STUDENTS ’ COGNITIVE PROCESSES IN SOLVING PROBLEM RELATED TO THE CONCEPT OF AREA CONSERVATION

This study aimed to determine the cognitive process employed in problem-solving related to the concept of area conservation for seventh graders. Two students with different mathematical ability were chosen to be the subjects of this research. Each of them was the representative of high achievers and low achievers based on a set of area conservation test. Results indicate that both samples performed more cyclic processes on formulating solution planning, regulating solution part and detecting and correcting error during the problem-solving. However, it was found that the high achiever student performed some processes than those of low achiever. Also, while the high achiever student did not predict any outcomes of his formulated strategies, the low achiever did not carry out the thought process after detecting errors of the initial solution gained. About the concept of area conservation, the finding also reveals that within the samples’ cognitive processes, the use of area formula come first before students decided to look for another strategy such as doing ‘cut-rotate-paste’ for the curved planes, which do not have any direct formula. The possible causes of the results were discussed to derive some recommendation for future studies.

latter concept is often isolated from the first two concepts when students solve an area-related problem (Kordaki, 2003).In addition, in the teaching of area measurement topic, the teacher tend to focus only on the use of formula.Unfortunately, most of the area measurement teaching gives the area formula too early for the students (e.g.Kordaki & Balomenou, 2006;Kospentaris et al., 2011;Papadopoulos, 2010).In fact, the idea of area conservation is deeper than finding the relation of area formula.The use of the formula in measuring the area of plane figure is considered as procedural algorithm only (Fauzan, 2002).In fact, relating to the area conservation skills, most of the pupils have difficulties in decomposing problems (Kordaki & Balomenou, 2006).They are unable to see that decomposing shape into another form would make the area of the figure invariant.Therefore, the students decide to the shortcut by only interested in the formula from which a nonmeaningful learning is resulted.This fact results that the students understanding on area measurement is limited to procedural only.In fact, remembering the formula is becoming the main problem on students learning not only in mathematics subject but also other science subjects.Therefore, the topic of area conservation plays an important role in the development of students reasoning on area measurement.
Area conservation can be defined as quantitative value of a certain area of figure remains unchanged after the figure is altered (Smith et al., 2011).Piaget, Inhelder & Szeminska (1960) stated that the term "conservation" means the invariance of the quantity value of the area of a plane while the plane may be transformed into a qualitatively different one.For example, students need to understand that when a shape is divided into several parts and these parts are re-arranged, the area remains the same.To state how wide the area of the figure is, a unit is selected and integrated until shape of the figure is fully covered.When arranging units into rows and columns, students can understand the area depending on the number of rows and the number of columns that there is a multiplicative relationship between these numbers.The studies of students' performance regarding the concept of area conservation have been reported by previous research with regard to some point of interest, such as students' error and misconception (Sisman & Aksu, 2016), students' solution strategies (Kospentaris, 2011), links between students' performance on the problems related to non-measurement and calculation tasks in area measurement (Tumová & Vondrová, 2017).The findings of studies is considered by many scholars as the preliminary step in understanding students' adequate mastering of area measurement (Clements & Stephans, 2004;Kospentaris et al, 2011).However, limited studies found to concern on how students perform their cognitive processes when solving area conservation-related problem.Therefore, In this study, we stress the need for the investigation into the nature of students' abilities by exploring their cognitive processes required for the improvement of students' performance on the topic of area conservation.
Cognitive processes may be described as online mental activities that are proactive in nature (the "to do" strategies) (Montague, Krawec, Enders, & Dietz, 2014).In a similar vein, cognitive processes are defined as the mental processes of an individual, with particular relation to a view that argues that the mind has internal mental states (such as beliefs, desires and intentions) and can be understood in terms of information processing, especially when a lot of abstraction or concretization is involved, or processes such as involving knowledge, expertise or learning.Some scholars have derived some stages of cognitive processes.For example, Montague, Warger and Morgan (2000) through the cognitive strategy instruction: Solve It! Believes that cognitive processes incorporate the activities of reading (identifying relevant/ irrelevant information), paraphrasing (rewording the information of the problem without changing the problem meaning), visualizing (transforming problem information to a representation that shows the relationships among problem parts), hypothesizing (setting up a plan to solve the problem by deciding on the type and order of operations), estimating (predicting the outcome based on the question/goal), computing (conducting the basic operations needed for solution), and checking (reviewing the accuracy of the process, procedures, and computation).Another cognitive process was offered by Montague (2002).The processes incorporates some stages: comprehending linguistic and numerical information in the problem, translating and transforming that information into mathematical notations, algorithms, and equations, observing relationships among the elements of the problem, formulating a plan to solve the problem, predicting the outcome, regulating the solution path as it is executed, and detecting and correcting errors during problem solution.
In this regard, the cognitive processes can be traced along the way how a learner process his/her thinking based on the types of reasoning mainly demanded by the tasks, i.e. non-measurement reasoning or measurement reasoning.Since in this study, we focus on measurement reasoning, the cognitive processes were measured following the stages of Battista (2007) from the use of numbers which not connected to unit iteration, the employment of unit iteration and enumeration which includes units properly located only along the sides/edges, the operation of numerical measurement, and the integration of measurement and nonmeasurement reasoning, such as understanding formulas for non-rectangular or composite shapes or determining the value of particular shapes based on a quantitative context inherent in the problem being solved .
In fact, according to Montague (2002), students simply may not know "what" to do or even "how" to think about beginning the problem.In addition, if students are not asked how they solve a particular problem and if the work and explanations that accompany their answers are not observed properly, a researcher learns a little about students' understanding and misunderstanding of mathematical ideas (Stylianou et.al, 2000).
Thus, this study took a part of carrying out an in-depth investigation of what students were thinking while they performed their cognitive processes on the problem related to area conservation.The cognitive processes model guiding this investigation is based on modification of Montugue's model of cognitive process, in which thought process and extending problem of Mason's (2015) model are added in the model.

