The Characteristic of the Process of Students ’ Metacognition in Solving Calculus Problems

This article is the result of research aims to describe the patterns and characteristics of the process of metacognition student of mathematics in solving calculus problems. Description was done by looking at changes in awareness, evaluation, and regulation as components of metacognition. The changes in components of metacognition seen by the emergence of indicators for each indicator were described in descriptors. To see the changes, the researcher used the instrument consisting of calculus problems, metacognition questionnaires, observation sheet, and interviews. Researcher gave calculus to 23 students who followed the course of differential calculus. The research data were in the form of the works of the students, transcript think-aloud, metacognition questionnaire results, observations using the observation sheet, and a transcript of the interview. Based on research data obtained, the research subjects were categorized in the high-ability students, medium, and low. Data were analyzed using constant comparison method of Glaser and Strauss. Based on data, analysis can be concluded that the pattern and characteristics of the change process awareness, evaluation and regulation mathematic students in solving calculus problems can be distinguished in the process of metacognition complete with the order, complete metacognition was not with the order, and metacognition incomplete.


Introduction
Mathematics as a part of the instruction at school has direct object and indirect object.Gagne (Soedjadi, 2000;Hudojo, 2008) states that mathematical direct object relates to fact, concept, operation, and principle.The fact is a convention or agreement used to make mathematical discussion go smoothly, such as symbol, notation.The symbol "4" has been understood as figure "four".If it is 4, people understand it is "four".On the other hand, if someone says the word "four", it is symbolized by "4".The concept is an abstract idea which can be used to classify a group of a certain objects.A concept in the field of mathematics is called mathematical concept."Triangulation" is a name of an abstract concept.With a concept, a group of objects can be classified as an example or not.The concept has a closely related to definition.Definition is an utterance which limits a concept.With it, someone can make illustration, picture, or symbol of defined concept.So that, it is clearer what is meant by a certain concept.Mathematical operation is procedures and as a process to find out a certain result.If the concept is an association, the possible operations are union, section, difference, or complement.Besides on association, it is known as addition operation, difference, multiplication, and division.The principle is the relation between various complex mathematical principle objects and consists of some facts.The right value principle is the principle has two or more concepts and states the relation between those concepts.For instance, the result of multiple of the figures p and q is zero if and if only p=0 or q=0.
The mathematical indirect object relates to the ability of logical thinking, solve problem, analytical thinking, positive thinking towards mathematics, careful, diligent, discipline and the other cases implicitly will be obtained if someone learns mathematics.Based on the opinion of Gagne above, it can conclude that one of the objects learned in mathematics is a concept.Mathematical concept orientation can be done through instruction.Djamarah (2008) states that someone who has possesses concept can do abstracting so that he/she can translate and call awareness and form mentally representation.During understanding the concept, someone needs the ability and strategy or certain way.Duffin and Simpson (2000) states that the ability possessed by someone during understanding the concept is expected to be able to re-express something communicated to the source of study, and finally, when the concept has been mastered, the given problems can answered.Polya (1988) mentions four steps in solving problem, i.e. understand the problem, plan the treatment, execute the plan, and review the treatment done.The steps expressed by Polya in solving problem are activities which can be during the instruction and indicator in determine instructional result completeness.According to Bloom taxonomy completeness of learning result is classified into the field of cognitive, affective, and psychomotor.Especially for the field of cognitive, Anderson & Krathwohl (2001) improve into cognition process dimension and knowledge dimension.Cognition process consists of remembering, understanding, applying, analyzing, evaluating, and creating.But, knowledge consists of factual, conceptual, procedural and metacognition.Metacognition comes from "meta" and "cognition".Meta is a Greek means after or behind.Cognition is a process of obtaining knowledge (Zahmeister & Neyberg, 1982).During its process, metacognition can be related to the activities of problem solving, knowledge, cognition process and the strategy used during the instruction.The term of metacognition used for the first time by Flavell in 1976.According to him, metacognition consists of knowledge, experience and regulation which function as the important element and contribute the success of the problem solving.
The facts expressed by some high students during solving the problem that they often meet the failures.The failures were caused by the lack of the understanding of metacognition aspects, especially, which are related to the steps done to solve problem (Schoenfeld, 1992;Goos, 1995).As the improvement of the study, especially relates to the process of metacognition, it is found out that metacognition can assist one's thinking process more effectively and opened (Wilson & Clarke, 2004).While Schoenfeld (1992) regards that the difficulty in solving problem is closely related to student's inability to observe and control the process of metacognition.Some studies related to metacognition and problem solving has been done.Among of them are Desoete (2001), Lioe et al. (2003), Wilson and Clarke (2004), Cromley (2005), Efklides (2006), Lesh (2007), Panauorra (2009), Kuzle (2011), Molenar (2011), Karan and Irizary (2011), Magiera andZawojewski (2011), In'am et al. (2012), Praba (2013), Zaenal andTajudin (2013).The studies generally discuss the process metacognition in the subject of the study.However, it has not brought up the student's metacognition process characteristic in solving problem yet.Relating to the case, the researcher does the research to analyse and describes the mathematical student's metacognition process characteristic in solving calculus problems.The student's metacognition process characteristic described based on the process of awareness, evaluation, and regulation which are the components of metacognition.

