Fragmentation of Thinking Structure ’ s Students to Solving the Problem of Application Definite Integral in Area

This study aims to reveal the fragmentation of thinking structure’s students in solving the problems of application definite integral in area. Fragmentation is a term on the computer (storage) that is highly relevant correlated with theoretical constructions that occur in the human brain (memory). Almost every student has a different way to construct a problem. That’s very interesting to finding a process of thinking students. Researcher works in three cases. The findings of this study were two in every case, which the fragmentation whole construction and fragmentation construction pseudo. Data this study a full description and in-depth exploration of the students majoring in mathematics education since high school that has been learned about the material Integral course and the area.


Introduction
Various types of errors in solving the problems presented by researchers integral (Kiat, 2005;Yost 2009;Dorko, 2009;Serhan, 2015) but only limited studies conducted to identify any errors that occur.Further searches related to how to structure the thinking of students when experiencing difficulties and mistakes in solving a given problem has not been investigated further.Kiat (2005) revealed that there are three types of errors in solving integral problems, including 1) a conceptual error (conceptual errors), 2) procedural error (procedural errors), and 3) technical errors (technical errors).At the conceptual errors that occur, there are two findings, namely, students are not able to determine the area of the curve that cuts the X-axis and the second in the student's inability to obtain integral gradient function of a curve function.On procedural errors were also found two cases, including the failure to put a constant c when determining the indefinite integral or failure in manipulating the constant c when needed and second, students are confused in distinguishing derivatives and integrals.On a technical error, there are some cases, such as in coordinate geometry, kinematic, algebra, and trigonometry.The third type of error is that a finding by the Kiat (2005) can be categorized as a research analysis, which requires follow up in more depth.Search through the research process thinking into suggestions proposed in this study and remedial efforts can be done.More than this suggestion, researchers are interested in targeting students as the subject for a student teacher candidates who will be assigned to studying integral material and the complexity related to student understanding more fully integral material so that the findings could be more varied.Serhan (2015) also expressed their continued research is needed to investigate the students thought process when solving a given problem.Hence, writer interested to reveal the thought processes of students in problem solving application definite integral in area, which is one of the topics raised by the Kiat (2005).
One interesting phenomenon discovered and described by Kiat (2005) related to solving problems of students at Integral material.Kiat (2005) provides a single issue, namely the "Determine the area between the curve 4 and X-axis from 0 to 5 " One subject of direct settlement, namely  (randomized), so this led to delays in making retrieval system.Cases that may occur as a continuation of the file C is deleted, so that there are holes sector which led to fragmentation in the files F getting worse.
Based on research conducted by Kiat (2005), Yost (2009), Dorko (2009) and Serhan (2015) on the application of problem solving definite integral in the area and analogies on computer-related fragmentation occurring, researchers interest to uncover how the fragmentation that occurred in the structure of thinking of students in solving problems in the application definite integral in area.Fragmentation can occur due to the construction pit (hole construction) and construction are apparent (pseudo construction).The term whole construction adapted from whole sector which occurred on a computer hard drive and construction pseudo adapted from non-full condition file in the storage space.

Research Design
This study is a qualitative study that investigated a social phenomenon or a human problem (Creswell, 2007).The research findings are not obtained through statistical procedures.In this type of study, researchers created a complex picture, studying words, a detailed report on the views of the subject of research, and conduct studies in a natural situation.Qualitative methods are a research procedure that produces descriptive data in the form of written and spoken on the subject of research related to the behavior observed.Yin (2011) states that qualitative research to further highlight the process and meaning in the perspective of the subject.

Participant
The research was conducted at the State University of Malang in force 2014/2015 second semester student.In his vote of mathematics education student who has been studying the concept of integral since high school and pursues back at his lectures, is assumed to have complete structure of thought and depth so that the process of exploration conducted by researchers associated with fragmentation search structure thinking it would be more visible.
Subjects selected using purposive sampling.In purposive sampling, researchers deliberately chose to study people and places that are rich in information.Individuals that are rich in information are intended as an individual who indicated structure fragmented thinking and has good communication skills.

