Modeling  is one of abilities  included  in studying  mathematical materials. Modeling 

is mostly used to express the events of everyday life , such as  modeling  of  tsunami

wave,  modeling  of  population  growth,  and  modeling  of  economics  growth. 

Therefore,  mathematical  modeling  is  an  important  ability  to  be  possessed  by

students. It is  in line  with the  opinion  of  Niss  (2010)  who  states  that  Modeling  is  a

Crucial  Aspect  of  Students'  Mathematical  Modeling.  Considering  its  significance, 

mathematical  modeling  should  be learned  by  students  since  they attend  elementary

school. For example,  Singapore  school  curriculum  has provided an opportunity  for 

students  to  learn  mathematical  modeling  abilities  since  they  attended  elementary

school  (Kaur  and  Dindyal,  2010).  On the other hand, it is a fact that  students  -  in

some  countries,  such  as  Germany,  England,  Romania,  Canada,  Czech  Republic, 

Mozambique,  Netherlands,  and  Japan  -  have difficulties in  mathematical  modeling

(Ikeda,  2007).  Therefore,  this paper  is intended to  discusse  some difficulties faced 

by  junior  high  school  student  in  conducting  mathematical  modeling  and  model 

interpretation,  particularly  in  the  materials  of    linear  equation  system  of  two 

variables. Data  were obtained from students in two junior high schools coming from

two and three cluster schools in the Bandung Municipality, covering research subject

as many as 151 students

Didi, Suhaedi

Tia, Purniati

Lumbung Pustaka UNY  (UNY Repository)

