RECOGNISING ROTATIONS AND REFLECTIONS

This first section is about paper-folding and tessellations.

Different points of view can arise most unexpectedly in classrooms, but when recognised as such, can be exploited. Consider this anecdote from a teacher in an 11–16 school.

‘The Vulture and the Mouse’, by John Hancock

The all-ability Y7 classes were due to do some work on rotations and angles. The material in the textbook seemed accessible and the scheme of work indi- cated how it could be differentiated. I decided we would all start at the first question (figure 10.1a) in the ‘Turning’ section of the book. I planned to tackle the vulture and the mouse problem orally and take very little time over it.

The class studied the question: I directed their attention to the text, diagram and picture with the intention of ensuring everyone could understand the prob- lem. I then asked for answers and the first two replies indicated differences of opinion about the correct direction. These differences of opinion were real: on asking for a show of hands it became apparent that there was a very large major- ity opinion in favour of one particular direction.

Students were becoming quite outspoken about the question and after a few minutes I invited them to form three groups: the clockwise and anti-clockwise factions faced one another across the room and two ‘don’t knows’ came and stood by my side.

In the light of the size of these groupings I was forced to reconsider my own answer: the very few allies I had in the class were some of the weakest students in the year and most of the strongest mathematicians had confidently expressed an opinion which contradicted my own view! Students clearly expected me to arbitrate in this intellectual dispute but I remained silent as to the correct answer. I invited students to talk and listen to one another’s answers. I was able to stand back and observed:

● Much on-task talking, listening and questioning. Opinions were expressed

loudly, immediately, vehemently and freely. Students tried to persuade and convince others of the correctness of their answers. Disagreements about what IS clockwise were in evidence.

● Students attempting to simulate the problem. Some were standing on desks

and others lying on the floor. There was a request to take down the clock. Some students built models. Students pointed at themselves and rotated their fingers and I was barely able to contain myself as they struggled to ‘get behind’ their hand in order to see the opposite view!

● Some changes of opinion, but not many. I remained, disconcertedly, a

member of a very small minority!

This activity continued for the whole lesson. It did not appear to be a waste of time as the class were totally engrossed in the problem. The lesson seemed to have been a success despite the departure from the original plan.

Later that day I heard students talking in the corridor about the problem. Staff approached me to ask why their lessons are being punctuated by references to vultures and mice. The next day some students were wearing badges which proclaimed the direction they thought was correct – ‘It’s ... ’

I have shared the vulture and the mouse question with a group of teachers. Their initial response – a polite silence – was disconcerting. Had I missed some- thing really obvious? Their reaction suggested they thought this to be a straightforward question and that students should have no difficulty in answer- ing it correctly. I suspect they would have made similar judgements to my own about how to use it in the classroom.

I find this anecdote worrying.In a school climate where I am supposed to have detailed prior knowledge of student performance I cannot explain why this question proved so controversial to the students. I am also at a loss to know as to how I could have quickly and meaningfully sorted out the misconceptions.

I find this anecdote encouraging.It suggests that meaningful mathematics can be enjoyed by all students of all levels of ability engaging in a common task. I was pleased with the way students at my school engaged in the task.

I find this anecdote challenging.The question revealed deep disagreements between students and I am concerned about how easy it would have been to skip over what appears to be a real failure of understanding.

The anecdote could of course be worthless. My experience might be just an isolated case which will not be replicated in other schools. I could also have got the wrong direction!

John S. Hancock is head of mathematics at Settlebeck High School, Sedbergh, Cumbria.

(Reprinted (with permission) from Mathematics Teaching, 185, December 2003)

Approaching geometry from a transformational point of view was introduced into the curriculum in the 1970s in an attempt to make the geometry of Euclid accessible to a wider group of learners. Since affine transformations are conveniently studied using matrices, these were intended as a unifying theme. However, the work needed to use matrices to deal with transformations was too demanding and the project failed.Transformations subsequently disappeared from GCSE examinations.

One of the major issues in constructing a geometry curriculum is to find a logical basis from which reasoning can build. Euclid uses very simple axioms, but it takes a long time to get to interesting results.Transformations are difficult to reduce to a few axioms, but more closely match how people see the world: indeed, the word theorem comes from Greek meaning ‘a way of seeing’.The fact that we operate in time and space, and that, while time can be described algebraically, space is described geometrically, means that geometry and algebra are complementary, even opposite sides of the same coin.

The building blocks of geometry are dynamic images. Indeed, geometry has been described as the study of the dynamics of the mind, for example by Caleb Gattegno (1987). The electronic world we inhabit is increasingly driven by images, both static and dynamic, and the study of geometry can help learners to develop their powers so as to be able to make sense and use of these images.

Every diagram in a geometry text is a particular instance of a general case. The work needed to make sense of it is to become aware of what is permitted to change and what relationships are necessarily invariant. The diagram provides a stable back- ground to enable visualisation or a sense-of generality, of properties which must hold, or which cannot hold, and of chains of reasoning using those properties. This is true whether thinking in Euclidean or in transformational terms.

LANGUAGE AND POINTS OF VIEW 177