Bal
Chandra Luitel
Curtin University of Technology
School-mathematics-curricula of Nepal have given emphasis on geometry learning from the beginning of schooling. The curricula have aimed at developing students’ understanding of intended geometric concepts at primary, lower secondary and secondary level. Moreover, the school curriculum has emphasized child-centred methods of teaching to promote the understanding of geometry learning. Similarly, according to the National Council of Teachers of Mathematics, geometry is one of the “content standard” of school mathematics, which aims at developing spatial reasoning, problem solving skills, and communication (Sellke, 1999). Thus, geometry is regarded as a core content area of school mathematics.
The teaching and learning situation is not the same in all schools of Nepal. On the one hand, the majority of government schools have been facing the problems of quality in teaching. On the other hand, some private schools have been implementing student-centred teaching strategies in mathematics teaching. As a result, geometry teaching and learning situations vary accordingly.
There are some examples of improving teaching-learning situations in Nepal. The “Dhulikhel experience” is one of the examples of improving quality of teaching and learning in the schools in which some activities were designed in order to improve geometry teaching and learning situations. These activities can be better examples for Nepalese schools.
Dhulikhel is one of the municipalities of Nepal, situated to the north of Kathmandu Valley at a distance of twenty-eight kilometres from the capital city. There are fourteen government schools, two colleges and one university (Mumaw, 2001). Since the quality of teaching and learning was not good in the government schools, Dhulikhel municipality and Kathmandu University reached an agreement to carry out a project, Quality Education Project to improve teaching and learning situation in the schools of Dhulikhel Municipality situated in the vicinity of the Municipality. The project started in 1997 when the Royal Danish Embassy (Danida) approved to provide grants.
The main focus of the project was to improve the teaching-learning situation in the schools through providing in-service training for teachers and improving physical facilities of the schools. The modality of the teacher training was school-based in which most of the training activities were conducted at schools. The “Quality Education Project” is the first of its kind in Nepal in which a university and a local authority have worked together to improve teaching and learning situation in the government schools. Now, this project is in the phase of extension to other municipalities and Village Development Committees in the surrounding area of Kathmandu University.
Mathematics was one of the subjects of teacher training programme in which the following activities were designed, such as classroom observation, demonstration teaching, case studies, students project, student and teacher writing, constructing teaching materials (manipulative) and teachers’ meeting. During the training programme, it was revealed that some extra support was necessary to improve the teaching and learning of geometry at lower secondary and secondary level. In response to this situation, some teaching and learning strategies were implemented in the schools of Dhulikhel municipality. This article will seek to explain the major issues of teaching and learning geometry and discuss the ways of improving geometry teaching in Nepalese Schools with reference to “Dhulikhel experience”.
There are mainly three issues in teaching and leaning geometry in reference to Nepalese Schools. These are: emphasis on learning geometry, contextualisation of learning geometry and change from the traditional one-way classroom to two-way interactive one.
Regarding an emphasis on learning geometry the secondary school curriculum has mentioned the four strands of learning such as knowledge, application, problem solving and comprehension (Curriculum Development Centre, 1999). “The knowledge strand” requires the learners to know definitions, facts and formulae and the emphasis of “application” is on transfer of learning into a novice situation (Curriculum Development Centre, 1999, p. 34). The “problem-solving strand” aims at developing an exposure of use of geometry to solve the day-to-day problems and the fourth strand aims at developing comprehension of geometric concepts, their relationships and structure (Curriculum Development Centre, 1999, p. 38).
While talking about the emphasis on teaching and learning geometry, the curricular objectives are still insufficient to address the two aspects of the changing context. Firstly, the curricula do not have a focus on “communication”. According to the National Council of Teachers of Mathematics, importance and use of communication in mathematics classroom, is necessary to increase students' reading, writing, discussing, representing, and modelling mathematics, because, when students communicate their ideas, they learn to clarify, refine, and consolidate their thinking (Perry, 2001). Secondly, the curricula also lack an emphasis on “spatial reasoning”. Spatial reasoning helps develop the understanding of everyday applications, for example, giving and receiving directions and reading maps, understanding of two and three dimensional objects, working with coordinates and graphing (Lindquist & Clements, 2001).
The second issue of geometry leaning is contextualisation. The term “contextualisation of learning” infers that learning can be promoted by meaningful contexts and relating instruction to the real-life situation. The learning in Nepalese schools is totally based on textbooks, which have been prepared according to the school curriculum. On the one hand, since the textbooks have been written in formal Nepali language, it is difficult for those students who have other language-speaking background than Nepali-in Nepal different local and ethnic languages are spoken for example, Newari, Maithili, Gurung, Rai, Limbu, Tamang, Sherpa and Magar. On the other hand, the teachers use the textbook as an ultimate means of teaching that does not provide the opportunity of relating their learning with local context. In order to minimise this problem, teachers and students can construct a set of glossary of local languages (Koirala, 2000), which can be useful to learn the geometric concepts.
