The use of geometric dynamic softwares by teachers in teaching mathematics at secondary school in Vietnam

THE USE OF GEOMETRIC DYNAMIC SOFTWARES BY TEACHERS IN TEACHING MATHEMATICS AT SECONDARY SCHOOL IN VIETNAM Assoc. Prof. Dr. NGUYEN Chi Thanh1 Abstract New technologies, in particular dynamic geometrical software, have been introduced in the teaching and learning of mathematics at secondary school in recent years in Vietnam. However, the institution of secondary education does not yet seem to be favourable to this introduction. In this paper, after describing the main characteristics of technological context and geometrical curriculum in secondary school in Vietnam, we analyse some examples of teacher use of dynamic geometry software in class activities. This analysis permits us to characterize different knowledge types as well as didactic functionalities of this software in teaching practice. Finally, we focus on a case study in order to bring out some elements of the genesis of technological professional use by one teacher. Keywords: Mathematics teaching, Dynamic Geometric software, teaching activity Introduction In Vietnam, the Ministry of Education and Training (MOET) pays a lot of attention on using new technologies for teaching and learning. The equipments is being deployed for a range of purposes including improving school information systems and teaching with new technologies. For instance, according to the MOET, at the end of 2013, about 96% of the upper secondary schools in Vietnam were connected to the Internet, large budgets were available for equipment and national conferences about using technology in education were regularly held. In mathematics, new technologies used are essentially non-graphic calculators and geometry software. For example, there is the introduction of using calculators such as the Casio Fx 500MS in the mathematics text books at grade 10. Since 2005, there are 3 sessions of 45 minutes each at the end of the school year on using calculators. In co- operation with the Casio calculator company, since 1996 the MOET organizes an annual national competition utilising the Casio calculator for “talented” students. The general aims of using the calculator are to verify results and to aid calculations (Nguyen N. H., 2009). Geometry softwares such as Cabri 2D, Cabri 3D have been introduced in 1 Faculty of Pedagogy, VNU University of Education, Vietnam National University, Hanoi. Email: thanhnc@vnu.edu.vn 22 KỶ YẾU HỘI THẢO QUỐC TẾ LẦN THỨ NHẤT VỀ ĐỔI MỚI ĐÀO TẠO GIÁO VIÊN classroom teachers’ practices in some urban regions of Vietnam since around 2000. In this paper, we would like to explore exactly how ICT has been used by teachers. What are the main characteristics of geometry in the mathematical curricula at upper secondary school in Vietnam? Regarding geometric softwares used by mathematic teachers, what are its main didactic functions? Curriculum of geometrical in space at upper secondary school in Vietnam The system of education in Vietnam underwent reforms in 1990, in 2000 and in 2005. The new secondary school curiculum (called the curriculum after 2015) will be implemted from the academic year 2020-2021. But “These reforms however seem to focus more on teaching content and organisation than on the transformations of teaching practices.” (Bessot A. and Comiti C., 2003). Content such as vectors in geometry, computational science, integral calculus and mathematical statistics were introduced. One of the aims of theses reforms was to reduce the highly theoretical nature of the curriculum and introduce aspects of applications of mathematics to practical real-life problems. This is captured in the teacher’s guidebook for Mathematics at grade 10 as “Learning mathematics at secondary schools should help pupils in training skills regarding solving mathematical problems and applying mathematics in real life”. The number of problems involving application of mathematics to real life situations is, however, still very small. For example, an analysis of the mathematics textbook at grade 10 reveals that in algebra there are only 5.4% and in geometry 2.5% of the problems are of an applications nature. The content of geometry has been introduced at grade 11 with two main chapters regarding respectively parallelism and orthogonality. The number of sessions devoted in the curriculum to each chapter is approximately equal. We show, for instance, the content of the orthogonality chapter in the following table: Content Session (45 minutes) §1. Two perpendicular lines §2. Line perpendicular to a plan §3. Two plans orthogonal §4. Distance §5. Angle §6. Revision 1 lesson + 1 training 1 + 1 2 + 1 2 + 1 2 + 1 4 Table 1. Contents and of corresponding time in the chapter “orthogonality” in the geometrical textbook at grade 11 in Vietnam (the program in 2000) We have found out 38 official exercises in these two chapters. The following table shows the type of tasks and the number of exercises corresponding to each: 23Phần 1: NHỮNG VẤN ĐỀ VỀ ĐỔI MỚI GIÁO DỤC VÀ ĐÀO TẠO GIÁO VIÊN Type of task Number To prove that two lines are perpendicular To prove that a line is perpendicular to a plan To prove that two plans are orthogonal To caculate distance, angle measurement To dertemine a locus 14 7 11 30 4 Table 2. Type of tasks and corresponding numbers in the two chapters of the geometry in space in text book at the grade 11 in Vietnam (the program in 2000) As there are always several types of task in an exercise, the total number of types of task (64) is more important than the number of exercises. This table also shows that the curriculum pays attention to the calculation type of task. The calculative method is even introduced in the demonstration problem as shown in the following example. Figure 1. An example of resolution given in the textbook at grade 11 (p.57) This example and others in the textbook show clearly that in accordance with the directive of curriculum, the text of the knowledge offered in the party “lesson” of the textbook of grade 11 is really organized to a classical hypothetical-deductive process. Indeed, axioms