Available online at http://www.pancaranpendidikan.or.id Pancaran Pendidikan FKIP Universitas Jember Vol. 7, No.1, Page 186 -193 , February 2018
PancaranPendidikan DOI:
10.25037/pancaran.v7i1.149
ISSN 0852-601X e-ISSN 2549-838X
Student’s Error in Writing Mathematical Problem Solving Associated With Corresponding Angles of The Similar Triangles R H Parastuti1*, B Usodo1, S Subanti1 1
Sebelas Maret University, Surakarta, Indonesia Email: *
[email protected]
ARTICLE INFO
ABSTRACT
Article History: Received Date: 1th January 2018 Received in Revised Form Date: 10thJanuary 2018 Accepted Date: 25thJanuary 2018 Published online Date: 01stFebruary 2018
The research method used is descriptive qualitative. The aim of this research is to describe the student’s error in writing the resolution about the similarity of the two triangles. The student has many problems in mathematics include analyzing a triangle. A polygon that has three sides, three angles, and three corners is called a triangle. The angle formed by the two sides of a triangle intersects with each other. The triangles are similar if the comparison lengths of the corresponding sides are same and the large of the corresponding angles are same. Three students are chosen as the subjects based on their score in the daily test about the similarity of two triangles. The students are given tests and then interviewed. Writing error by students in solving problems is analyzed based on Newman’s theory. The result obtained in this research is that error writing occurs because of the misconceptions when transforming the information from the problem. The students make mistakes in writing occurs when they have to solve the problems.
Key Words: Student error, Mathematical Problem Solving, corresponding angles, Similar Triangles
Copyright © Parastuti et al, 2018, this is an open access article distributed under the terms of the PancaranPendidikan Journal license, which permits unrestricted use, distribution and reproduction in any medium, provided the original work is properly cited
INTRODUCTION Mathematics as one of the foundations of some science is needed by the human. Students are required to think deductively and logically in learning mathematics. The general purpose of learning mathematics is stated as making an individual acquire the mathematical knowledge needed in a daily basis, teaching how to solve problems, making the students have a method of solving problems and acquiring reasoning methods [1]. Mathematics tests the students’ intelligence to think and solve daily life problems. The ability to remember concepts, memorize facts, manipulate numbers and solve the mental puzzle is driven by the memory power of mathematics [2]. The
187 ___________ ©PancaranPendidikan, Vol. 7, No.1, page186-193, February 2018 mastery of the mathematical concepts and the skills of problem-solving will support the students to make fewer mistakes in trying to solve a lot of problems [3]. Newman’s research generated a large amount of evidence emphasizing that far more children experienced difficulty with the semantic structures, the vocabulary, and the symbolism of mathematics than with the standard algorithms [4]. There are five hierarchical areas of Newman theory namely reading, comprehension, transformation, process skills and encoding that can be used to help students identify those mistakes [5]. A reading error occurs when it is written that the failure to pursue the course of problem-solution, misunderstanding occurs when students fail to understand the requirements, causing them to be wrong or failed to solve the problem, a transformation error occurs when the students completely understand the requirement but failed to identify the course of the problem solving, a skill process error occurs when the operation to be used to solve the problem has been identified but the student fails to properly perform the procedure, whereas coding errors occur when students fail to provide a properly written mathematical answers [6]. Students with problem-solving strategy appear to have better performance on test more than students who do not have a problem-solving strategy [7]. The strategy can be implemented in the form of writing problem-solving clearly, coherently, logically, and systematically. Mathematical detailing is a linguistic action in which an individual refers to given information for a mathematical context task, and integrates such information to support a claim. As such, this grammatical resource applies to a particular form of writing in mathematics [8]. Student’s ability to write can be seen from four aspects: the idea that comes from reading the word problem, drawing graphs, writing to design data analysis and presenting data [9]. Drawing graphs in the junior high school usually just drawing a polygon such as a square, a triangle, a circle, a trapezoid, etc. A simple triangle is one of the forms of a polygon it has three sides, three angles, and three corners. The angle formed by two sides that match in one vertex or usually named the point of angle. Two triangles are congruent if the three corresponding sides have the same length and the three corresponding angles have the same degree. If the three corresponding sides have the same of comparison length and the three corresponding angles have the same degree, then the two triangles are similar. The analyze of the similarity of two triangles more difficult then congruently of two triangles. Errors often occur when analyzing the two triangles are similar. METHODS The purpose of this study is to find out the students’ mistakes in writing the corresponding angles between two similar triangles. The approach used in this study is a qualitative approach. Subject selection techniques used with purposive sampling. The procedure in this study begins by determining the subjects based on daily test score on triangle similarity and grouped into three categories namely high, medium, and low. Furthermore, the selected subjects are given problems in the form of test questions to be solved. Interviews are also conducted to find out how students solved the problem verbally. The research subject consisted of three students from class IX H SMP Negeri 1 Plaosan Magetan district of East Java Indonesia in October 2017. The three students are selected from 29 students who take daily tests on the material of triangle congruence
Parastuti et al: Student’s Error in Writing Mathematical Problem Solving ... _____188 and similarity. The research subject consisted of one student with a high score, one student with the medium score, and one student with the low score. RESULTS AND DISCUSSION Error when doing mathematics often occurs in students, even students who have highest test score. The evidence of student’s error in writing the problem solving will be shown on the results of the student’s written test which will then be compared with the interview transcript. Students must write the corresponding angles of the similar triangles (the problem shown in Figure 1).
