DISCERNING AND STRESSING

There is a great deal of literature on the nature of mathematical discourse, and particu- larly on how children come to an understanding of mathematical language and reasoning, and the difficulties they experience. To take just one example of difficulties that arise, there are many words in mathematics that are also used in everyday language but with a different meaning; sometimes radically different, sometimes rather close. An amusing, but nevertheless illuminating, example is

when a child describes the first angle shown in Figure 2.1a as a ‘right angle’ (having just been taught the term), then describes the second angle as a ‘left angle’ or the third angle as a ‘wrong angle’.

The child is discerning and stressing more than

the adult. Another example is the word ‘base’, which has a number of different mean- ings within the world of mathematics itself (in area formulae for certain shapes, in vector geometry, in number representation) as well as its different everyday meanings. Moreover, the term ‘base’ when used in area formulae is a relational concept: its mean- ing depends on its relation to something else. There are many such words in mathematical vocabulary, perhaps not surprisingly, since mathematics is primarily concerned with relationships. For example, to describe a line as ‘a perpendicular line’ is not meaningful in itself. It must be perpendicular to something. To return to the term ‘base’, when using the formula ‘Area of triangle = Base × Height’ it is under- stood that the base and height referred to are a ‘corresponding pair’: for any side that is taken as base, there is a corresponding height. This can be a source of difficulty for learners for whom the concept of base is firmly rooted

in its everyday connotation of being the ‘bottom’ of something. So the diagram on the left in Figure 2.1b apparently presents no problem, whereas the one on the right is not seen as including a base-height pair.

1 – 2

Figure 2.1a

Some of the important relational concepts encountered in Chapter 1 included congruent, similar, and parallel. Terms for shapes such as hexagon, quadrilateral and parallelogram involve relational concepts in their definitions. Even a seemingly obvi- ous concept like ‘equal’ may need to be interpreted in terms of the context. For example, when developing the concept of fractions, the ‘whole’ is often referred to as being divided into ‘equal parts’. This sounds

innocuous enough, but consider the two diagrams in Figure 2.1c.

The two shaded triangles in the rectangle on the left are both (of the area), but a young child is

quite likely to argue that the two parts are not equal (identical in shape), so how can they both be ? They are not equal in the sense of being congruent. The child might eventually be convinced by the diagram on the right, which shows that each triangle is , and all the small triangles are congruent. In what sense then, are the two triangles on the left equal? Clearly, there is an implicit assumption that the equality refers to equal areas, rather than congruent shapes. Meaning depends on context. When a learner says something that seems wrong to the adult, it may be that the learner is stressing something that the adult is ignoring, and vice versa.

This chapter looks more closely at some of these concepts. In particular, having investigated the behaviour of shapes under given constraints, you need to understand

why they behave in that way. And this in turn means identifying the properties of the

shapes. As before, many such explorations in geometry can be greatly facilitated by using dynamic geometry software.

2.2 CONGRUENCE

You came across many examples of congruence in the previous chapter but what exactly is meant by ‘congruence’, and how, beyond the purely perceptual, can you convince yourself that two shapes are actually congruent? A simple physical interpre- tation would be to say that if you cut out two shapes and you can place one on top of the other so that they overlap exactly, then the two shapes are congruent. In practice, this is not always feasible as a form of reasoning.

2 – 8 1 – 4 1 – 4 20 BLOCK 1 Figure 2.1c Reflection 2.1

Think of some geometric words that have a different meaning in everyday life. What commonalities are there between the geometric meaning and the everyday meaning?

Task 2.2.1 Corresponding

Try to be precise about the meaning of ‘corresponding’ in the statement that ‘two polygons are congru- ent if corresponding angles and sides are equal’.

Be critical of your definition. Get someone else to see if they can make sense of it.

Comment

Did you think to specify that, for example, if two successive sides of one polygon corre- spond to two successive sides of the other, then the angles formed also have to correspond? Or perhaps it doesn’t matter? This is the sort of question that is taken up below.

Do you need to check that every pair of edges, taken in the corresponding order, all need to be equal, in order to be sure of congruence? Is the same true for angles? And would checking edge lengths or angles alone be sufficient? In the case of triangles, some thought will show that if you know that two pairs of angles are equal there is no need to check the third pair; they must also be equal since the sum of the angles is 180° in every triangle. So the important question arises, ‘what is a minimum set of conditions for determining congruence?’ To simplify matters, it is helpful to concen- trate on triangles to begin with.

Uniqueness

Congruence is a property involving relationships between two objects; uniqueness is a property of a collection of information about a shape. What information about a shape determines that shape uniquely?

Just as with previous tasks in this book that have used the phrase ‘how many different … ?’, you need to ask what is meant by ‘different’. Imagine the following incident:

The teacher asks the group: ‘What information do you require in order to draw exactly the same triangle as one that I have already drawn?’ The triangle is not shown to the group but, in order to commu-

nicate efficiently, a ‘generic’ triangle labelled ABC (as shown on the left in Figure 2.2a) is drawn on the board. The teacher points out that his/her triangle may of course look like the one on the right.

The teacher continues: ‘You can now ask me for an item of data (for example, length of BC, angle B and so on) and after each item is given, we will check to see whether you have enough information to draw my triangle’.

By working in this way, checking at each stage, the idea of minimum conditions arises naturally.When a successful construction is completed, the process can be repeated for another triangle but with the added proviso that the same set of data cannot be asked for.At this stage, one point that will almost certainly arise and be discussed is whether, say, asking for AB, BC and ∠B is effectively different from AC, BC and ∠C.

Notice how attention shifts from specific information (particular sides and angle) to relationships (two sides and the included angle).This leads to the idea of classifying LANGUAGE AND POINTS OF VIEW 21