Direct Proportion: The Application of Dienes' Theories to a Traditional TopicAuthor(s): Tom LeddySource: Mathematics in School, Vol. 8, No. 1 (Jan., 1979), pp. 24-25Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30211716 .

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Direct proportion The application of Dienes' Theories to a traditional topic by Tom Leddy, City of Liverpool College of Higher Education

This article is based on a paper prepared by the author for his teacher-training students as an example of how a traditional topic in mathematics may be approached using Dienes' theories.

For many teachers the name Dienes conjures up a picture of blocks of wood being used by children in junior schools. To the uninitiated, the materials (Multi-based Arithmetic Blocks) just appear to be expensive tools for teaching addition and subtraction. To those familiar with Dienes' theories they are but one example of how materials may be specially constructed so that principles underlying arithmetical operations may be experienced in concrete form by children.

But to concentrate on MAB is to obscure the overall con- tribution Dienes has made to understanding the process of learning mathematics. It is not the purpose of this article to give a full account of his theories. These may be found in the references. In the writer's opinion some of Dienes' more recent publications are rather "way out", but this does not detract from the value of his earlier work.

Two aspects ofDienes' work have much to offer to the teacher:

1. He directs attention to children's interest in games. This can be harnessed to motivate the learning of mathematics. Now it may be said that this is nothing new. Teachers have used games such as dominoes and bingo for many a year. This is true, but such games have generally been used to anchor facts or to give practice in skills.

Dienes sees games being used to reveal new mathematical concepts or structures, with the pupils some times demon- strating their real understanding of a structure by going as far as modifying the rules of the game themselves.

2. He describes steps by which mathematics can be learned. Some of the important steps are: (i) The play stage. This involves a number of games

and activities all of which embody the same mathe- matical structure. Dienes emphasises the importance of new structures being presented in different concrete situations, i.e. multi-embodiment. One of his basic principles is the Perceptual Variability Principle (PVP) which states that a new structure cannot be properly understood unless it is derived from a variety of situa- tions.

(ii) Abstraction. This takes place following the play stage.

(iii) Symbolisation. A structure needs to be represented in some way, usually in symbols, before it can be dis- cussed in abstract terms.

(iv) Anchorage. Structures need to be firmly anchored, through practice and further experience, before pro- ceeding to new ones.

(v) The formal operational stage. The abstraction having been made, the new structure can be analysed and developed without reference to concrete situations.

Let us now examine the question "What use is all this theory when one is trying to teach 2C the intricacies of, for example, Direct Proportion?" A fair question. Here is one answer.

As Dienes points out in his writings, there is an on-going interplay between the various mathematical stages of structured play, abstraction, etc. Thus although attention is drawn, in each of the sections below, to particular aspects of the learning process the others'are often present.

1. The play stage Structured games The appeal of games is used to introduce pupils to situations

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which involve direct proportion. The use of different games illustrates the PVP and the multi-embodiment principle.

(i) Direct proportion bingo Rules: Each player takes one of the cards, each of which is labelled DPC 3 (or 4 or 5 .. .). The DPC is your Direct Proportion Constant for this game and tells you what you have to multiply the dice total by, when you have your turn. On the card are also nine multiples of the DPC.

Each player in turn throws a pair of dice and multiplies the total by their DPC. If the result is on their card and not yet covered then one of the counters is placed on it. The first to cover all the numbers is the winner.

Example card DPC 5

15 5 35

40 25 50 55 20 30

(ii) Direct proportion race Each player takes a card on which is a statement such as: "Your DPC is 11/2. On squared paper draw a line 30 units long." (N.B. Each line must be in direct proportion to DPC.)

Each player draws his line. Then, in turn, a die is thrown and the player multiplies the die number by his DPC. He then draws in a jump of this distance along his line.

For example: A player with a DPC of 11/2 who threw a 4 then a 6 would have drawn the following: (Pupils could be asked to record throws and jumps.)

1 5 10 15 20 25 30

First player to reach the end wins. (Note: Fractions should not be used with children who have difficulty with them.)

(iii) "200 up" game A pack of cards on the faces of which are written, for example: 3 chairs; 5 men; 4 stools; 2 spiders; are placed face down on the table. Each player takes a card in turn and works out the number of "feet". For example, 3 chairs give 12 "feet"; 4 stools give 12 "feet". The card is returned to the bottom of the pack.

