1 Talking and learning in mathematics

This is the account of a teacher who tried Activity 1 with her secondary students.

I made sure that I told the class right at the start why they were doing this activity. I told them that it was really important that they could identify and talk about all the different parts of a triangle. They would then be able to understand better the questions they are asked to do in their textbook or in the exams. I also told them that at the end of the lesson, I would ask them to check that they could use the conventional terminology for every part of a triangle, so it was important that they made sure they understood and could use the names correctly.

As I have quite a large class of students, I asked them to work in groups of three on this exercise. One student from each group came out and picked up three sticks; I had to hold onto them so that they did not know if they were picking long or short sticks.

Then I asked them to make a triangle with their three sticks meeting at the ends and to hold up their hands when they had done that. Some groups very quickly held up their hands, but there was a great deal of trying this way and that in two of the groups. I told everyone to stop and asked the groups that had made a triangle to sit down.

Then I asked the two groups that were still standing what the matter was. ‘We cannot make a triangle, Miss,’ they said. ‘Why not?’ I asked. At first they started to say things like ‘the sticks won’t meet’ or ‘we cannot make a point’. I stopped them and asked the class for what the sticks and points were called when in a triangle. Danna eventually came up with ‘they are sides and vertices,’ so I congratulated her on using the correct terminology – reminding them of the purpose of the lesson. After much talk and help from others, the two groups were able to say that ‘in order to form a triangle, the sum of the lengths of the two shorter sides must be longer than the length of the longer side’. It was real class endeavour, so I felt pretty certain that they had all contributed and understood this point. They were then allowed to swap one of their sticks with another group and we went onto the second part.

I told the students they had to introduce their triangle to the rest of the class using conventional mathematical language. I wrote the words on the blackboard that they had to use, using the textbook to check I had all that they needed. I gave them ten minutes to name their triangle and devise an introduction for it using as many of the words on the blackboard as possible.

Because I’d written down the list of words, the students could speak mathematically; also, I made a point of getting every student say something about the triangles they had made, so that everyone had to make some contribution to the discussion.

This was interesting because some of the students came up with very unusual descriptions for their triangle. Sona said that she had made ‘an old triangle’. When I asked her what she meant, she said it was bending. I asked her to use a mathematical term and she couldn’t think of a word for it, so I asked her to draw it on the blackboard and asked the others what they would call it. Ravi at once said that it was ‘obtuse angled’. We had a discussion about these triangles and, as a result, we discussed a lot of things. As each group introduced their triangle I ticked off the words ‘hypotenuse’, ‘opposite’, ‘acute’, ‘obtuse’, etc. on the blackboard, and told them how successful they had been. No one quite used all of the words, but it came pretty close.