  Proceeding of  International Conference On Research, Implementation And Education  Of Mathematics And Sciences 2014, Yogyakarta State University, 18-20 May 2014            ME-583  AN ANALYSIS OF DIFFICULTIES ON MATHEMATICAL MODEL INTERPRETATION OF JUNIOR HIGH SCHOOL STUDENTS  ON THE MATERIALS OF TWO-VARIABLE LINEAR EQUATION SYSTEM  Didi Suhaedi1, Tia Purniati2  1Departement of Mathematics, Bandung Islamic University, dsuhaedi@hotmail.com 2Department of Mathematics Education, Indonesia University of Education, tpurniati@yahoo.com     Abstract Modeling is one of abilities included in studying mathematical materials. Modeling is mostly used to express the events of everyday life, such as modeling of tsunami wave, modeling of population growth, and modeling of economics growth. Therefore, mathematical modeling is an important ability to be possessed by students. It is in line with the opinion of Niss (2010) who states that Modeling is a Crucial Aspect of Students' Mathematical Modeling. Considering its significance, mathematical modeling should be learned by students since they attend elementary school. For example, Singapore school curriculum has provided an opportunity for students to learn mathematical modeling abilities since they attended elementary school (Kaur and Dindyal, 2010). On the other hand, it is a fact that students - in some countries, such as Germany, England, Romania, Canada, Czech Republic, Mozambique, Netherlands, and Japan - have difficulties in mathematical modeling (Ikeda, 2007). Therefore, this paper is intended to discusse some difficulties faced by junior high school student in conducting mathematical modeling and model interpretation, particularly in the materials of  linear equation system of two variables. Data were obtained from students in two junior high schools coming from two and three cluster schools in the Bandung Municipality, covering research subject as many as 151 students.  Key words: analysis of the students’ difficulties, two-variables linear equations system   INTRODUCTION Algebra is one of fundamental subjects thought in mathematics in many countries. Algebra content is considered important to be mastered because it is used in everyday life implicitly and explicitly.  Syntax is used to make many things, such as web address, e-mail, searching engine in internet; outomatization of TV remote control, radio, LCD; tsunami wave modeling, population growth modeling, economics growth modeling, etc.; all of them needs logics of algebra. Therefore, it is not excessive if Katz (2007) entitles his writing, Algebra: Gateway to a Technological Future, also algebra as gatekeeper for future education as well as for employment opportunities (Moses & Coob, 2001; National Research Council [NRC], 1998). Unfortunately, it is a fact that many students have barriers in learning algebra and experience misconceptions for material of algebra (Asquith at al, 2007; Kriegler & Lee, 2007; Knuth et al, 2006; Knuth et al, 2005; Kieran, 2004; MSP, 2003; Falkner et al, 1999; Alibali, ME - 75 Didi Suhaedi & Tia Purniati / An Analysis of..                                                                        ISBN.  ME-584  1999). Even, NRC (1998:1) says that algebra courses in United States of America is characterized as "an unmitigated disaster for most students". This fact is not different from the fact in Indonesian junior high schools although there has not been a research conducted about it. Indeed, which aspect is most difficult in algebra for students? Which is most essential causing algebra difficult for students? The most contributing aspect for students’ failure in learning algebra is a fundamental difference of thinking between arithmetic which is concrete and algebra which is abstract (Lawrence and Hennessy, 2002:ix). Transition from arithmetic thinking to algebraic thinking must not be considered easy. This paper is intended to discusse some difficulties faced by junior high school student in conducting mathematical modeling and model interpretation, particularly in the materials of  linear equation system of two variables.  RESEARCH METHOD The research uses two steps. First step is prelementary study and conceptual formulation of instrument prototype to measure four indicators of activity. 1) expressed everyday situations to mathematical models; 2) transforming a mathematical model to other mathematical models; 3) determine solution of a mathematical model of mathematical problems; 4) make interpretation of the results of a mathematical problem solving. Second step, data collection that is done through written tests from students in two junior high schools coming from two and three cluster schools in the Bandung Municipality, covering research subject as many as 151 students.  