Figure 1. A Given Problem Based on the problem on Figure 1, data from the subject has been obtained. There are several types of writing errors occurring in the subject of study. The following will be presented the students’ work and part of the interview transcript. Written test and interviews that have been done, generate the results of the student’s work in written and interview transcript as a result of the oral test with the same material and type of questions. Interviews are conducted in the Indonesian language for easier implementation. The interview has been recorded and transcribed. The results of written test and interviews are presented based on each subject category that is high, medium, and low. Interview transcripts are presented only in certain parts of the problem. The first subject is a student with the highest daily test score for triangle similarity material. Further, a subject to this category will be given HS (High Subject) code. The result of this subject written test is shown in Figure 2 below.
Figure 2. High Subject wrote test results The part of interview transcript between the interviewer and the HS is as follows. (IInterviewer, HS- High Subject) I HS I HS I HS I HS I HS I
:”Tell me which pair of angles is the same degrees?” :”Angle ACB and angle BDE” :”What is the reason?” :”It is already stated in the problem” :”Then is there anything else?” :”Angle CAB with angle BED.” :”Why?” : (in silence while thinking of the reason) “I don’t know” :”Okay, what else?” :”Angle ABC with angle DBE.” :”Why?”
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HS
:”Because the angles are overlapping.”
The second subject is a student with medium daily test score (medium grade in class) for triangle similarity material. Further, a subject to this category will be given MS (Medium Subject) code. The result of this subject written test is shown in Figure 3 below.
Figure 3. Medium Subject wrote test results The part of interview transcript between the interviewer and the MS is as follows. (I Interviewer, MS - Medium Subject) I :”The third problem, which triangle is a similar triangle?” MS :”Triangle ABC and triangle BDE.” I :”Now, we look at your work, why do you write this symbol?” (Yellow circle) MS :” That's a symbol for a similar triangle, right?” I :”And then what is this symbol for?” (red circle) MS :”To show that the angle is the same.” I :”Do you know what the symbol for a congruent triangle is?” MS : (silent for a while) “it seems is my symbol is wrong, isn’t it?” I :”what do you think?” MS :” I think it's wrong. “ The third subject is the student with low daily test score for triangle similarity material. Further, a subject to this category will be given LS (Low Subject) code. The result of this subject written test is shown in Figure 4 below.
