As before, imagine drawing a rectangle inside a given rectangle. Now imagine drawing another rectangle inside this one and continuing the process. What conclusion can
you draw about this sequence of rectangles?
Open the interactive file ‘1e Interior Rectangles2’ and drag the point A until your starting rectangle is a square as shown in Figure 1.4d. This is a very special case.
Now watch what happens when you drag the point A by a very small amount. The effect is quite dramatic and illustrates how the ratio of length to width varies for the interior rectangles.
Comment
This task certainly lends itself to investigating with dynamic geometry software.
In the case of the square, the ratio of length to width of inner squares remains 1:1 (that is, is invariant). However, you will have found that the ratio is certainly not invariant when you start with a rectangle that is not a square.
Figure 1.4d
Reflection 1.4
What has been varied so as to provide you with experience of discerning details, recognising relation- ships, perceiving properties and reasoning with properties?
On rotating the board through 90° so that A has the same orientation that B had before, reactions have included: ‘Oh, now it is!’ What is the learner stressing, what is taken as invari- ant and what is seen as a permissible change? Simply subjecting learners to variation may not be sufficient for them to become aware of what the teacher sees as invariant, what they are permitted to change, and in what way. In order to learn from experience, it is not usually sufficient just to have experi- ences. Some sort of reflection is at least beneficial, and usually necessary. In the reflection tasks in this chapter you were sometimes asked to look back and consider what role was played by ‘invariance in the midst of change’ – what sorts of things were permitted to vary, and in what ways, and what sorts of things remained invari- ant. Becoming sensitised to notice invariance in any mathematical task is good preparation for getting the most out of any task and the activity that arises from it.
Reference has been made to the fact that different people discern different details in geometrical figures (and in situations generally). Sometimes learners focus on recognising relationships (two angles equal, two sides equal, two shapes the same), sometimes on perceiving properties (isosceles means two angles equal and two corre- sponding sides equal) and sometimes on using properties to reason (since two sides are equal, two angles are equal, which means … ). Noticing yourself focusing on dif- ferent details at different times sensitises you to notice when learners are attending differently and so to adjust your teaching accordingly.
Pedagogic Strategies
The most effective pedagogic strategies for promoting learning are based around pro- voking learners into using their natural powers to make sense of the world, listening to what learners have to say as they do this and engaging in discussion with them. In short, effective teaching involves being mathematical with, and in front of, learners. A variety of strategies have been illustrated already in this chapter.
You may have noticed that instead of the usual ‘can you find … ?’ or ‘find a … ’, many of the tasks asked ‘in how many ways can you … ?’ Studies of classrooms around the world have revealed that where learners’ attention is directed to multiple methods, they engage more fully, more creatively and more readily than when just asked to find an answer. In this text, the number of questions asked of you, the reader, has been restricted because it is not possible to modify the text on the basis of what you say and think. But the more you stop and think, question and challenge, the more you will get from doing the tasks presented in the text.
Many of the tasks in this chapter have been deliberately ambiguous. They have required you to recognise and make choices. This is especially true in counting tasks where what ‘counts as the same’ has to be decided by the person counting.Young learners may at first be frustrated by ambiguity, preferring to be told ‘what is permit- ted’, but once they become used to it, they tend to value it.
Asking learners ‘what is the same and what different about’ two or more objects or two or more figures is often fruitful, for it reveals what they are discerning and the relationships they are recognising. By hearing what others are saying they can see, learners have the opportunity to expand their own perceptions.
INVARIANCE 17
Dragging in Dynamic Geometry Software
Some of the interactive files you have opened have illustrated the power of the drag- ging tool in dynamic geometry software.
A construction can be explored by dragging relatively free elements in order to seek relationships in the figure that remain invariant. These can then be treated as properties and checked in other instances. To take a very simple example, learners might be asked to drag the vertices of a triangle, measure the interior angles and find the sum. Dragging the vertices of the triangle to different positions is likely to lead to a conjecture that the sum of the angles is invariant and is 180°. This is an empirical result, based on the large number of examples that one can quickly generate using dynamic geometry software. It may be intuitively convincing, but it cannot rule out the possibility of some extreme counter-example.
To be certain the conjecture is valid, you need to make use of underlying structural relationships by reasoning on the basis of agreed and accepted properties. Searching for structural relationships as the basis for reasons to explain why some relationship must always hold, and articulating the conditions under which the invariance is pre- served, are fundamental to becoming a competent citizen as well as to appreciating mathematics (Bishop 1991).
Although dynamic geometry software is a powerful medium through which to encounter geometric relationships and properties, well-established principles recom- mend exposure of learners to physical objects. Handling physical objects (shapes, geoboards, geostrips, Polydron©, and so on) is an invaluable part of learning because it
links physical movement and touch with geometric shapes rather than relying exclu- sively on sight and hand–eye co-ordination when using software.
18 BLOCK 1
Reflection 1.5
What struck you about the work in this chapter?
What do the following terms mean to you currently? Try telling someone else, or writing down some- thing about them:
congruence, similarity, counter-example, construction, corresponding angles, properties.
What aspects of the tasks gave you some pleasure? What happened inside you? How might that happen for your learners?