RESULT AND DISCUSSION Algebra is one of the topics studied in school mathematics, and it is a difficult material to be mastered by students (Asquith et al, 2007; Knuth et al, 2006; Knuth et al, 2005; MSP, 2003). Linear equation systems of two variables is one of the algebra materials that studied by junior high school students. On this occasion, the author presents the results of exploratory student difficulties in learning linear equation system of two variables. Measurement of student difficulties in the material of linear equation system of two variables is done using several indicators, the indicators are made with reference to the definition of school algebra. These indicators are as follows: expressed everyday situations to mathematical models (IND-1);  transforming a mathematical model to other mathematical models (IND-2); determine solution of a mathematical model of mathematical problems (IND-3); make interpretation of the results of a mathematical problem solving (IND-4). The ability of students to master certain material expressed by degree of success. Education Ministry determines degree of success of students gained at least 60% (Depdiknas, 2008: 4). The data of degree of success (DS) of 151 students in do the problems of linear equation system of two variables for each indicator are presented in the following table.  Tabel 1. Data of degree of success for each indicator Learning Difficulties Indicator Number of problem IMS Average Score Students DS IND-1 1(a) 3 2,272 75,7% IND-2 2(a) 3 2,132 71,1% IND-3 1(b) 2 1,152 57,6% 2(b) 2 0,874 43,7% 1(b) and 2(b) 4 2,026 50,7% IND-4 1(c) 2 0,722 36,1%   IMS = Ideal Maximum Score  Table 1 shows that most students have a major weakness in the third and fourth indicators. This is indicated by the percentage of degree of success of students in the third   Proceeding of  International Conference On Research, Implementation And Education  Of Mathematics And Sciences 2014, Yogyakarta State University, 18-20 May 2014            ME-585 indicator amounted to 50,7%, and fourth indicator with a degree of success of 36,1%. Both the degree of success is below the minimum degree of success (for 60%) set by Education Ministry. This shows the student's major weak point in the mastery of linear equation systems of two variables is about determining solution of a mathematical model of mathematical problems (three indicators) and make interpretation of the results of a mathematical problem solving (four indicators). Therefore, the following discussion focuses only on the third and fourth indicators. Here the authors present examples of a mistake of student answer in solving problem number 1 (for the fourth indicator) and problem number 2 (for the third indicator).  Problem 1.  Pertambahan tinggi pohon A dan pohon B selalu tetap untuk setiap minggunya, dengan data sebagai berikut: Pohon Tinggi pohon (cm) Minggu ke-6 Minggu ke-17 A 20 42 B 15 48  a. Buatlah suatu sistem persamaan linier dua variabel untuk pohon A dan pohon B, yang menyatakan hubungan antara waktu (minggu) dengan tinggi pohon !   b. Tentukan penyelesaian dari sistem persamaan linier dua variabel tersebut ! c. Kapankah kedua pohon tersebut memiliki tinggi yang sama dan berapa tingginya? Berilah penjelasan!  Most of the students can do on question 1 (a). It can be seen from degree of success of student for 75,7% (above the minimum degree of success set by the Education Ministry). The biggest mistake made by most students occurs when students determines the interpretation of the solution of a mathematical model that they had found the question answer 1 (b). Example of the error is shown in the following the figure 1. Figure 1 shows that the student can do and complete the mathematical model (linear equation systems of two variables), which states the relationship between the time and the height of the tree. However, the student makes a mistake in answering question 1 (c), the student makes a mistake in giving the interpretation of the answer which he had obtained in part 1 (b). The student is not able to say that at week 11 trees A and B have the same height, namely 30 cm. This condition is an indicator that the student has a weak ability to make interpretation of the results of a mathematical problem solving (four indicators). Mistake of student answer shown figure 1 (and similar mistakes done by other students) support and complement earlier findings that students have difficulty in using and interpreting literal symbols/variables (Goos, 2007: 234; Knuth et al., 2005) and have misconceptions over algebra materials (Alibali et al, 2007; Asquith et al, 2007; MSP, 2003). By looking at the problems provided in the National Examination and also the problems on "student textbook", it is known that the problems presented slight requires students to provide an explanation/ interpretation of the results of the solution of a mathematical problem. These conditions greatly affect the unfamiliarity of students to provide arguments (explanation/interpretation), so the impact on student weakness in this ability. Didi Suhaedi & Tia Purniati / An Analysis of..                                                                        ISBN.  ME-586     Figure 1.  Example of student answer error for problem No. 1 (c)   Problem 2.  Perhatikan gambar berikut ini. yx(0, 5)(5, 0)(0, 2)(-4, 0)A a. Tentukan persamaan kedua garis pada grafik tersebut! b. Jika koordinat titik A adalah (a, b), tentukanlah a dan b !       Proceeding of  International Conference On Research, Implementation And Education  Of Mathematics And Sciences 2014, Yogyakarta State University, 18-20 May 2014            ME-587 Most of students can do on question 2 (a). It can be seen from the degree of success of student for 71.1% (above the minimum of the degree of success of student set by the Education Ministry). But most students wrong in finding solutions of the mathematical models that have it get the answer 2 (a). It can be seen from the degree of success of student of 43.7% (below the minimum the degree of success set by the Education Ministry). Example of this error is shown in the following the figure 2.   