Figure 4. Low Subject wrote test results The part of interview transcript between the interviewer and the LS is as follows. (I Interviewer, LS - Low Subject) I :”What is the symbol used for the similar triangle?” (yellow circle)” LS :”Like this.” (while writing the symbol) I :”Then, what symbol used for the congruent triangle?” LS :”It uses the equal symbol.” I :”What symbol used for an angle with an angle degree?” LS :(long silence) “It also use equal symbol” I :”Which other angles are to the same degree?” LS :”Angle CBA and angle EBD are in the same degree because they are overlapping.” I :”Then why do you write angle CEA equal to angle EBD?” (red circle)
Parastuti et al: Student’s Error in Writing Mathematical Problem Solving ... _____190 LS :(looking at his/her work) “This I think I misunderstood the image. It should be CBA.” The results from HS work is shown in Figure 2 and the data obtained that HS write the corresponding angles of two similar triangles that are ∆ABC and ∆BED. HS read and illustrate the problem well so that this subject has no difficulty in analyzing the similar triangles. HS has pinpointed the corresponding angles of two similar angles but does not write the reason why the angle is to the same degree when presenting the solution. Transformation error occurs to HS because she does not write the reason why the angle is equal. Transformation error is seen when the interviewer asks the reason she chooses CAB with BED, this subject does not know what the exact reason for choosing the angles. The data presentation of the problem analyzes results are seen in Figure 1 where HS write excellently the problem solving although HS does not write the reasons for selecting the corresponding angles. This subject in reading a problem is drawing the picture about the problem, and analyzing data she can write it correctly. This subject does not write the reasons when presenting data that means she does transforming error. So, this subject’s mistake in writing the corresponding angles on two similar triangles is that this subject does not write down the reason why HS chose that angle. The results from MS work is shown in Figure 3 and the data obtained is that MS write the corresponding angle of ∆ABC and ∆BED that are similar. MS read and illustrate the problem well shown in this Figure. Analyzed data of MS is good shown by the transcript of the interview. The corresponding angles are shown correctly but this subject does not give the reason why the angles are to the same degree causing the transformation error when presenting the solution. The error in writing similar and equal symbol occurs in MS when presenting data (encoding error shown by yellow and red circle in Figure 2). The equal angle should be marked “=” but the equal symbol is written with “ ” and “ symbol is written with “ ”. The misconception in writing the symbol that occurred in MS resulted in this subject fail to solve the problem. Misunderstandings occur in administrating the symbol or coding resulted in the result of the subject’s answer is wrong. The results of the previous research indicate that high school students perform some misconceptions and lack knowledge related to geometry [10]. There had been a misconception on this subject when writing the symbol of angles in the similar triangles. This subject in the reading a problem, illustrate the problem and analyzing data she can write it correctly. This subject when presenting data does not write the reasons and the symbol of are written is wrong. Transforming and encoding errors occur when presenting the data. So, the subject's mistake in writing the corresponding angle on two similar triangles is that the student does not write down the reason why MS chose that angle and wrong in writing the mathematical symbols. The result of LS work is shown in Figure 4 and the data obtained is that the encoding error (shown by a yellow circle in Figure 3) occur on this subject caused by the “ ” symbol being written as “ ” symbol. The problem given is not read well by LS and she is not careful in seeing the picture. As a result, LS is wrong in writing the corresponding angles. is written incorrectly because the CEA does not form an angle that shown by a red circle in Figure 3. Because is not an angle, does not correspond to any angle. The writing of the angle is wrong because from the interview it appears that LS is wrong in seeing the picture and read the problem. Reading error also appears to form the writing of when it is clear on
191 ___________ ©PancaranPendidikan, Vol. 7, No.1, page186-193, February 2018 the picture that the known angle is not the angle mentioned. There are not written reasons why the angles are corresponding caused by transformation error when analyzing and presenting the data from the problem. LS is failed in performing problemsolving procedure so the process skill error occurs. Thus, LS cannot show the corresponding angles of ∆ABC and ∆DEB. This subject in the reading a problem occur comprehension error. When illustrating the problem has been right, but when analyzing data she cannot write the angle correctly because of comprehension error. This subject when presenting data does not write the reasons (transformation error) and the symbol of are written is wrong (encoding error). This subject in write the corresponding angles of the similar triangles had done reading error, comprehensive error, transforming error, and encoding error. So, this subject's mistake in writing the corresponding angle on two similar triangles is that the student does not write down the reason why LS chose the angle, misstated in mathematical symbols, and misrepresented the angle of the triangle. CONCLUSION The conclusion of the result that had been obtained was the student with the highest score still made writing error caused by transformation error. A student with the medium score there had been a misconception when encoding error and transforming error. A student with low score did more errors namely reading error, comprehension error, transformation error, and encoding error. The results of this study reveal that there was a misconception when transforming information in analyzing and presenting the data of the problem. So, students made the mistake of writing the corresponding angles on two similar triangles that occur because of misconceptions when solving the problem. Teachers need to consolidate every concept and angle properties in all students, especially to students with the medium and low score. Students must know the writing error done in solving triangle similarity problem so that students could correct their mistakes in writing. How the thinking process of students when write the problemsolving. The analyze of the problem in this research could be used for another researcher.
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