Mid-points of Quadrilaterals

Using dynamic geometry software, draw any quadrilateral and then join the mid-points of the sides. This gives a new quadrilateral inside the original one. What can you say about this interior quadrilateral?

Comment

When dealing with any general proposition concerning quadrilaterals it is always of interest to consider special cases too (after all, there are so many special quadrilaterals!).Think about a square, rectangle, parallelogram, kite and so on as your starting quadrilateral.What happens to the interior quadrilateral?

Figure 2.4b

the example of the interior rectangle in his interesting article about the different functions of proof.

While investigating Task 2.4.3, you may have dragged the vertices of the quadrilat- eral into a concave position and even into a ‘cross-over’ position as shown in Figure 2.4d.You may have been surprised to see that such dragging did not affect a parallelo- gram being formed but now you can understand why.

In order to reason on the basis of properties, it is necessary to use mathematical lan- guage precisely, and to follow consequences. Sometimes it is necessary to insert or construct extra elements that enable you to discern sub-figures whose known rela- tionships (for example, similar triangles and ratios and parallelism) you can use to make deductions about other relationships. Notice also that it is important at the end of a chain of reasoning to look back at what was invariant, and to clarify what it is that is allowed to change and in what ways.

2.5 PEDAGOGIC PERSPECTIVES

The introduction to this chapter discussed some aspects of language in relation to math- ematical concepts and relationships. Many of the tasks have encouraged you to think about these concepts and relationships in different ways and to try to come to an understanding of their properties. Such understanding is intimately connected with the language that is used to describe the ideas.This is not only a question of formal defini- tions. Indeed, in moving along the path towards understanding, formal definitions may be the least useful vehicle to use.As David Ausubel (1963) explains, one way to measure understanding is to see if the learner can express an idea in their own words rather than reciting a definition verbatim. There are two aspects in particular that have pedagogic implications and are therefore useful to focus on. One is the idea of ‘negotiating mean- ing’ and the other concerns what teachers can learn about children’s understanding by

listening to them talk about mathematical ideas. Both of these aspects once again illus-

trate the importance of discussion in the learning of mathematics.

Children’s Descriptions

Getting children to talk about mathematical ideas and concepts usually requires some form of stimulus from the teacher, although sometimes it may be spontaneously initi- ated by the children themselves asking a question. The following example illustrates LANGUAGE AND POINTS OF VIEW 35

Figure 2.4d

Reflection 2.4

Look back over the reasoning given in this section and consider what particular features of language are involved in reasoning on the basis of properties. How has changing your point of view supported you in seeing why some relationship must always hold?

how the stimulus of asking for descriptions of geometric figures can give rise to important insights about children’s understanding.This in turn could lead to valuable discussion about the concepts involved.

Some groups of children were asked to write a description of a number of geometric figures. They were then asked to write an alternative description for each figure. This was to emphasise that there is no single answer to the question; it is a matter of how you see it. Before reading further, write your own descriptions for the diagrams illustrated in Figure 2.5a, and get some children to write, or tell you, their descriptions.

Although there are no ‘right’ answers for this task, sometimes descriptions demon- strate some misunderstanding, or incomplete understanding, about certain concepts. To this extent, such a task can act as a diagnostic tool, in that it may uncover such problems. On the other hand, a perfectly good description does not necessarily imply complete understanding. Consider these two descriptions given by the same student:

This shows a triangle with a perpendicular to of the base (diagram 2). This shows a triangle with a perpendicular to of the base (diagram 4).

What does this tell you about the student’s concept of perpendicular? Another stu- dent gave the following identical description for diagrams 1 and 3.

One triangle is enlarged to another by a scale factor from the vertex.

The important thing to consider here is not just the responses and whether they seem interesting or not, but on how you would follow up on and discuss the responses. For example, how would you react to the following descriptions?

This shows two similar triangles, one enclosed inside the other (diagram 3). Here are two similar trapeziums (diagram 5).

Ask yourself whether the first of these could be true. What would it require? If the second statement was supported by the argument ‘It’s just like diagram 1’ how would you respond?

To give a flavour of the range of different responses produced by this task, here are some descriptions for diagram 1. In each case, consider how you would try to develop the subsequent discussion.

It is a triangle and a parallelogram.

It is a triangle with 3 sharp edges and 2 parallel lines.

Two triangles with same base and height of different measurement. Two triangles with one same angle.

It has a double adjacent.