The Definition of Metacognition
Friedrichs and Hoyt (1976) mention metacognition in term of metamemory.While Veenman (2012) describes metacognition as two main parts: the knowledge of metacognition and regulation or the observation of metacognition.The knowledge of metacognition is an interaction offering between someone's knowledge and the ability to do someone's assignments, characters of assignment, and the strategy which can be used to do the assignments.The regulation or metacognition observation is the activities relate to planning, monitoring, someone's evaluating, and the process of cognition to control the process.Metacognition as a process has four important aspects.According to Baker & Brown (1984) those aspects are self-controlling, planning, evaluating, and monitoring.Wellman (1985) states that metacognition is as form of cognition or two or more thinking process which include the control of the cognition activity.Therefore, metacognition can be said as thinking of someone's thinking of him/her-self or someone's knowledge of his/her cognition.
Besides having four aspects, according to Schoenfeld (1992), metacognition is as the process of someone's thinking process of what he/she has thought and as the interaction between the three important aspects, namely the knowledge of thinking process, self-controlling, and intuition.The interaction is very important for the knowledge of cognition process can assist and control the cases around us and select the strategies to improve our next cognition ability.According to Schoenfeld, the process of metacognition includes the ability to ask a question, and answer the question about a case, topic and problem, time allocation used to learn a certain topic, strategy, method and tactic used, level being learn by the student, fault done by the student, and revise the next plan.Livingstone (1997) defines metacognition as thinking about thinking.In the other word, metacognition is someone thinking ability about what is thought.So that, metacognition object is thinking process happened to someone.Biryukov (2003) says that metacognition is someone thinking hypothesis of what has been thought before and includes knowledge, skill and experience.Knowledge is awareness of what is known, skill in the form of awareness of something done, and experience is awareness of cognition ability possessed.Davidson and Sternberg (1998) state that metacognition has an important functions and contributes the success of problem solving that enable someone to identify and work strategically.Matlin (1998) states that metacognition is knowledge relates to awareness and cognition process.Wellman (1985) states that metacognition is as a form of cognition, or two or more thinking process including controlling of cognition activity.Therefore, metacognition can be said as someone thinking of self-thinking or someone cognition of self-cognition.Tan (2003) states that metacognition is thoughtfulness refers to think of self-thinking, self-checking, and information process and how to process information effectively.Lioe (2003) states that metacognition is someone awareness about cognition process and selfness to reach a certain goal.Metacognition appears in problem solving whose components are attitude, skill, concept, process, and metacognition.
Peirce ( 2003) defines metacognition generally and specifically.Generally, metacognition is thinking of thinking.While specifically, Peirce quotes the definition of metacognition by Taylor, which states that metacognition is the appreciation of what has known which relates to the ability to make a right conclusion about how to apply someone's strategy knowledge of certain situation, and to do it efficiently and accurately.Taccasu ( 2008) describes metacognition as a part of planning, monitoring and evaluating the instructional process, thinking what we have known or unknown and control how the instruction include both awareness and aware control someone's learn so that it is effective.Mokos and Kafoussi (2013) states that metacognition is someone ability to observe and control him/her-self toward the case known.During the mathematical instruction, the important thing is the process of the study of student's metacognition in doing the problem more focused to the problem solving field relates to mathematics.
Based on some definition stated above, we can identify the main meanings of metacognition: (1) is soul ability in cognition group, ( 2) is ability to aware, know, cognition process happened in oneself, is ability to direct cognition process in oneself, ( 3) is an ability of how the instruction done including the process of planning, observing, and evaluation, ( 4) is an activity of high level thinking activity for its activity can control thinking process happened in oneself in the present time, and (5) relates to student's thinking process to find out suitable strategy in solving problem, (6) metacognition skill is very important in solving mathematical problem, so the skill needs to improve.To improve metacognition skill, student needs awareness in his/her thinking steps.Student awareness in thinking is needed to finish a problem.