Sample of Research
Prior to selecting the student interviewees participating in the interview session, as many as 25 seventh graders with various background in terms of gender, mathematical ability, and verbal communication from a private junior high school in Surabaya city participated a test consisting of five items examining their mathematical ability particularly around the concept of area conservation on two dimensional figure.They were asked to do the test in 45 minutes.They were also informed that their work would not be graded so that they could use their own methods to solve the tasks.
The result of the test informed that approximately half of them were in the group of high achievers (score >60 out of 100) and half in the group of low achievers (score ≤ 60 out of 100) based on their written test performance.As many as two samples were recruited from each of those two groups as the representatives by considering same gender as the control variable, ease of verbal communication based on information from their mathematics teacher and willingness to participate.Beside, to ensure the subject, we also confirm with data of students' mathematics performance.Thus, we had one male student having good score/High Achiever Student (code as HAS) and the other one male student having low score/Low Achiever Student (code as LAS).The data were analyzed qualitatively.

Instrument and Procedures
Data were collected from the samples' work on written test which is different from the test given in the initial stage of selecting samples and follow-up interviews.First, students worked on two area conservation-related tasks in 30 minutes.The first task was arranged by the authors in quantitative approach in which the real-world situation was embedded in the tasks, while the second task was developed by the authors relying on students' quantitative approach without any real-world situation.
Furthermore, those two tasks were developed around the view of Euclid's elements, in which the practice of measuring area is the use of "additivity axiom", i.e. dividing one figure into some parts which rearranged would form another figure, in order to prove the area equivalence of the figures (Freudenthal, 1986).Thus, instead of only focusing static perspective of area measurements, the tasks also focus a dynamic perspective where the qualitative approach: emphasizing the conservation of area without the use of numbers (Hiebert, 1981).Those tasks were then validated by experts in terms of content, construct, and language as well as by learners, i.e. students' aside the samples to examine the practicality such as the ease of and presentation of picture and table.See those two task at Figure 1.