From a 3-d view, a triangle-based pyramid with the top part horizontally cut off. 1 – 4 1 – 2 36 BLOCK 1 Figure 2.5a

In many of these cases, your first task would be to try to get the learner to explain his or her description a little more. Indeed, you may not be at all sure for some of the descriptions, just what is meant. However, as long as you do not try to impose your view, but rather take the student’s description as a starting point for further thinking, the outcome is likely to be fruitful. In fact, this now leads to the first aspect men- tioned earlier, namely, negotiating meaning.

Negotiating Meaning

Any form of genuine discussion involves the negotiation of meaning between partici- pants. Unfortunately, many of the interactions that take place between teacher and students are not at all like this but, rather, take the form of ‘transmission of knowl- edge’ from one to the other or of ‘guessing what is in the teacher’s mind’. However, in using the idea of negotiation, it is not the case that everything is ‘up for grabs’ and that mathematical concepts can be interpreted in any way one wishes.The anarchy of Humpty Dumpty’s ‘Looking Glass’ logic would not be very helpful!

‘When I use a word,’ Humpty Dumpty said, in rather a scornful tone, ‘it means just what I choose it to mean – neither more nor less.’

‘The question is,’ said Alice, ‘whether you can make words mean so many differ- ent things.’

‘The question is,’ said Humpty Dumpty,‘which is to be master – that’s all.’ (Lewis Carroll, Through the Looking-Glass) The point here is that the very act of discussing, involving the sometimes tentative use of new words, is part of the process of arriving at meaning and understanding. And it is the teacher’s duty to help children achieve this understanding for themselves, by starting from their initial position or by challenging them to think

about a mathematical idea from a different viewpoint.You saw something of this in Chapter 1, with the discussion of exterior angles for concave polygons. Consider the following example, arising from a geometric problem in the classroom. The stu- dents had been asked to find the length of arc PAQ in terms of

x, the circumference of the circle (Figure 2.5b).

Prior to this problem, the students had discovered that arc lengths are proportional to the angles at the centre, and to the

angles at the circumference, subtended by the arc. They had also discovered the important relationship that the angle at the centre is twice the angle at the circumfer- ence. This is what the teacher was expecting them to use. However, one student produced the following solution, which puzzled the teacher.

arc PAQ 60° 1

––––––– = –––– = –– Hence arc PAQ = x/3

x 180° 3

When discussing this solution, the student explained that, having found the angle subtended at the circumference by arc PAQ was 60°, he wanted to compare this to the

angle subtended at the circumference by the circumference. He then argued that the circum-

ference was composed of arcs PAQ, QBR and RCP and they subtended angles 60°, LANGUAGE AND POINTS OF VIEW 37

R C P 80 40 A Q B Figure 2.5b

80° and 40° respectively. The sum of these angles would be the required angle at the circumference subtended by the circumference.

The italicised phrase used by the student is probably one that a teacher would not use.What meaning can be attached to such a phrase? The student endowed it with his own meaning. By going with the learner’s idea, a fruitful discussion, even a debate, could arise, for example: by considering other partitions of the circumference into arcs; by investigating the idea of adding the angles at the circumference even when they are at different points on the circumference; by comparing this situation to the angle subtended at the centre by the circumference; by taking a ‘limit’ approach. The interactive file ‘2i Circumference’ takes this latter approach.

This chapter concludes with one more example that illustrates the process of negotiating meaning. It is not discussed in detail but is more of a stimulus for thought. Consider the concept of parallel lines. Railway lines are often given as a real- life example. Children may make the observation from this that railway lines sometimes curve. So can you have parallel curves? In particular, can you have parallel circles? If you go to a dictionary (for example, the Oxford English Dictionary) to resolve this question you would find the phrase ‘continuously equidistant’ used to define par- allelism of lines, and the definition of line includes the phrase ‘straight or curved’. Well, there would certainly be trouble if railway lines were not always the same dis- tance apart! Does this mean that two concentric circles can be described as ‘parallel’ circles? Imagine the inner circle getting smaller and smaller until it becomes a point. Does this mean a circle can be described as a line parallel to a point? Perhaps all this sounds a little bizarre, but whenever definitions are introduced these are the kind of problems that must be faced. For example, what exactly is meant by equidistant in this context? How do you measure the distance between two straight lines or between two curves? You could follow up on the idea of concentric circles as parallel by con- sidering a transversal and see if the same properties hold for this as for parallel straight lines. In fact, the concept of parallelism holds a very significant place in the history of mathematics. If you are interested to know more about this, type in the phrase ‘Euclid’s Parallel Postulate’ into a web search engine.

38 BLOCK 1

Reflection 2.5

What struck you about the work in this chapter?

What do the following terms mean to you after working on this chapter? How has your understanding changed: