Making (part of) a Hyperbolic Paraboloid

Imagine holding a metre ruler just in front of your eyes so that it makes an angle of about 30º to the horizontal. Now slowly move it away from your eyes and as you move it steadily rotate it clockwise so that at first it is horizontal and then when it is at arms length it makes an angle of 30º the other way (that is, you have rotated it through 60º). Now imagine doing this again and think about the surface that the ruler would sweep out. Does it help to use a physical ruler?

Comment

When imagining things, it is often hard to keep hold of all the features as you focus on something particular. It is better to let go of other features but to refresh their presence every so often, rather than trying to retain them all as in a picture.When you have a picture to look at, you ignore many features in order to focus on others.The difference is that you know the other features are available when you choose to refocus, whereas with imagery those features might have to be reconstructed.

Stretches, Shears and More

Just as one-way stretches in two dimensions can be composed to produce two-way stretches, so in three dimensions you can have three-way stretches. Imagine a sphere (the three-dimensional analogue of a circle). Imagine stretching each of the co- ordinate axes by a different factor, say a, b and c respectively. The sphere will be stretched differently in each direction to form an ellipsoid: a shape which is elliptical in each cross-section. It looks like a rugby ball.Then as in the area of an ellipse in the two-dimensional case, the volume of any shape will be scaled by a factor of abc. The volume of a sphere is 4πr3_{/3 and, more generally, the volume of a three-way-stretched}

sphere, an ellipsoid, with semi-axes a, b and c, is 4πabc/3.

Shearing in three dimensions involves sliding planes sideways proportional to their distance from an axis plane, and shears can be composed. Furthermore, shears can be composed with stretches. Analysis of all the possibilities is most easily accomplished using matrices and linear algebra.

9.5 PEDAGOGIC PERSPECTIVES

In this chapter you have considered transformations as actions upon objects and as actions upon the whole plane (and even the whole of three-dimensional space). Thinking about transformations as objects leads in several directions: to composing them to form classes of transformations; to seeking invariants which characterise vari- ous transformations (length, angle, area, parallelism); to seeing relationships between shapes (congruence, similarity, area preservation) as determined by the collection of transformations permitted (isometries, similarities, shears). Similar work can be done with orthogonal projections and point projections.

There has been an alternation between task-driven experiences and exposition in order for you to reflect upon when one or the other might be more appropriate when working with learners. A steady diet of all one or all the other can become numbing, as perhaps you have noticed. Learners do not always know what to do with tasks (they need to re-learn to become independent of explicit and totally clear instructions in order to re-develop a problem-solving approach to situations).There is also a risk of learners getting stuck and being unable to complete the task; various strategies are helpful in overcoming this such as specialising in order to re-generalise; finding a physical presentation from which to get a sense of what goes on; seeking relationships, and so on.

Being text-based, the tasks presented here are often in the form ‘imagine … ’. Increasingly you have been expected to draw your own diagrams and to work men- tally on those diagrams. There have been suggestions to use interactive files, and opportunities to cut out shapes and to move them about. Working with physical objects can focus your attention on achieving results without actually developing imagining powers. Prompting learners simply to look at physical objects and diagrams 170 BLOCK 3

Reflection 9.4

What role did the practical examples play for you in thinking about projections, including using a sheet of paper to make a cone, and using a ruler to imagine the hyperbolic paraboloid? What happened when you switched from imagining to interpreting the diagrams provided?

and to imagine parts moving but preserving the structure, or superimposing further elements makes use of learners’ powers and helps those powers to develop. There is more on these ideas in Chapters 13 and 14.

Visual–Aural Preferences

It is important to remember that some people find it easier to respond to mental imagery invitations than others.This is because some people ‘see’ in their minds, or at least respond readily to the language of ‘seeing’, while others are more auditory in focus, hearing words and finding them come to mind more readily. Everyone has the powers necessary which can be developed, but it may take time. Getting learners to discuss their ways of thinking can be very effective in opening up possibilities that they have previously overlooked.

There is a wider dimension to the visual–aural preferences which some learners present. Imagine the mismatch in the classroom where some pupils and a teacher might be discussing ways of working:

I think in pictures and diagrams I teach with words

I want the whole picture I want you to look at the details I want to synthesise I want you to analyse

I want to know what It is important that you know why I make intuitive leaps Every small step is important I like challenging puzzles I want you to master the basic facts

Learners do not usually articulate these differences, indeed they may not be aware of them, but in most classes some learners think differently from others and from the teacher. One coping method is to set tasks that can be approached in different ways, then notice, acknowledge and celebrate differences.

Attention

A related issue is the form of support given. When learners are temporarily stuck, it can be tempting to try to give them a push by telling them something that they could have worked out for themselves. By being aware of what you are attending to, and trying to work out, perhaps by asking questions, what they are attending to, it often emerges that there is a mismatch. Learners’ thinking can be influenced by attracting or focusing attention, and by prompting relevant kinds of attention: discerning details and features, recognising relationships between features, perceiving relationships as properties so as to be able to reason on the basis of established and agreed properties. Often what is problematic is the form of learners’ attention.

As a teacher it can be difficult to anticipate how learners will begin a task.Very often there will be surprises and new approaches. A further difficulty is to recognise whether an approach is legitimate, and to accept that when it is not, some good geometry and good problem-solving might still be learned.

Learners often answer the question ‘why (does this property always hold)?’ with the reply ‘it just is’ or ‘it’s obvious’.This sort of reply is what might be expected by the van Hiele (van Hiele and van Hiele-Geldof, 1986) approach and by thinking in terms of the structure of learners’ attention. Learners may be focusing on relationships and properties but not yet isolating some properties as the basis for reasoning and TRANSFORMATIONS AND INVARIANTS 171

other properties as susceptible to reasoning. Of course, learners may also sometimes say ‘it’s obvious’ when they are trying to cover the fact that they are not sure and are only guessing.

Learning to reason involves both constructing your own chains of reasoning and also learning to listen to and critique other peoples’ reasoning. Getting learners to work alone briefly, then in pairs, then in fours before having a plenary discussion can enrich the discussion at each level, and alert individuals to ways of thinking that had not yet occurred to them, as well as recognising that others are struggling in similar ways to themselves.

Learners at school may not have ever thought about a one-way stretch in geometry, but they would have informal ideas linked to it; and they may confuse changing a square into a rhombus with a shear, but they would still have experience of both transformations.

Articulating

The tasks in this chapter often assumed that you would stop and develop an image (which might be mostly kinaesthetic–aural or mostly pictorial). You were often expected to draw your own diagram or even to arrange some physical manifestation. The point about finding something which is confidence-inspiring to manipulate is not to answer the specific question, but to enable you to get a sense of what might be going on, to recognise some relevant relationships and perceive appropriate proper- ties. As this sense develops, it becomes easier and easier to express those relationships and properties in diagrams, words and symbols. As you become more articulate you 172 BLOCK 3