An exercise using the computer as a teaching-learning tool in an under-resourced environment
Anthea Roberts (Grassy Park High)
Shaheeda Jaffer (Mathematics Education Project, UCT)
Cabri geometre is a dynamic drawing package which allows users to construct geometrical figures. The package is DOS-based and can even be used on unsophisticated computers such as XT’s and 286 computers. The program is very easy to use, allowing pupils who are not computer literate to adapt to the program very quickly.
In Cabri you can set up a construction and then move the figure around the screen. By creating movement the programme brings static figures to life This makes it a powerful tool in establishing and understanding geometric relationships. Many teachers can identify with the frustration when they see pupils refer to a line as "angle FE". By working with angles and lines on screen, and combining them to make geometric figures, pupils gain a deeper, clearer understanding of the difference between a line and an angle. Pupils can generate a variety of permutations of the same figure through manipulation.
Textbooks normally represent standard sketches in standard orientations. For example, an isosceles triangle is often represented with its two equal angles at the base. Subsequently, pupils develop misconceptions about geometrical figures and experience difficulty in analysing sketches which are drawn in orientations which they are not familiar with. Many pupils lack the skill of visualisation and the ability to transform shapes in their minds. Using Cabri enables the pupils to manipulate the sketch, to turn it upside down or onto its side. They can also drag points which leads to a change in angle or line measurements, thus producing several variations of the same geometrical figure. At the end of the day the pupils have a far deeper perspective of the geometric relationships and are encouraged to make conjectures and generalisations about geometric figures.
One of the serious limitations of the package is its inability to do calculations. Screen measurements are rounded off measures which often lead to inaccurate results when for example calculating the sum of the angles of a triangle. This aspect could lead to problems when attempting to convince pupils of a particular geometrical result.
The school in which the project was carried out is under-resourced in terms of space and equipment. The school possesses a limited number of computers: 10 x 286’s and 2 x 486’s. However these are used mainly by teachers for administrative purposes, and to a limited extent, for computer literacy classes. The computers have been used sporadically for academic purposes and the school does not have a set policy about the use of computers in the learning and teaching of academic subjects. However, the school is in the process of developing a policy.
The school draws its pupils mainly from the lower socio-economic areas. As a result these pupils do not generally have access to computers and the school does not stand in line to gain equipment through parent donations. Because of the limited facilities and large class groups, we split the class so that one half worked on the computers while the other half engaged in a desk bound activity.
The class with which we worked consisted of 46 pupils. Many of them failed maths in Std 6 and have a very low self-esteem with regard to the subject - especially geometry. It is extremely difficult to motivate them or to keep their attention for an extended period of time.
Our focus was congruency which is a difficult component of geometry to teach, and for pupils to understand.
1. Our starting point was to focus on the construction of triangles. Our reason for this was to lead pupils to the realisation that a triangle consists of 6 parts (3 angles and 3 sides), but that one only needs to know 3 parts to be able to construct a triangle. Pupils then generated different combinations for the construction of triangles e.g. 3 sides (SSS) , 3 angles (AAA), 2 sides and 1 angle (SSA, SAS), 2 angles and 1 side.(AAS, ASA).
2. Pupils also investigated the relationship between the sides and angles of a triangle on the computer. The aim of this activity was to establish that the sum of the two shorter sides of a triangle can never be less than the longest side and that the largest angle is opposite the longest side. This point was re-emphasised in the construction of triangles using three sides.
3. A desk-bound activity was to establish the meaning of the word ‘congruency’ and to explore the difference between similarity and congruency.
4. The plan initially was to use the different combinations pupils had established, to construct triangles and to lead them to making conjectures about which combinations would result in a case of congruency. However, early in the project we realised that this was too time-consuming and that pupils were becoming so bogged down in the construction activity that they were losing sight of the aim of the activity. We then modified the activity so that pupils worked with ready-made constructions in order to make conjectures about which cases would result in congruency.
Each activity spanned two lessons because the class was divided into two groups. The fact that whole class was unable to do a computer-based activity at the same time, had tremendous implications for planning. In order to ensure that time was used effectively, desk-bound activities had to be developed so that both groups were occupied.
Pupils were then further grouped into 3’s so that there were 3 pupils to a computer. Within each group there was a leader who would ensure that each member took a turn at the keyboard. Pupils were grouped so that each group had a strong pupil and a very weak pupil.
1. Space is a major problem. The room in which the computers are housed is a small area which leads off from the main classroom. The computers are very closely arranged and to get 3 chairs around one computer was quite a squeeze.
2. Not all the computers were in operational order. Subsequently we only had 6 computers at our disposal. This was later reduced to 5. This highlights the need for reliable technical back-up.
3. Pupils had not been given enough opportunity to experience the Cabri package. They therefore needed lots of guidance initially. This meant leaving the group in the main classroom unattended, which often led to discipline problems. As pupils became more familiar with the package they worked better on their own. Furthermore, the pupils in the classroom were introduced to investigations which required the use of other apparatus e.g. using geostrips to investigate the properties of quads. This is important in order to establish a variety of resources and images which pupils can rely on in order to improve their understanding of geometry. We would emphasise that the computer should not be the only vehicle through which pupils learn geometry.
1. Although the computer room was small, the activities were so exciting that they did not find the space overwhelmingly problematic.
2. Pupils became very positive towards mathematics. They were keen to co-operate with the teacher and even came to work on their activities during intervals.
3. Pupils were very excited by the fact that they could manipulate sketches. This often led to healthy debate amongst members of the group which went beyond what they were asked to investigate on the worksheets.
In closing, may we add that the response to these exercises after exposure to the Cabri activities was very encouraging and gave us as educators a very warm feeling of having achieved success.