3 Coping with daunting mathematical writing

Activity 3 was given to the students as an independent exercise and I went around observing how they were able to do it. They identified almost all the formulae well and wrote down the shapes that they represented, but when it came down to drawing and writing down in their own words what it meant to them, they had some problems. To help them become more aware of their own learning, and to be able to pinpoint what it was they were stuck on, I asked the students to note down their thoughts about their problems at whatever point they were, so that they could contribute to a discussion. The main issue appeared to be about drawing three-dimensional solids. Because I wanted the students to know that there is not just one correct way of doing this, I called the students who had been able to draw a certain figure to come and draw it on the blackboard.

Once the students had some ideas about how to draw a three-dimensional solid, and had practised it themselves, we went on to discuss the explanations given. I asked all those who had different explanations about a certain formulae to share their thoughts and ideas so that all of the students could hear ideas and think about what makes a question easy or difficult.

We did Activity 4 over two periods because they got so engaged with the activity. They worked on their own but talked to a classmate about the choices they had made. They used their dictionary and formulae booklet without prompting and I did notice students using their fingers to point and keep track of what part of the example related to which formula. I also saw students covering up part of the drawings in order to ignore the bits they were not working on, so that they could focus on the parts the calculations were about.

Mona said if only they could take such a dictionary and formulae booklet into the exam! We then had a discussion about how to try to remember the formulae using logical thinking. Sushant suggested that it could be helpful to think of a cylinder whose lateral surface area would be the circumference (that is, the perimeter for a circle) multiplied by its height and its volume the base area multiplied by its height. Sushant then said that you could then think about how the solid you were working with was different from the cylinder and adapt the formulae accordingly. We also discussed how this relates to going from two to three dimensions, and why some questions were difficult for some and why others were easier. Ramona said they were all easy, so I asked her to work on the last question to try to make it harder.

I gave them the first part of Activity 5 as a homework assignment and I told them that they had to prepare the test questions for their classmates, although who would get to solve whose would be a mystery. They came back the next day enthusiastically with their questions, happy that they – rather than me, or an examination board – were setting the test. The next day I distributed their questions randomly, although I did try to match the difficulty level to individual students’ attainment. I had to swap two papers when I got to the end as I found that I was giving Mona her own question back, and a couple of students had to double up as that day there were more students in class. They settled down to work on the problems. The class especially liked that the test was marked by the originator of the questions and they enjoyed doing the marking.

The activity allowed them to pinpoint what made a question hard. Instead of saying the whole topic was difficult to do, they agreed it was only the questions involving a frustum – and not because of its shape but because of its complicated formula, which is nearly impossible to memorise! We discussed how we could avoid memorising the formula, especially as so many mistakes are made with writing it down from memory. We talked about how the formula was derived from other, simpler formulae, and that working through that logical process means you do not have to memorise the impossible-to-remember formula. I think this helped some students, but there still were those who insisted on learning the formula.

Doing all these activities did take a considerable amount of time, but I think it was well worth it. The students learned a lot of mathematics, seemed more relaxed and in control of their learning, and engaged actively with the tasks. All students, whatever their attainment, could do the task and could learn at their own pace and level. They had to think, be creative and make their own decisions. They really seemed to enjoy doing the mathematics and there were smiling faces and even laughter in the classroom – which I absolutely loved. I think they will also remember the mathematics they have learned more, which will save me time in the long run because I will not have to revisit the topic as often!