The Components of Metacognition
Magiera and Zawojewski (2011) find that metacognition activity happened during giving assignment in the class.Metacognition happening in the students has three components, e.g.awareness, evaluation, and regulation.During the process of metacognition, it can be seen the appearing activities in every components of metacognition called as types of metacognition activities.The types of awareness consist of what the students know, what the students need to solve the problem, what the students must do, where the students solve the problem.The types of evaluation consist of result evaluation, students' difficulty problem of evaluation, progress, ability or understanding.The types of regulation consist of planning strategy, selecting strategy of problem solving, formulating the goal.
Awareness, according to Wilson andClarke (2002, 2004), relates to someone's awareness in the process of learning or in the process of solving problem, the content of specific knowledge owned, and someone's knowledge in learning or strategy in solving problem.It also includes someone's knowledge about what is needed to do, and what has done, and what can be done in a certain learning or situation in solving the problem.Evaluation refers to evaluation made by someone about thinking process, ability and limitation, such as working in a certain situation or as a self-complication.For example, someone can make evaluation about thinking effectiveness done or strategy chosen.Regulation in metacognition happens when someone uses his/her skill of metacognition to direct knowledge and thought and refers to individual knowledge in the form of strategy, such as how and why using certain strategies, as well as skill, such as planning, self-correction, decide the goal to optimal the usage of their own cognition source.
Metacognition components stated by Wilson andClarke (2002, 2004) and Magiera and Zawojewski (2011) have indicators as variable and measurement.However, how the process of changes among the components of metacognition has not analysed deeper yet.Sriraman (2003) has considered students about the relationship between their knowledge and what is needed in problem situation given, as Stillman and Gabraith (1998).
Evaluation has been described and studied concerning the students explicit reflection during the process of solving problem, the function of evaluation in determining the strategy in solving problem.Make a decision in systematic evaluation, alternative plan, and strategy in solving problem (Lester, 1980;Garofalo & Kroll, 1989).Regulation has been clarified in the form of student flexibility in choosing a solution plan, choosing strategy, and plan implementation improved by Lester (1989), Zan (2000).

Method
The study is a descriptive qualitative research so that the method used in the study methods and analysis of qualitative data.To obtain research data, researcher undertook some steps as follows: (1) gave to the subject matter of research as many as 23 students who have taken mathematics courses differential calculus.Problems were given as a matter of application of the derivative function of the variable to determine the maximum area of a field.Problem has been validated by the mathematician; (2) the subject of research resolved the matter in the think-aloud for 60 minutes.Researcher corrected students' works to classify them into categories of students capable of high, medium, and low.Think-aloud activity recorded in the form of video footage to be observed metacognition process; (3) once the subjects solved the problem, researcher gave questionnaires metacognition form of 14 statements about metacognition and every statement given alternatives replied "yes", "no" or "unsure".Statement in the questionnaire consisted of 6 statements related to the process of awareness, 5 statements related to the evaluation process, and the 3 statements related to the regulation process.If the study subjects answered "yes", it meant that the indicator component of metacognition would appear, if did not mean metacognition, indicator would not appear, and when the subject answered "unsure" followed up with interviews; (4) researcher studied the video footage and recorded the results in the observation sheet which included the indicators and descriptors awareness, evaluation, and regulation.In the awareness process observation sheet contained 5 indicators with 30 dekriptors, evaluation contain 5 indicators with 23 descriptors, and regulation contained 4 indicators with 19 descriptors; (5) conducted interviews, interviews aimed to explore the emergence of indicators and descriptors awareness, evaluation, and regulation; (6) transcription research data.Transcription was made after researcher get the data needed in the research.Transcripts of data in the study was the transcript of the think-aloud and interview transcripts; (7) performed data reduction.Data reduction was done by making abstractions core summarizing data, processes and statements that needed to be maintained to remain in it.Compiled data in units which was further categorized by making the code; (8) data analysis, process analysis metacognition each research subject was through each of the indicators awareness, evaluation, and regulation.Analysis was conducted to describe the activity of thinking the subject of research in accordance with the indicators and descriptors in every process of awareness, evaluation, and regulation; (9) data validation, data validation was done by triangulation techniques and inspection peers through discussion and research seminar; (10) summed up the results of the study.Conclusions obtained through a comparison between the works of the students, metacognition questionnaire, think-aloud transcripts, observation using observation sheets, and a transcript of the interview each subject of study.