Task 1
Eko gets an assignment from his father to give the suitable price tag on the piece of wood he will sell.His father gave a standard price for a rectangular piece of wood measuring 12 cm x 10 cm, which is IDR 24,000 In each of the following pieces of wood, give the suitable price in the available price column.
Task 2 Look at the parallelogram below.which of the figures below having the same area with the above parallelogram?Explain your reason.

Figure 1. The dynamic area conservation tasks used in this study
In the following day, we interviewed and videotaped the two samples.Table 1 describes the interview protocol that guided the interviewer to collect data.However, this protocol does not mean to guide the interviewers used all the question items too rigidly.Rather, it plays role as the tool to confirm some particular subject's responses.This is to keep the subject reveal their thinking processes as naturally as possible.Thus, when the responses of the subject did not indicate particular cognitive process to occur during the interview, the interviewers did not ask such processes further.

Data Analysis
Data of interview were analysed by firstly reducing data, displaying data, and finally drawing conclusions and verification (Miles & Huberman, 1994).The conclusion was sought to understand the most dominant pattern of cognitive processes performed by samples within their problem-solving activities on the tasks.To analyze data interview, we employed a modification of cognitive processes from Montugue (2002).The modification regards to the addition of one more stage as the last stage following the recommendation of Mason in which in the last stage, a solver should not only accentuate an analysis of answers, but also carry out the thought process and problem extension (Mason, 2015).Figure 2 shows the stages which possibly occur during solving a mathematics problem.The arrow direction indicated in figure 2 points out that a solver may follow a cyclic process where the solver moves back and forth, perhaps getting stuck and having to take steps back along the way (Mason, 2015).
For instance, there is a possibility that a solver moves back to the stage of observing relationships among information when he/she gets stuck in formulating a solution.In addition, the cycling process can occur for more than two times depending on the degree of his/her confidence and plausibly of solution strategies obtained.
Furthermore, the arrows presented in the stages in figure 2 indicate the logical progression from one process to another although it is possible for a student to skip any of these processes, or they can just jump from one process to another process when they change their solution process.For example, when trying to regulate the solution using a plan the student has derived, he/she may be directly arrive at the stage of regulating the solution path as it is executed.Thus, he/she skip any activities indicated in the stage of predicting any outcome.The model of using arrows in analysing the stages that might occur on student's cognitive process are proven as a helpful tool for keep track student's behaviors (Yeo & Yeap, 2010).While the model of analysing students' cognitive processes employing Montugue's ( 2002) model has been used by Jones (2006) to track the existence of the Montugue's stages of cognitive processes, there are still lack findings reporting the Montugue's stages which consider both the existence and the order of process of the stages.Thus, in this study, the modified Montugue's model in terms of the dynamic processes which might occur during student solution process indicated by the arrows of the stages were used as a tool of analysis shown in Figure 2.

Figure 2. Framework of analysing students' cognitive processes
With the same way, he get a square for figure no 10 and 11.For the third group of figures, when he tried examining figure no 7 and 9, he found it was easy and decided that these two figures also have the same area with.He argued, "Same with figure no 6, I cut off these two parts and paste them so that the figure becomes a square."However, when he examined figure no 2, which is a kite-shaped plane, he got stuck.It was observed that he found difficulties in determining the place where he should cut the figure in his mind.After for more than 1 minute, he finally revealed that he need to cut the plane twice, rotated the cut plane, and paste it so that it forms a square.The only figure that he did not any idea to solve is figure no 14.He said, "It is very difficult to find where to cut off this plane, it might have another method, and I don't know." To complete his solution process, he was asked to compare which method he should use when finding a similar problem in the future.He argued, "I found some difficulties when using a formula since I sometimes forget with the formula.Therefore, I have to use another method such as by cutting particular parts of the figure and move the cut parts to the other part of the figure so that it becomes a square."In brief, his cognitive processes are illustrated in Figure 3b.Interestingly, HAS carried out the thought process on both the two tasks, which are proven as important processes to convince the correctness and the most effective method to derive the solution.
Also, for HAS, the process of predicting any outcome does not seem likely becomes a crucial process in the initial cognitive processes.As evidence, this stage was not carried out in the first task, while this was carried out in the second task, but after the thought process, instead of between the process of determining relationship among information and formulating solution planning.