are explicitly named and differentiated from theorems or properties. The order of presentation satisfies fully the cannons of a hypothetical-deductive presentation. For the grade 11 textbook, theorems are all demonstrated, and a theorem is a consequence of axioms and previously deduced theorems. Therefore, the geometry aspects of the curriculum at grade 11 are taught in a theoretical way in which calculative methods are emphasized. National examination at the end of secondary school Vietnam has an extensive state-controlled network of schools, colleges and universities but the number of privately-run and mixed public and private institutions is also growing. There has been an increase in the number of students completing general education, although the limited number of universities makes it difficult for them to continue further education after completing school. Secondary school education ends with a national examination for 24 KỶ YẾU HỘI THẢO QUỐC TẾ LẦN THỨ NHẤT VỀ ĐỔI MỚI ĐÀO TẠO GIÁO VIÊN which the success rate has been between 70% and 80% in recent years (75.96% in 2008). Students also need to pass competitive examinations to enter universities and the success rate is about 25 %. These examinations are content-driven and have a high level of difficulty. In order to prepare students for these examinations, supplementary courses are given by teachers at universities. The competitive nature of these examinations and the emphasis on manipulative skills of complexly composed problems, requiring “exact” rather than approximate answers, mitigates against the use of calculators. The geometry content of the university entrance examination deals with analytical geometry. We assume that the use of geometry software remains optional during mathematical activities in teacher’s practice. Teacher’s pratice in using geometrical sofwares In this session we describe some activities of the lesson entitled “two lines which are not at the same plane and two parallel lines” in the first chapter about parallelism. This lesson has been taught by a teacher at Chu Van An upper secondary school at Hanoi. The teacher has designed 4 activities within a 45 minutes session in the computer room. Cabri 3D has been installed on each computer. These activities and some teacher’s notes are given in the following table: Activity 1 (10 minutes). To construct a cube ABCDA’B’C’D’ with Cabri 3D and to comment on the relative position couples of lines AB and BC, AB and CD, AB and DD’? To make some remarks about the common point of these lines.Note: From this activity I will give the definition of two lines which are in the same plane and the one of two lines which are not the same plane. And then I ask student to find out other couples of line in the given cube which are not the same plane. Activity 2 (5 minutes). Euclid axiom in the space. Note: I will guide students to construct a line passing a point and parallel to a given line by using Cabri 3D and then for demonstration of this theorem, I will remind students the Euclid axiom in the geometry in plane. Cabri 3D is used to illustrate the Euclid axiom in the space. Activity 3 (10 minutes). Theorem: “The intersection line of three planes”. To construct a cube by using Cabri 3D. To consider any three planes determined by the given cube and to construct the intersection of these planes and then comment on this intersection. And I will read the theorem given in the textbook. Note: I will guide students to discover the theorem about the intersection of three planes in space. Activity 4 (20 minutes). Example. Given a pyramid SABCD. The base ABCD is a parallelogram. 1- (Without Cabri 3D) To construct the intersection line of the planes (SAB) and (SCD). 2- M is a point of the segment SA. To construct the intersection beween the pyramid and the plane (MBC). 3- To use Cabri 3D to verify the results. Note: If we have not got access to Cabri 3D, usually we give this exercise to students, therefore they receive it in a passive manner. With Cabri 3D, students are not only the ones who solve the problem but also the ones who discover the problem. This is very important in developing students’ thinking and in stimulating motivation. 25Phần 1: NHỮNG VẤN ĐỀ VỀ ĐỔI MỚI GIÁO DỤC VÀ ĐÀO TẠO GIÁO VIÊN We can see that the first use of the geometrical software Cabri 3D consists essentially of illustrating properties, axioms and demonstrations in geometry. The second use is to help students to discover a theorem and geometric properties through the dragging mode and “sphere of glass” functionality of Cabri 3D. These two uses of Cabri 3D are somehow representative in the teacher’s practice in Vietnam, at least in urban areas where ICT use is possible. But the question of instrumental genesis (Lagrange 2011) of Cabri 3D in a teacher’s practice is raised because after each figure constructed using Cabri 3D, this teacher has noted that“As the 3D softwareconsiders ob- jects to be virtual objects and not as the rep- resentative figure therefore the opposite fig- ure does not follow the rules of presentation in perspective”. Figure 2. An example of activities with Cabri 3D given by a teacher Conclusions New technologies, mathematical softwares, have been introduced in teaching and learning of mathematics at secondary school in recent years in Vietnam.However, much needs to be done to realise the potential of ICT within mathematics teaching and learning. Some of the issues needing attention are: the incorporation of digital technologies for mathematics teaching and learning in pre-service teacher education courses at universities (Tapan 2011); changes to mathematics curriculum and examination content at the end of upper secondary school which take due consideration of digital technologies; change in assessment practices, awareness-building and familiarisation of practising teachers and educational decision-makers of the benefits of digital technologies for teaching and learning mathematics and researchers to connect with other researchers and research activities across the world. References 1. 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