While talking about the contextualisation of geometry learning, it is important to identify the socio-economic background of the students. Since ninety percent of Nepalese’ occupation is about agriculture, the activities of geometry teaching and learning should be made contextual with agricultural community. Similarly, it is important to identify the extent of contextualisation of the curricular contents. (Taylor & Mulhall, 1997).
The third issue is related to the ways of teaching. In most of the Nepalese schools students have less chance to interact with their peers and teachers. They have to listen to the teachers’ idea. The crowded classroom is one of the major problems of implementing interactive teaching and learning situation. Of course, an interactive classroom should provide opportunity to the students to talk, to question, to present their ideas. Regarding interactive mathematics classroom, Alper et al (1995) have mentioned as:
“All the desks are turned to face each other. The students are writing with felt-tipped markers on butcher paper. They are looking at one another’s graphing calculators. The teacher is nowhere to be seen. Oh, there he is, sitting down with one of the groups. A student has walked away to from her group to confer with another group. She never asked for permission.” (Alper, Fendal, Fraser, & Resek, 1995; p. 1)
The aforementioned situation reflects the scenes of an interactive classroom of one of the schools of the United States. On the contrary, the classroom situation of Nepalese schools has been mentioned by Luitel (1999) as:
“The classroom was appropriate for thirty students. However, there were more than fifty students. …The teacher was talking and no sound was listened from the side of students in thirty minutes.” (Luitel, 1999, p 6)
By comparing these two cases, it reveals that Nepalese schools need to manage the classrooms to make interactive ones.
In order to improve teaching and learning situation, the aforementioned issues should be addressed by suggesting possible teaching learning strategies. As a consequence of Dhulikhel experience, four strategies are discussed to improve geometry teaching and learning situation in Nepalese schools such as: group investigation, writing in geometry, problem solving and use of locally available materials.
Group-investigation is one of the strategies to be implemented for improving teaching and learning situation in Nepalese schools. There are three advantages of group investigation such as, group investigation helps the students get into the learning situation; it gives an opportunity for the learners to develop more socially accepted ideas; and it promotes communication and interaction among the learners. Furthermore, when students work together on mathematical problems, they express the positive views about mathematics, are more likely to be engaged, and show internal motivation to do well (Steele, 1993; cited in Pandiscio, 2001).
In the case of schools of Dhulikhel Municipality, the group investigation was carried out as one of the strategies to change the situation of geometry learning. It comprised of four steps. Firstly, the “investigation-groups” were formed consisting of four to six students with different ability levels. The groups were named after the name of renowned mathematicians such as Euclid, Newton, Leibnitz, and Gauss. Secondly, the teacher had prepared the different tasks to be investigated and was assigned to each group. Thirdly, the students carried out the main investigation. Normally, they were given two weeks to complete the investigation. Finally, they had to make a presentation about their findings, which was followed by a discussion. Some of the tasks of such group investigation were:
q Angle sum of a triangle
q Relationship of the opposite and adjacent angles of parallelogram
q Exterior angle, interior angle and angle sum of regular polygon
q Area of triangle, rectangle, circle, trapezoid, parallelogram
q Symmetry
q Tessellation of pentagon and hexagon
Several project activities were carried to develop the students learning in geometry (Brown, 1999). The students were assigned the “project” to investigate, prepare write-ups and make presentation. The main topics of the “project” were: similarity of triangles by side-angle-side, similarity and area, reflection of a geometric shape, trapezoid and their properties, and angle-angle similarity (Brown, 1999). Two cases of group investigation mentioned by Wohlhuter (1998) were carried out by high school students. Both cases had been conducted in a sequence of group investigations, group presentations and guided discussions (Wohlhuter, 1998, p. 2).
Comparing with other cases of group investigation, Dhulikhel experience was different in three aspects. Firstly, the investigating groups were the same for the whole academic year. Secondly, the students were given a longer time and thirdly they had to carry out the entire task outside the classroom.
This experience may help change teaching and learning scenario if it is carried out as a regular classroom activity in other schools of Nepal. The effectiveness of the group investigation depends upon the selection of tasks and process of investigation. The teachers should be active in monitoring and facilitating their activities rather than in transmitting “the knowledge”.
Writing is a unique mode of learning because it connects three major tenses of our experiences to make meaning by shuttling between past, present and future (Emig, 1983 cited in Menon, 1998, p.19). In general, there are two types of writing in mathematics known as formal and informal. The formal writing comprises of writing geometric proofs, description of experimental verification and solution of mathematical problems. In formal writing, the students should use appropriate symbols and metaphors to represent their ideas. Of course, the expected outcome of school curricula is to develop the skill of formal writing. The informal writing does not seek to use the appropriate format as in formal writing. In most of the cases the informal writings are all about the reflection of understanding of learning. In the case of “Dhulikhel experience”, it was emphasised that students write their feeling about ongoing classroom activities, geometric concepts and the investigation activities.