The Cognitive Processes of LAS
On the first task, LAS began his cognitive processes by claiming that he ever faced similar problem and mentioned the information either known or unknown.He then re-drew the information from the problem to show the meaning of magnitude notation as circumference of rectangle as part of the process of 'transforming information into mathematics notation, algorithm and equation.
Afterwards, he observed the relationship among element of the problem by finding the unit price as basis to find total price.He continued with formulating solution planning.He determined three plans such as 1) divided the given price of pieces of wood to explore the price for the unit magnitude, 2) calculating the circumference of each plane by counting the number of square that were covered by the explored plane, 3) determining the price of each piece of wood by multiplying the price of unit magnitude with the circumference of each plane.He skipped doing predicting outcome and continued with regulating the solution path as is it executed by calculating the price of each 'cm'.However, LAS could not sure about the result of his calculation but he did not prove his prediction.This process was coded as predicting outcome process.After he did prediction, he formulated a plan again to solve the problem by doing revision on the plan of determining unit price and delete his initial idea of calculating the rectangle circumference.He tend to doing multiplication 2 (12x10) = 240 cm.Furthermore, another solution path was regulated such as by calculating the unit price of every 'cm', though he did a little error in calculation.In determining the price of unit piece of woods, a unique order of work on plane was as follows: squarerhombusparallelogramtrapezoid -rectangle with arc modificationtrapezoid with arc modification.For all planes, he counted the number of 'box/square' that cover the planes.
The cognitive process continued with comparing the price of wood in the form of parallelogram and rectangle.By this, he shared his uncertainty and did checking since he found for bigger form of wood is cheaper than those smaller one.This process was coded as detecting and correcting error during problem solution.Afterwards, another activity of formulating a plan to solve the problem was executed by revising the unit price.As a result, regulating the solution path as it is executed process appeared again by repeating the calculation for the price of each.However, LAS found difficulties in determining the price for rectangle with arc modification and trapezoid with arc modification.He felt uncertain with his strategy to solve the problem.Therefore, the following process were not performed.By considering the framework of analyzing students' cognitive process, the cognitive process in solving are  (2002), the two students managed to obtain mathematics' solution and describing their process.The common feature finding was those students follow the hierarchical step of cognitive process except the thought that was not elaborated by Low Achiever Students (LAS) .The initial strategy used to find the price was by elaborating 'unit price'.Cramer et.al (1993) suggested that unit rate approach was the most popular strategy and responsible for the largest number of correct answers.However, LAS got stuck after detected errors and repeating the cyclic of formulating solution planning and regulating solution path.He still fell uncertain with his used strategy.On the contrary, High Achiever Students (HAS) performed more cyclic cognitive process on the task with quantitative approach.The cyclic started after he detected errors and he tried to revise his errors.He turned back the process until he re-determined several elements of the problem, re-formulating solution planning and re-regulating solution path.In addition, the thought process were elaborated by HAS to arrive to his final solution.These findings in line with previous research that suggested that high academic achievers and low

Figure 3 Figure 3 .
Figure 3 compares the cognitive processes of HAS when solving two area conservation tasks.It indicates that there are some repeating processes done for the two tasks, primarily from the process of determining relationship among element of problems to detecting and correcting errors of the solution resulted.

Figure 4 .
Figure 4.The cognitive process of LAS

Table 1 .
Protocol for interviewing the subjects Please read the question to me.If you don't know certain word, say it.Tell me what the question is asking you to do.