3 Embodying mathematics
‘Body mathematics’ is a technique that requires the student to physically experience the mathematics that they are learning about. This is also known as ‘embodiment’, or giving concrete form to an abstract concept. The concept of embodied cognition is that of using the body to shape the thinking of the mind (Dreyfus, 1996; Gibbs, 2006). Embodying mathematics can:
- overcome the barrier of seeing mathematics as a theoretical subject, far removed from the student’s life experiences
- build more imagery for mathematical concepts
- get an emotional, and playful, link with mathematical properties.
In Activity 2 you worked with your students on large measurements. The next activity takes this a step further: working outside with large measurements again, but also requiring students to be the mathematics – to represent mathematics in a bodily form. Students will be asked to represent a point on the circumference of a circle and then to construct the centre of that circle precisely. Consequently each student/point has to be convinced they actually are at the same distance from the centre as all the others. The activity requires students to stand up for their rights and let their voice be heard when they think they are being discriminated against by not being the same distance from the centre as all the other circumference points. This type of activity is not easy to manage the first time you try it, but if you persist your students will gain more sense of mathematical properties.
Activity 3: Being mathematics
For this task you will need enough space for all of the students to hold hands and stand in a circle, with some more space outside the circle to be able to enact the construction of its centre. This task is to embody constructing the centre of a circle using ‘a perpendicular bisector of a chord pass[ing] through the centre of a circle’, as discussed in the textbooks for Class IX. If you think the group is too large to be able to complete the discussions they are asked to have in this task, it might be a good idea to ask them to discuss their ideas first in groups of three before sharing them with the whole class. You might need some rope or string, a ball, and a whistle.
- Tell your students: ‘The purpose of this activity is to identify the student that can run the fastest in the class.’Ask your students to all hold hands and stretch as much as they can to form the largest circular arrangement possible.
- Once the students are standing in a circular formation, pick a student at random and ask: ‘Can you put a ball inside the circle so that it is at the same distance from every student, as precisely as possible? If you place the ball at the wrong spot, you cannot compete later on in this activity.’ (But they could still act as ‘judges’ in the competition.)
- As soon as the ball is placed, tell the other students: ‘It is in your interest to reduce competition if you want to win. Can you convince us that the ball is not at the same distance from every one on the circumference so the student who placed the ball can be eliminated from the competition?’
- After a couple of students have been eliminated, pose the following question: ‘How can you locate with more precision and certainty the centre of the circle that you have formed? Can we come up jointly with a way to place the ball exactly at the centre?’
- Ask them to discuss this in twos and threes; then ask some groups to share their ideas with the whole class. After each group shares its ideas, ask the class whether they would be convinced that this is a good method. There is no need to ask all the groups to tell the class their ideas; after two or three have spoken, ask if anyone has a very different idea to those already given. If no one thinks of using construction with chords and perpendicular bisectors, it might be useful to give them a hint (but without telling all!) such as: ‘Now there was this kind of magical construction tool, called a bisector …’
- Ask them how they would do that construction, and what assumptions they would need to make because they are doing it outside using their bodies and on a large scale; then actually do it.
- When the ball has been placed in the centre of the circle correctly, blow a whistle to signal that the remaining students should run towards it and kick it. The first one to kick the ball wins the contest.
Case Study 3: Mrs Bhatia reflects on using Activity 3
Oh, they loved the competition element of this task! It made them so eager to think and come up with ideas. They all thought it would be so easy and simple to find the centre of the circle.
It was real problem solving – there were problems to be solved, both by thinking and by coming up with ideas to enact/embody the construction. There was a lot of getting stuck, and thinking about how to get unstuck. When they realised after several goes at placing the ball in the centre that it was difficult to be exact, they were kind of stunned. They did find it hard to come up with a fool-proof method, and again, I was tempted to walk them through the chapters relating to circles.
But the hint of the bisector did the trick – although thinking of using a chord or tangent to act as line segment took some time. But they got there on their own, and the sense of achievement, joy and pride, that they had thought of it themselves, was so much worth waiting for. I think that self-discovery and the ‘struggle’ to get there will also help them remember it much better – perhaps even in the exam.
Actually, in that process of thinking of what method would work, they came up with nearly all the different properties and theorems related to the circle. So it was good revision at the same time, together with the realisation that these theorems and properties might help them solve the real problem they were facing in the playground.
Pause for thought