Results
The data of the study are in the form of the result of works, the transcript of think-aloud, result of the observation using observation sheet, answers of metacognition questionnaire and the interview transcript were studied and analysed qualitatively.The theory design built by the researcher was analysing the characteristic of the student metacognition process in finishing calculus problems through indicators of awareness, evaluation, and regulation.Based on the theory design, the result of analyse of the study was grouped into category of characteristic of student metacognition process of high, middle and low ability.
The result of the study is from 23 students of Mathematics Education Department who had Differential Calculus.They were given calculus problems and during finishing it, the components of awareness, evaluation, and regulation appeared.Besides the metacognition appearance components completely, the subject of the study were given similar patterns so the next analyse was to classify the subjects into high, middle, and low ability.Based on it, 23 subjects of the study were classified into 6 for high ability, 9 for middle ability, and 8 for low ability.
Based on the result of data analyse using the method of constant comparison by Glaser and Strauss and data validation using triangulation in which each of the subject of the study in each group had relatively same characteristic.So that data explanation was done to 6 subject of the study.They were S-1 and S-2 of high ability group, S-3 and S-4 of middle ability group, and S-5 and S-6 of low ability group.The next step was studying the characteristic of metacognition process of those three groups of subject of the study through their component indicators of awareness, evaluation, and regulation.

Conclusion
Based on the data analyse, it can be concluded that the high ability student metacognition process characteristic in solving calculus problem is complete and in order, the middle ability student is complete but disorder, and the low ability student is incomplete.
The characteristic of complete and in order metacognition process is in S-1 can be explained as follows: The component of awareness with A1 brings up 6 characteristics, A2 brings up 7 characteristics, A3 brings up 4 characteristics, A4 brings up 8 characteristics A5 brings up 1 characteristic.The component of evaluation with the indicator E1 brings up 2 characteristics, E2 brings up 5characteristics, E3 brings up 6 characteristics, E4 brings up 4 characteristics, and E5 brings up 4 characteristics.The component of regulation with the indicator R1 brings up 3 characteristics, R2 brings up 3 characteristics, R3 brings up 3 characteristics, and R4 brings up 5 characteristics.
The characteristic of complete and in order metacognition process is in S-2 can be explained as follows: The component of awareness with A1 brings up 4 characteristics, A2 brings up 7 characteristics, A3 brings up 4 characteristics, A4 brings up 8 characteristics A5 brings up 1 characteristic.The component of evaluation with the indicator E1 brings up 3 characteristics, E2 brings up 3 characteristics, E3 brings up 6 characteristics, E4 brings up 4 characteristics, and E5 brings up 4 characteristics.The component of regulation with the indicator R1 brings up 3 characteristics, R2 brings up 3 characteristics, R3 brings up 3 characteristics, and R4 brings up 5characteristics.
The characteristic of complete but disorder metacognition process is in S-3 can be explained as follows: The component of awareness with A1 brings up 5 characteristics, A2 brings up 5 characteristics, A3 brings up 4 characteristics, A4 brings up 6 characteristics A5 brings up 1 characteristic.The component of evaluation with the indicator E1 brings up 2 characteristics, E2 brings up 4 characteristics, E3 brings up 5 characteristics, E4 brings up 3 characteristics, and E5 brings up 3 characteristics.The component of regulation with the indicator R1 brings up 2 characteristics, R2 brings up 2 characteristics, R3 brings up 2 characteristics, and R4 brings up 4 characteristics The characteristic of complete but disorder metacognition process is in S-4 can be explained as follows: The component of awareness with A1 brings up 4 characteristics, A2 brings up 6 characteristics, A3 brings up 4 characteristics, A4 brings up 7 characteristics A5 brings up 1 characteristic.The component of evaluation with the indicator E1 brings up 2 characteristics, E2 brings up 4 characteristics, E3 brings up 5 characteristics, E4 brings up 3 characteristics, and E5 brings up 2 characteristics.The component of regulation with the indicator R1 brings up 2 characteristics, R2 brings up 3 characteristics, R3 brings up 1 characteristic, and R4 brings up 4 characteristics The characteristic of uncomplete metacognition process is in S-5 can be explained as follows: The component of awareness with A1 brings up 5 characteristics, A2 brings up 3 characteristics, A3 brings up 4 characteristics, A4 brings up 5 characteristics A5 brings up 1 characteristic.The component of evaluation with the indicator E1 brings up 2 characteristics, E2 brings up 3 characteristics, E3 brings up 4 characteristics, E4 brings up 3 characteristics, and E5 brings up 0 characteristic.The component of regulation with the indicator R1 brings up 3 characteristics, R2 brings up 3 characteristics, R3 brings up 3 characteristics, and R4 brings up 3 characteristics The characteristic of uncomplete metacognition process is in S-6 can be explained as follows: The component of awareness with A1 brings up 4 characteristics, A2 brings up 4 characteristics, A3 brings up 5 characteristics, A4 brings up 5 characteristics A5 brings up 0 characteristic.The component of evaluation with the indicator E1 brings up 2 characteristics, E2 brings up 4 characteristics, E3 brings up 5 characteristics, E4 brings up 2 characteristics, and E5 brings up 2 characteristics.The component of regulation with the indicator R1 brings up 2 characteristics, R2 brings up 2 characteristics, R3 brings up 3 characteristics, and R4 brings up 2 characteristics The whole characteristics which appearing at each of the subject of the study can be explained in the following Table 1.

