European Journal of Education Studies ISSN: 2501 - 1111 ISSN-L: 2501 - 1111 Available on-line at: www.oapub.org/edu 10.5281/zenodo.375965 Special Issue - Basic and Advanced Concepts, Theories and Methods Applicable on Modern Mathematics Education LIMITATIONS OF SECONDARY SCHOOL STUDENTS IN SOLVING A TYPE OF TASK RELATING TO THE EQUATION OF A CIRCLE: AN INVESTIGATION IN VIETNAM Nguyen Phu Loc1, Le Thai Bao Thien Trung2, Le Viet Minh Triet3 School of Education, Can Tho University, Vietnam 1 Ho Chi Minh City University of Education, Vietnam 2 Pacific College, Can Tho City, Vietnam 3 Abstract: In Vietnam, secondary school students learn the equation of a circle in Grade 10. Based on how to present this equation in the textbook Geometry and types of task for students, we believe that some limitations happen to students when they solve problems related to the equation of a circle. This paper reports the investigation of 845 students from the Mekong Delta-Vietnam. The results show that our prediction is correct. Keywords: the equation of circle, type of task, errors in solving problem, mathematics education 1. Theoretical background 1.1 A mathematics task The definition of mathematical task is a classroom activity, project, question, problem, construction, application or exercise in which the purpose is to focus the students learning on a specific mathematical concept (Cai et al., 2010; Stein et al., 1996). According to Gahamanyi, a mathematical task can be viewed, in general terms, as any piece of mathematical work to be done by an individual or a group. In mathematics education, especially in teaching-learning context, a mathematical task normally refers to mathematical work or problems that are assigned to students, teachers or other concerned people (such as parents and mathematics curriculum makers) to be performed for the purpose of societal knowledge development in the subject of Copyright © The Author(s). All Rights Reserved Published by Open Access Publishing Group ©2015. 103 Nguyen Phu Loc, Le Thai Bao Thien Trung, Le Viet Minh Triet LIMITATIONS OF SECONDARY SCHOOL STUDENTS IN SOLVING A TYPE OF TASK RELATING TO THE EQUATION OF A CIRCLE: AN INVESTIGATION IN VIETNAM mathematics. Chevallard (1999) considered a mathematical activity as a human activity situated in an institutional setting and any mathematics activity can be subsumed as a system of tasks. 1.2 The role of mathematics task Mathematical tasks shape what students learn, how students think about mathematics, how students develop their understanding of mathematical concepts, and how students make sense of mathematics. A task is represented in multiple ways, including as it appears in the curriculum, how a teacher adapts the task and how the task is carried out during instruction (Stein et al., 1996). Mathematical tasks provide the contexts in which they learn to think about mathematics, and different tasks may place differing cognitive demands on students (Henningsen & M. K. Stein, 1997). . The equation of a circle in Geometry of Vietnam 2.1 How to introduce The equation of a circle in Geometry of Vietnam The equation of a circle In the textbook Geometry Hao et al 7 , the equation of a circle is presented in two forms: The first form: In the Oxy , given the circle (C) with center I (a; b) and radius R . The equation of (C) is: ( x  a)2  ( y  b)2  R2 (1). The second form: x2  y 2  2ax  2by  c  0 is the equation of a circle with center I (a; b) and radius R  a 2  b2  c  a2  b2  c  0. 2.2 Types of related task ‛asing on exercises in the Textbook Geometry , we generalized classified them into types of tasks as follows: A. Type 1: Verify whether a quadratic equation x2  y 2  2ax  2by  c  0 (1) is the circle equation or not? Technique for solving: - Calculating: a2 + b2- c. - Basing the sign of a2 + b2- c to reach conclusion. If a 2  b2  c  0 , (1) is the equation of a circle with center I(a; b) and radius. If a 2  b2  c  0 , (1) is not the equation of any circle. European Journal of Education Studies - Special Issue Basic and Advanced Concepts, Theories and Methods Applicable on Modern Mathematics Education 104 Nguyen Phu Loc, Le Thai Bao Thien Trung, Le Viet Minh Triet LIMITATIONS OF SECONDARY SCHOOL STUDENTS IN SOLVING A TYPE OF TASK RELATING TO THE EQUATION OF A CIRCLE: AN INVESTIGATION IN VIETNAM B. Type 2: Verify whether quadratic equation kx2  ky 2  2ax  2by  c  0 (2) is the circle equation or not? Technique for solving: - Converting Type 2 to a b c x2  y 2  2 x  2 y   0 . k k k Type 1 by diving both sides of (2) by k: C. Type 3: Find values of a real parameter m in order that x2  y 2  px  qy  r  0 (3) becomes an equation of a circle Technique for solving:  p  q  - Building the inequation:       r  0 .  