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 10” 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.  Article visualizations

Minh Triet, Le Viet

Phu Loc, Nguyen

Thien Trung, Le Thai Bao

English

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 10” 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

Nguyen Phu Loc

Le Thai Bao Thien Trung

Le Viet Minh Triet

NEUROSURGERY ENTHUSIASTIC WOMEN SOCIETY

LIMITATIONS OF SECONDARY SCHOOL STUDENTS IN  SOLVING A TYPE OF TASK RELATING TO THE EQUATION  OF A CIRCLE: AN INVESTIGATION IN VIETNAM

European Journals of Education Studies

  European Journal of Education Studies ISSN: 2501 - 1111 ISSN-L: 2501 - 1111 Available on-line at: www.oapub.org/edu  Copyright © The Author(s). All Rights Reserved                                                                                                                 103 Published by Open Access Publishing Group ©2015. 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 1School of Education, Can Tho University, Vietnam 2Ho Chi Minh City University of Education, Vietnam 3Pacific College, Can Tho City, Vietnam  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 10‛ 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 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 European Journal of Education Studies - Special Issue Basic and Advanced Concepts, Theories and Methods Applicable on Modern Mathematics Education                 104 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).   2. The equation of a circle in “Geometry 10” of Vietnam  2.1 How to introduce “The equation of a circle” in “Geometry 10” of Vietnam The equation of a circle In the textbook ‚Geometry 10‛ (Hao et al 2007), the equation of a circle is presented in two forms: The first form:  In theOxy , given the circle (C) with center ( ; )I a b  and radius R . The equation of (C) is: 2 2 2( ) ( )x a y b R     (1).                               The second form:  2 2 2 2 0x y ax by c      is the equation of a circle with center ( ;b)I a  and radius  2 2R a b c    2 2 0.a b c     2.2   Types of  related task  Basing on exercises in the Textbook ‚Geometry 10‛, we generalized classified them into types of tasks as follows:  A. Type 1: Verify whether a quadratic equation  2 2 2 2 0x y ax by c       (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 2 2 0a b c    ,  (1) is the equation of a circle with center I(a; b) and radius. If 2 2 0a b c    , (1) is not the equation of any circle.  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 European Journal of Education Studies - Special Issue Basic and Advanced Concepts, Theories and Methods Applicable on Modern Mathematics Education                 105 B. Type 2: Verify whether quadratic equation 2 2 2 2 0kx ky ax by c      (2)   is the circle equation or not?    Technique for solving:  - Converting Type 2 to Type 1 by diving both sides of (2) by k: 2 2 2 2 0a b cx y x yk k k     .  C. Type 3: Find values of a real parameter m in order that 2 2 0x y px qy r      (3) becomes an equation of a circle Technique for solving: - Building the inequation: 2 202 2p qr            . - Finding m by solving the inequation2 202 2p qr            .  D. Type 4: Find coordinates of center and radius of a circle (C):  2 2 0x y px qy r      Technique for solving: - Determining the coordinates of the center Iof (C):  ,2 2p qI     . - Computing the radius of (C):2 22 2p qR r            .  E. Type 5: Find coordinates of center and radius of a circle (C):                  2 2 0x y px qy r      . Technique for solving: - Dividing both sides of (4) by : 2 2 0p q rx y x y       . - (C) has the center ,2 2p qI      , and the radius is 2 22 2p q rR              .  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 2 2( ) ( )A AR IA x a y b     . - The equation of C) is: 2 2 2( ) ( )x a y b R    . 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 European Journal of Education Studies - Special Issue Basic and Advanced Concepts, Theories and Methods Applicable on Modern Mathematics Education                 106 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): ( ; )2 2A B A Bx x y yI .    - Computing the radius of (C):2 2( ) ( )2 2B A B Ax x y yABR    . - The equation of (C) is: 2 2 2( ) ( )2 2A B A Bx x y yx y R     .  H. Type 8 : Write the equation of a circle (C) tangent to the straight line   : 2 20  (A +B 0)Ax By C    .           and whose center is I (a; b). Technique for solving: - Computing2 2. .( , )Aa Bb CR d IA B   . - The equation of (C) is the one after plugging the values of a, b, R into the equation: 2 2 2( ) ( )x a y b R     .  I. Type 9 : Write the equation of a circle (C) whose center I  is on the straight line  : 2 20  (A +B 0)Ax By C    .  