Title
Indicator Characteristic

A1
Rethink of what is known of calculus problem given.
The subject of the study: 1.
Read the given problem repeatedly and give marks at the words which are regarded as the key-words.

2.
Note the important cases of problems by underlining the words which are regarded as the key-words.

3.
Check the figure in the problem and represent the length and width of the figure as variable x and y.

4.
Read the table of problem and note it as the known case and differentiate the volume for the field as part of problem known.

5.
Make an important note and conclude known cases as the requirement to determine the way of solving the problem given.

6.
Make a correlation ship between the cases known in the problem and each questions which will be answered at the problem given

A2
Rethink the questions of calculus problems and similar problems which are finished before.
The subject of the study: 1. Read each question repeatedly and relates them to the previous questions of problems as known cases.
2. Give marks at the words which are regarded as the key-words and each questions and conclude cases asked in problems.
3. Make relationship question (a) in the problem relating to the area of rectangular.The relationship between area (L) and length and width is L=xy or L=5xy depends on the assumption done for the length and width which are assumed before.
4. Make relationship between the length of available barbed wire (s) on the problem and the fanced stable in the form of equation s=x+6y or s=5x+6y Since the length of barbed wire is 240 meters, there are two equations can be written: 240=x+6y or 240=5x+6y.
5. State the area of rectangular at the problem known as one function consisting of long and wide variable known as area function variable.
6. Change the function of rectangular area on the problem given as one function variable by determining the maximum requirement of a field area.
7. Conclude the cases asked in the problem based the note made.

A3
Rethink of cases which have not been finished in the last time yet when finishing calculus problem given.
The subject of the study: 1. Make the relationship between the length of barbed wire and wired stable part as equation 240=x+6y or 240=5x+6y.The Subject of the study : 1. Rewatch cases known in the problem.
2. Chose the way used to do the problem given based on the knowledge got before.
3. Determine variable value decided after downwarding the area function stated in one changer function.
4. Do variable substitution which has got into the previous function, i.e. 240=x+6y or 240=5x+6y 5. Find out the whole stable area and each stable as the answer of the problem given.
6. Reread table at the problem given to check the truth of the answers got.
7. Write the area comparison of each stable and the amount of goats.
8. Write the fee of stable building material and the fee of weekly operational as stated in During solving the problem I found a mistake and corrected it (if "Yes", describe the mistake) 10.I thought about how I was going 11.I tried different approaches for solving the problem 12.I asked myself whether my answer made sense 13.I checked my calculations to make sure they were correct 14.I thought whether there was something in the information that was given in the problem that needed special attention (if "Yes", describe it) Figu ocess, E is eva Figure 4. awareness pro

2.
State the area of stable in form of one changer function as L=xy into L=x(240-6x) 3. State the area of stable in the form of one changes function as L=5xy into L=5(240-6y/5)y 4. Do the substitution of area function in one changer and determine downward (derivative) of the function.5. Write downward of one changer function and determine the requirement of maximum and minimum value of a function as dL/dx=0 or dL/dy=0A4Rethink the next step to do to finish calculus problem given.
. Measure Mr.Syarif's stable to make all stables as wide as possible!b.State the comparation between the amount of goat in each stable and the width of each of stable!c.Count the fee of weekly operational needed by Mr.Syarif and the fee of the whole stable!(Adapted and modified from the Book of Calculus volume I by Edwin J. Purcell, Dale Verberg, dan Steven E, Rigdon in 2004 page 172).
table of problem given.a