2   2  2 2  p  q  - Finding m by solving the inequation       r  0 .  2   2  2 2 D. Type 4: Find coordinates of center and radius of a circle (C): x2  y 2  px  qy  r  0 Technique for solving:  p q  - Determining the coordinates of the center Iof (C): I  ,  .  2 2   p  q  - Computing the radius of (C): R        r .  2   2  2 E. Type 5: Find coordinates  x2   y 2  px  qy  r  0 . 2 of center Technique for solving: - Dividing both sides of (4) by  : x 2  y 2   p x  q y and  r radius of a circle (C): 0. r q   p   q   p - (C) has the center I  ,     .  , and the radius is R    2   2    2 2  2 2 F. Type 6: Write the equation of a circle (C) with center I (a; b) and through a point A (xA; yA) Technique for solving: - Computing the radius of the circle R  IA  ( xA  a)2  ( y A  b)2 . - The equation of C) is: ( x  a)2  ( y  b)2  R2 . European Journal of Education Studies - Special Issue Basic and Advanced Concepts, Theories and Methods Applicable on Modern Mathematics Education 105 Nguyen Phu Loc, Le Thai Bao Thien Trung, Le Viet Minh Triet LIMITATIONS OF SECONDARY SCHOOL STUDENTS IN SOLVING A TYPE OF TASK RELATING TO THE EQUATION OF A CIRCLE: AN INVESTIGATION IN VIETNAM G. Type 7 : Given A (xA; yA) and B (xB; yB). Write the equation of a circle (C) with diameter AB. Technique for solving: - Determining the coordinates of center of (C): I ( xA  xB y A  yB ; ). 2 2 ( xB  xA )2  ( yB  y A )2 AB . - Computing the radius of (C): R   2 2 x x y  yB 2 - The equation of (C) is: ( x  A B )2  ( y  A )  R2 . 2 2 H. Type 8 : Write the equation of a circle (C) tangent to the straight line  : Ax  By  C  0 (A2 +B2  0) . and whose center is I (a; b). Technique for solving: - Computing R  d ( I , )  A.a  B.b  C A2  B 2 . - The equation of (C) is the one after plugging the values of a, b, R into the equation: ( x  a)2  ( y  b)2  R2 . I. Type 9 : Write the equation of a circle (C) whose center I is on the straight line  : Ax  By  C  0 (A2 +B2  0) . and through two points A and B. Technique for solving: - General form of (C) is: x2  y 2  2ax  2by  c  0 . - Plugging coordinates of A, B into: x2  y 2  2ax  2by  c  0 and coordinates of I into: Ax  By  C  0 (A2 +B2  0) , we obtain system of 3 equations in three unknowns: a, b, c. Solving this system to find out a, b, c. J. Type 10: Write the equation of a circle (C) passing through three points A, B and C. Technique for solving: - General form of (C): x2  y 2  2ax  2by  c  0 . - Plugging coordinates of A, B, C into: x2  y 2  2ax  2by  c  0 , we obtain system of 3 equations in three unknowns: a, b, c. Solving this system to find a, b, c. K. Type 11: In coordinate plane Oxy, find a set of points which satisfies property p, where the solution is a circle. European Journal of Education Studies - Special Issue Basic and Advanced Concepts, Theories and Methods Applicable on Modern Mathematics Education 106 Nguyen Phu Loc, Le Thai Bao Thien Trung, Le Viet Minh Triet LIMITATIONS OF SECONDARY SCHOOL STUDENTS IN SOLVING A TYPE OF TASK RELATING TO THE EQUATION OF A CIRCLE: AN INVESTIGATION IN VIETNAM Technique for solving: - Let M(x; y) be in the set of points to find out. - Make use of the property p, we can deduce an equation F(x; y) = 0. - Prove F(x; y) = 0 to be the equation of a circle. 3. Research hypothesis Basing on how to present the equation of a circle (but not to show in which case an equation ( x  a)2  ( y  b)2  k becomes an equation of a circle) and types of related task as above, we formulated the research hypothesis as follows: Hypothesis: For the problem Find the values of real parameter k such that the equation ( x  a)2  ( y  b)2  k are the circle equations , most students will convert it to the form x2  y 2  2ax  2by  c  0 to solve; if not, they will commit errors or not to know how to solve. 4. Methodology The problem to investigate: In order verify the above hypothesis, we used the following problem: Find the values of real parameter ( x  1)  ( y  2)  m  2 are the circle equations? 2 such that the equation 2 The investigated students: 845 students from 5 secondary schools in the Mekong Delta – Vietnam. Time: The investigation was carried out in 2015. 5. The results and discussion After analyzing the students answers, we found out three following strategies used: A. Strategy 1 (correct): ( x  1)2  ( y  2)2  m  2 are equations of circle with center I (1, 2) and radius R  m  2 if and only if m  2  0  m  2 . B. Strategy 2 (correct): ( x  1)2  ( y  2)2  m  2  x2  y 2  2 x  4 y  m  3  0 (*) (*) are the equations of circle a2  b2  c  0  m  2  0  m  2 ( a  1, b  2, c  m  3 ). C. Strategy 3 (incorrect): ( x  1)2  ( y  2)2  m  2 are equations of circle with center I (1, 2) and radius R  m  2 if and only if m  2  0  m  2 . This strategy is not correct European Journal of Education Studies - Special Issue Basic and Advanced Concepts, Theories and Methods Applicable on Modern Mathematics Education 107 Nguyen Phu Loc, Le Thai Bao Thien Trung, Le Viet Minh Triet LIMITATIONS OF SECONDARY SCHOOL STUDENTS IN SOLVING A TYPE OF TASK RELATING TO THE EQUATION OF A CIRCLE: AN INVESTIGATION IN VIETNAM x  1 because in the case of m  2 , ( x  1)2  ( y  2)2  0 (3)   . Therefore, (3) is not an y  2 equation of a circle. Table 1: The results of students answers Answers of Strategy Strategy Strategy No solution/don’t find out any students 1 2 3 strategy to solve The number of 180 295 89 222 845 students (21.3%) (34. 91%) (10. 53%) (26. 27%) (100%) Sum Statistical table 1 shows that strategy 2 was dominant strategy where 34.91% of students, namely 295 (out of 845) students, used to solve the problem, which means that 2 2 they converted the given equation into the form x  y  2ax  2by  c  0 , then computed m. Below is the answers presented by students (see Figure 1-2). Figure 1: Written answer of T.M.T Class , Secondary School ‛ình Thủy , Can Tho Figure 2: Written answers of N.P.N.T (Class 10, Secondary School Trương Định , Tien Giang European Journal of Education Studies - Special Issue Basic and Advanced Concepts, Theories and Methods Applicable on Modern Mathematics Education 108 Nguyen Phu Loc, Le Thai Bao Thien Trung, Le Viet Minh Triet LIMITATIONS OF SECONDARY SCHOOL STUDENTS IN SOLVING A TYPE OF TASK RELATING TO THE EQUATION OF A CIRCLE: AN INVESTIGATION IN VIETNAM Table 1 gave fact that many students (89/845 (10.53%)) used strategy 3 to solve (see Figure3). This strategy led to m  2 (not true if m  2 ). Figure 3: Written answers of T.H.M. Class , Secondary School Trương Định , Tien Giang Table 1 also showed that 222 (26, 27%) students did not know how to solve the given problem. From the above analysis, the research hypothesis is true. It means that to solve that problem Find the values of real parameter such that the equation ( x  1)  ( y  2)  m  2 are the circle equations? , most students will convert it to the form 2 2 x2  y 2  2ax  2by  c  0 to solve; if not, they will commit errors or not to know how to solve. 6. Conclusion The results of the above investigation, in our opinion, students made some errors and chose the strategy 2 to solve the given problem (not optimal) because of the following reasons: 1. The teachers depended much on contents of knowledge presented in the textbook. They do not actively complement the knowledge and skills necessary for teaching students in a mathematical knowledge; in this case, the teacher should guides 2 2 their students how to find values ( x  a)  ( y  b)  k becomes an the equation of a circle. of k such that 2. Students often have a habit of doing what the teacher teach and inability to apply flexibly in specific situations. Therefore, in the process of teaching mathematics, teachers should have pedagogical measures to overcome the above limitations in order to increase effectiveness of their own teaching. European Journal of Education Studies - Special Issue Basic and Advanced Concepts, Theories and Methods Applicable on Modern Mathematics Education 109 Nguyen Phu Loc, Le Thai Bao Thien Trung, Le Viet Minh Triet LIMITATIONS OF SECONDARY SCHOOL STUDENTS IN SOLVING A TYPE OF TASK RELATING TO THE EQUATION OF A CIRCLE: AN INVESTIGATION IN VIETNAM References 1. Bessot, A. et al, (2010), Những yếu tố cơ bản của Didactic toán, NX‛ Đại học quốc gia TP. Hồ Chí Minh In Vietnamese 2. Cai, J., Moyer, J. C., Nie, B., & Wang, N. (2010) Learning mathematics from classroom instruction using standards-based and traditional curricula: An analysis of instructional tasks. Paper presented at the annual meeting of the American Educational Research Association. Denver, CO, April 29-May 3, 2010. 3. Chevallard, Y. 999 . 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Received date January 10, 2017 Accepted date March 2, 2017 Publication date March 9, 2017 European Journal of Education Studies - Special Issue Basic and Advanced Concepts, Theories and Methods Applicable on Modern Mathematics Education 110 Nguyen Phu Loc, Le Thai Bao Thien Trung, Le Viet Minh Triet LIMITATIONS OF SECONDARY SCHOOL STUDENTS IN SOLVING A TYPE OF TASK RELATING TO THE EQUATION OF A CIRCLE: AN INVESTIGATION IN VIETNAM Creative Commons licensing terms Author(s) will retain the copyright of their published articles agreeing that a Creative Commons Attribution 4.0 International License (CC BY 4.0) terms will be applied to their work. 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