The “Dhulikhel experience” on students’ writing has revealed three important consequences. Firstly, the student writing helps assess themselves. Secondly, it is a basis of getting into a classroom discourse. Thirdly, it is a way of improving their formal writing because the informal writing is useful in selecting appropriate metaphors for mathematical communication.
A similar case of writing has been reviewed by Shield (1998). He found that students understanding of mathematics was improved as a result of writing activities (Shield, 1998; p 3). Similarly, students writing in the classroom can be promoted through the problems and problematic situation (Manester & Schlesinger, 1999). In Dhulikhel experience, the focus of informal writing was mainly on students’ reflection about their particular learning activities and geometric concepts. In one student’s writing it was mentioned that, “ Today was the turn of Newton-group. Their presentation on properties of triangle was very good. …. I found my definition of triangle was wrong…”
A second experience was about the teachers’ writing. However, only a few teachers had participated in this activity for the whole academic year. There was no ultimate writing format. Teachers, who continued to write, mentioned about their daily classes, students’ understanding, and colleagues’ opinion regarding the teaching issues. In case of geometry teaching and learning situation one teacher expressed by writing “This year my students has become more active. When I remembered last year, it was very difficult for me to make speak them.”
There are different types of teachers’ writing. “Episode writing”(Schifter, 1999) is useful to develop a set of teaching methods. Similarly, the episodes can be good examples for the novice teachers. The teachers writing also presents the way of developing their teaching style and it helps bridge the gap between theory and practice (Schifter, 1999). In Nepalese contexts the teachers writing can be a very important ways of developing teaching techniques in geometry. Since the teachers write their reflection about day-to-day classes, this becomes a useful resource for the other teachers, especially, for novice ones.
One of the objectives of school-mathematics is to develop a problem solving abilities of the learners (Curriculum Development Centre, 1999). Problem solving is regarded as a learning activity in which the learner is provided an opportunity to apply the understanding of concepts and to be trained to think creatively (Schoenfeld, 1992). Regarding the importance of a mathematical problem Stanic and Kilpatric (cited in Schoenfeld, 1992) have identified five roles that problems play, which are: as a justification of teaching mathematics, as specific motivation for subject topics, as recreation, as a means of developing new skills and as practice (Schoenfeld, 1992, p 338). Furthermore problem solving is one of the dominant strategies used in mathematics teaching and learning for many years.
In the case of the schools of Dhulikhel municipality, problem-solving strategy was identified and implemented as a regular activity of geometry teaching and learning. The students were encouraged to put the problems and to find the solutions. Eventually, this activity was reported to be very useful in developing the conceptual understanding. Besides this, problem solving helps introduce the geometry in day-to-day life.
Broadly speaking, problem-solving strategy was carried out in four steps. Firstly, problems or problematic situation was created, secondly students had to work in their group to find the solution, thirdly they put their solution ahead of the class and fourthly, a discussion was carried out to discuss about the problem and solution. In such a case sometimes the problems used to be open-ended; and students had to investigate for the multiple solutions.
In a case of Pandiscio (2001), the students were given a grid paper, and they were asked to divide it into exactly two polygons of equal area (Pandiscio, 2001, p. 1,). Furthermore, it has also been suggested that the open ended problems help develop the “creativity”(Carroll, 1998, cited in Pandiscio, 2001). For instance, the problems, like, “construct a triangle and divide it in a half way” could be very interesting.
Comparing the Dhulikhel experience with the aforementioned case, it was not much distinct in terms of the aims of problem-solving activities. The difficulties of problem-solving activities in the schools of Dhulikhel municipality were related to the number of students and the teachers’ traditional beliefs. The number of students in a classroom was about sixty, which was very difficult for managing such activities. The teachers were trained in the traditional way of teaching. They had different conceptions about problem-solving activities. For example, they thought to start the activity by giving a sample solution and they did not have the idea of presentation of the students. Despite the difficulties, all the teachers and students gained from the problem solving strategies.
The immediate solution of crowded classrooms, limited space and lack of teaching-learning multimedia and software was the use of locally available materials. The wax-match-box, newspapers, bamboo, wooden models were the examples of locally available materials. An activity planner was prepared to collect, develop and store the teaching materials. Students collected and constructed different solid models. Similarly from the bamboo, they prepared a set of virtual-two-dimensional shapes.
This activity had another aspect such as, each students had to maintain a “math bag” in which they had to bring manipulatives which were essential in learning activities. Most of the manipulatives were constructed from the low-cost and no-cost materials. For instance, in geometry class, they often had to bring scissors, virtual-two-dimensional shapes, solid models and mirrors, straightedge, protractor.
In my experience, this is an effective localised strategy of the geometry learning in Nepalese schools. Most importantly, it is a way of motivating the students to learn geometry.
The three issues of geometry learning in Nepalese schools are: emphasis of geometry learning, contextualisation of learning and change from the traditional one-way classroom to two-way interactive classrooms. Those issues have been addressed on the basis of the experience of Quality Education Project in Dhulikhel Municipality, which has been labelled as “Dhulikhel experience”.
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Bal Chandra Luitel
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