Instruments
Instruments in this study is the researchers themselves and assisted with job sheets and semi-structured interviews.As for the issue raised is Find the area between the curve 4 and the -axis from 0 to 5.
Carilah luas daerah antara kurva 4 dan sumbu , dari Structural problems that used to capture the fragmented structure student thinking in solving problems in the application of integral course the area is based on Polya step.As for the structure of the problem in question is:

Results
In the present study found three variants of students in solving problems in the broad application of integral course area.3 cases have been explored will be described by naming the first case on the subject 1 or S1, second cases on the subject 2 or S2, and third cases on the subject or S3.

Case 1 on S1
S1 resolve a given problem without sketching graphs.S1 only imagine the extent of existing areas but not visualized in graphical form.
S1 started working on the given problem by looking at the curve and its limits.Then create a strategy or plan that this problem can be solved by using the integral rules.Here are excerpts of interviews conducted R : What did you think when You first read about this problem?
S1 : The first time I think sir are the curve and limits.Here was told to find the area bounded by the curve and these axes.So I imagine the Cartesian diagram.There is a curve defined by x = 0 and x = 5, so x = 0 x = 5 is a curve (pointing first step workmanship).Continue using the integral rules to calculate it.

R : Why is integral?
S1 : Because I had studied ever before integral.
S1 can understand what is known and ask nicely.S1 is also trying to make a plan with integral necessarily apply to search the area.But the plan made by S1 is not complete, resulting in an error in determining the extent of the area and boundaries.
In settling, S1 is able to integrate properly means S1 has a good ability to apply the integral method is simple on a given problem.

R : How you do it?
S1 : First is distributed then use the integral nature of the reduction.Then using indefinite integrals, first

I4b
Implement selected step by substituting the limits of 0 to 5 to , obtained 25 50

D1
Understanding the area with a limit of 0 to 4 F Understanding the area below the x-axis is negative for upper curve -the curve below 1 0

Symbol Explanation
Fragmentation pseudo construction: the boundaries of the area of integration are not seen as the side of the area being searched.
Fragmentation hole construction: not necessarily interlinked concepts integral to the area because there are holes in the area of integration which construction has not been determined Fragmentation pseudo construction: a positive sign in the area simply understood as "necessity" and not based on the high concept of an area of integration Hole construction: when a student's inability to understand the area as an integral course (not captured the area as an integral course)

Case 2 on S2
S2 solve the problem given by sketching graphs, but the sketch made of no help in finding the right solution.
From sketches made S2 led to a new confusion resulting S2 does not solve the problem to find the answers.
S2 initiate settlement by drawing a curve in the Cartesian diagram, search for the point of intersection and determine the cusp.Then S2 thought to determine the extent of areas with known boundaries.Here is an interview with a researcher S2.

R : What did you think when you first face this matter, resolving this problem?
S2 : We were told to finding restricted area of this curve.In the -axis, 0 and 5. First I think, illustrates the curve, meet like this (pointing to the image created on the task sheet).That cut 0 and 4 continues its peak point at 2, 4 .I continue to be confused here, the limit.Maybe if the is that I know the boundaries to sign less than or greater than.If it was just like this, (pointing drawings made on the assignment sheet), LHA, I confused this or this (pointing image/area curve).R : Why did you think that we should draw?S2 : I want to know first the shape.Continue after know its shape, it limits like this, keeps me confused.If the limit is usually, less than or more than, like, the shape inequalities.But after I picture shuttlecock shape like this.More confused.Areas that fulfill that which, area that meets the where deciding.
S2 seemed confused when determining the extent of the area on a chart created sketches.S2 think that it should be the limits of the known form of inequality.Here is a sketch of the graph of the results of S2.

Hole cons boundary integration explored f the problem
The thoug in the form through th

Figure
Figure 5. Sk It turns out there is empty space between file E and C. The fourth case file, placed F file size larger than the empty space between the E and the file C file, and it turns out the remaining part of the file is placed in sector after sector occupied by D. file fragmentation occurs in the fourth case, where the F file located not in one sector

Table 1 .
Adoption instruments task sheet Instruments task sheetKiat (2005)Instruments task sheet for this research