition.
The explaining category, mathematics students can use knowledge and skills of metacognition in understanding the concept of indefinite integrals, while the definite integral concepts only use metacognition skills. Meanwhile, mathematics education students can use knowledge and skills of metacognition in understanding the concept of both indefinite and definite integrals.