Figure 2.  Example of student answer error for problem No. 2 (b)  Figure 2 shows that the student can do a linear equation systems of two variables based on a given graph. But most students make mistakes in algebra operations during the process of completing a linear equation systems of two variables that have it get in answering question 2 (a). These findings support and complement earlier findings that students have difficulty in completing the equation (Asquith et al, 2007) and in line with the findings Kriegler and Lee (2007) that only 22% of eighth grade students in California who demonstrate proficiency in equivalent in a algebra course.  CONCLUSION  The above discussion provides information that students have difficulty mastering the material of linear equation system of two variables in the indicator of determine solution of a mathematical model of mathematical problems (three indicators) and make interpretation of the results of a mathematical problem solving (four indicators). The findings about the weakness of students in determining solution of a mathematical model of mathematical problems, support and complement earlier findings that students have difficulty in completing the equation (Asquith et al, 2007). Meanwhile, students weakness of in explaining or interpreting the results of a mathematical problem solving, this happens because during learning, students are not stimulated and facilitated (teaching materials) that can develop such capabilities. Questions of national examination and questions of the student textbook is not conducive to growth and Didi Suhaedi & Tia Purniati / An Analysis of..                                                                        ISBN.  ME-588  development of students' skills in explaining or interpreting the results of a mathematical problem solving.  REFERENCES  [1] Alibali, M.W. (1999).  “How Children Change Their Minds: Strategy Change Can Be Gradual or Abrupt”,  Developmental Psycholog; 35; 127–145. [2] Asquith, P., Stephens, A.C., Knuth, E.J. (2007).  “Middle School Mathematics Teachers’ Knowledge of Students’ Understanding of Core Algebraic Concepts: Equal Sign and Variable”, Mathematical Thinking and Learning; 9(3); 249 – 272. [3] Departemen Pendidikan Nasional [Depdiknas]. (2008). Kriteria dan Indikator Keberhasilan Pembelajaran, Depdiknas, Jakarta. [4] Falkner, K.P., Levi, L., and Carpenter, T.P.  (1999).  “Children’s Understanding of Equality: A Foundation for Algebra”,  Teaching Children Mathematics; 6; 232 – 236. [5] Goos, M. (2007).  Teaching Secondary School Mathematics : Research and Practice for the 21st Century, Allen & Unwin, Singapore. [6] Ikeda, T. (2007). “Possibilities for, and Obstacles to Teaching Applications and Modelling in the Lower Secondary Levels”. Dalam Blum, W., et al. Modelling and Applications in Mathematics Education: The 14th ICMI Study (pp. 457-462), Kluwer Academic Publishers, New York. [7] Katz, V. J.  (2007).  Algebra: Gateway to a Technological Future, University of the District of Columbia, Columbia. [8] Kaur, B., & Dindyal, J. (2010). Mathematical Applications And Modelling oleh Association of Mathematics Educators, World Scientific, Singapore. [9] Kieran, C.  (2004).  “The Core of Algebra: Replections on its Main Activities”. Dalam Stacey, K., et al.  The Future of the Teaching and Learning of Algebra: The 12th ICMI Study (pp. 21–33), Kluwer Academic Publishers, New York. [10] Knuth, E.J., Stephens, A.C., McNeil, N.M. dan Alibali, M.W. (2006). “Does Understanding the Equal Sign Matter? Evidence from Solving Equations”,  Journal for Research in Mathematics Education; 37(4); 297–312. [11] Knuth, E. J., Alibali, M. W., McNeil, N. M., Weinberg, A., Stephens, A. C. (2005). “Middle School Students’ Understanding of Core Algebraic Concepts: Equivalence & Variable”,  ZDM; 37(1); 68 – 76.  [12] Kriegler, S., and Lee, T. (2007). Using Standardized Test Data as Guidance for Placement into 8th Grade Algebra, University of California, Los Angeles. [13] Lawrence, A., and Hennessy, C.  (2002).  Lessons for Algebraic Thinking: Grade 6 – 8, Math Solutions Publications, Sausalito. [14] Moses, R., and Cobb, C.  (2001).  Radical Equations: Math Literacy and Civil Right., Beacon Press, Boston MA. [15] Mathematics Study Panel (MSP). (2003). Mathematical Proficiency for All Students: Toward a Strategic Research and Development Program in Mathematics Education, RAND, Santa Monica CA. [16] National Research Council [NRC]. (1998). The Nature and Role of Algebra in the K-14 Curriculum.  Washington, D.C : National Academy Press. [17] Niss, M. (2010). “Modeling a Crucial Aspect of Students’ Mathematical Modeling”. Dalam Lesh, R., et al. Modeling Students’ Mathematical Modeling Competencies (pp. 43-59), Springer, New York. 

AN ANALYSIS OF DIFFICULTIES ON MATHEMATICAL MODEL

INTERPRETATION OF JUNIOR HIGH SCHOOL STUDENTS  ON THE MATERIALS

OF TWO-VARIABLE LINEAR EQUATION SYSTEM

AN ANALYSIS OF DIFFICULTIES ON MATHEMATICAL MODEL INTERPRETATION OF JUNIOR HIGH SCHOOL STUDENTS ON THE MATERIALS OF TWO-VARIABLE LINEAR EQUATION SYSTEM

Abstract

Similar works

Full text

Available Versions

Lumbung Pustaka UNY (UNY Repository)