and through two points A and B. Technique for solving: - General form of (C) is: 2 2 2 2 0x y ax by c     . - Plugging coordinates of A, B into: 2 2 2 2 0x y ax by c      and coordinates of I into:  2 20  (A +B 0)Ax By C    , 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): 2 2 2 2 0x y ax by c     . - Plugging coordinates of A, B, C into:  2 2 2 2 0x y ax by c     , 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.  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 European Journal of Education Studies - Special Issue Basic and Advanced Concepts, Theories and Methods Applicable on Modern Mathematics Education                 107 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 2 2( ) ( )x a y b 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 equation2 2( ) ( )x a y b k     are the circle equations‛, most students will convert it to the form2 2 2 2 0x y ax by c      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 such that the equation 2 2( 1) ( 2) 2x y m      are the circle equations?‛ 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):  2 2( 1) ( 2) 2x y m      are equations of circle with center (1,2)I  and radius 2R m    if and only if 2 0 2m m     . B. Strategy 2 (correct): 2 2( 1) ( 2) 2x y m     2 2 2 4 3 0x y x y m        (*) (*) are the equations of circle 2 2 0 2 0 2a b c m m          ( 1, 2, 3a b c m     ).  C. Strategy 3 (incorrect): 2 2( 1) ( 2) 2x y m     are equations of circle with center (1,2)I  and radius 2R m    if and only if 2 0 2m m     . This strategy is not correct 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 European Journal of Education Studies - Special Issue Basic and Advanced Concepts, Theories and Methods Applicable on Modern Mathematics Education                 108 because in the case of 2m    , 2 21( 1) ( 2) 0 (3) .2xx yy    Therefore, (3) is not an equation of a circle.  Table 1:  The results of students’ answers Answers of students Strategy 1 Strategy 2 Strategy 3 No solution/don’t find out any strategy to solve Sum The number of students 180 (21.3%) 295 (34. 91%) 89 (10. 53%) 222 (26. 27%) 845 (100%)  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 they converted the given equation into the form 2 2 2 2 0x y ax by c     , then computed m. Below is the answers presented by students (see Figure 1-2).   Figure 1: Written answer of T.M.T   (Class 10, Secondary School ‚Bình Thủy‛, Can Tho)  Figure 2:  Written answers of N.P.N.T (Class 10, Secondary School ‚Trương Định‛, Tien Giang) 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 European Journal of Education Studies - Special Issue Basic and Advanced Concepts, Theories and Methods Applicable on Modern Mathematics Education                 109 Table 1 gave fact that many students (89/845 (10.53%)) used strategy 3 to solve (see Figure3). This strategy led to 2m    (not true if 2m   ).   Figure 3: Written answers of T.H.M.  (Class 10, 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 2 2( 1) ( 2) 2x y m      are the circle equations?”, most students will convert it to the form2 2 2 2 0x y ax by c      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 their students how to find values of k such that 2 2( ) ( )x a y b k     becomes an the equation of a circle. 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.   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 European Journal of Education Studies - Special Issue Basic and Advanced Concepts, Theories and Methods Applicable on Modern Mathematics Education                 110 References  1. Bessot, A. et al, (2010), Những yếu tố cơ bản của Didactic toán, NXB 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. (1999). L ‘analyse des pratiques enseignantes en théorie anthropologique du didactique. Recherches en Didactique des Mathématiques, 19(2), 221–266.  4. Gahamanyi, M. (2010). A Study of Mathematical Organisations in Rwandan Workplaces and Educational Settings. Linköping Studies in Behavioural Science No. 150 Linköping University, Department of Behavioural Sciences and Learning. Linköping 2010  5. Hạo, T.V,Hy, N. M, Đo|nh, N.V., Huyên, T.Đ (2007), Hình Học 10. Publishing house Gi{o dục (in Vietnamese). 6. Henningsen, M. & Stein, M.K.(1997). Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning. Journal for Research in Mathematics Education, 28(5), 524-549.  7. Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. Educational Research Journal, 33(2), 455-488.           Received date January 10, 2017 Accepted date March 2, 2017 Publication date March 9, 2017 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 European Journal of Education Studies - Special Issue Basic and Advanced Concepts, Theories and Methods Applicable on Modern Mathematics Education                 111                                   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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LIMITATIONS OF SECONDARY SCHOOL STUDENTS IN SOLVING A TYPE OF TASK RELATING TO THE EQUATION OF A CIRCLE: AN INVESTIGATION IN VIETNAM

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