All players keep running totals for every player. First player to reach 200 wins.

Comment: It could be argued that the above games are merely ways of giving children practice with their tables. They could indeed be used as such! The objective of the games is to draw children's attention to pairs of numbers, the second of which is a constant multiple of the first. This multiple is labelled as the DPC to familiarise the children with the phrase before bringing out its full meaning later.

2. Abstraction Work sheet activities

By relating to the real world in different contexts (i.e. using PVP) the concept of direct proportion is gradually abstracted.

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(i) Number of cars Number of wheels

2 3 4 6 8 9

10

8

O O O O O

N

Complete this column

What is the DPC? (

(ii) People 1 2 3

The DPC is

Eyes

4

O

Number of Heads Feet

1

6 E2 0

Toes

20

0

Teeth

0

Discussion at this point emphasises that the second number in each case equals the DPC times the first number, in most of the cases examined.

We say the second number is directly proportional to the first number e.g. number of hands is directly proportional to the number of people. Discuss the contra-example of teeth and ask for others. (iii) State whether true or false

(a) The number of hands is directly proportional to the number of people (........) (b) The number of toes is directly proportional to the number offeet (........) (c)

(f) The total cost of some "Mars" bars is directly proportional to the number bought (........) (g) The total cost of different comics bought is directly pro- portional to the number bought (........) (h) The total cost of same comics bought is directly proportional to the number bought (........)

These worksheet activities enable the teacher to see whether the child has abstracted the underlying structure from the above concrete cases.

These could include examples of the following types: Fill in the missing numbers

(a) DPC= 3: (1, 3), (2, 6), (3, 0), (5, 0), (D, 21) (b) DPC= 0: (2, 8), (3, 12), (5, 20) ...

Clearly the different types could be introduced gradually e.g. might start with examples which only require the second numbers to be entered.

3. Symbolism Discussion on the lines indicated in Section 2 above could lead children to suggest the letters F and S to symbolise the first and second numbers of a pair.

Using the examples from 2 above, children could be asked to say and write, for example, for (a) (i) The second number is three times the first (ii) S=3xF (iii) The second number is directly proportional to the first with DPC = 3 (iv) S is directly proportional to F with DPC = 3 (v) S+F=3

Plenty of practice is now needed in which children are given one or two of (i) to (v) and have to complete the rest.

4. Activities to anchor the concept and its symbolic representation This is done by presenting real situations different from those so far discussed.

(i) The sun casts a shadow for various objects. Here are two:

2m I 4m

40m

80m

Complete the following (a) Shadow length is N times the object height

(b) S = OH (c) The DPC = 0 What are the shadow lengths for objects of height (a) 10 m (b) etc. (ii) If the sun in (i) drops lower in the sky so the DPC becomes 3, what will be the shadow lengths for: (a) the man (b) the church, etc.? (iii) Potatoes cost 10p per lb. Write down some ordered pairs for amounts bought, A, and their cost C. (1, 10), (2, 20), ... Complete the following (a) C= NA (b) C is ........ to A Repeat for potatoes costing 8p per lb. (iv) Draw (freehand) some small triangles. Count how many triangles, call it T. Count how many sides altogether, call it S. Do some more. Each time count T and S. Write down the ordered pairs. (T, S), (1, 3), ( , ) ...... Complete the following

(a) S= OT

S (c)

S (b) ==T

(d) S is ........ to T

Repeat with squares and corners hands and fingers persons and toes

Comment: Care must be taken here: (i) that the demands made on reading ability are not forgotten; (ii) not to involve difficult number products which would obscure the meaning of direct proportion; (iii) not to be misled, by a child's ability to cope with sections 3 and 4, into thinking he understands direct proportion. These could be done by "rule of thumb". Section 5 should expose lack of understanding.

5. Formal operational A discussion on these lines should reveal whether a child is ready to abandon reference to concrete examples. For class use a question sheet could be used. e.g. (i) If Y is directly proportional to X, what kind of "thing" should fill the box in Y = OX (ii) If A is directly proportional to B and A is 10 when B is 2 complete the following A= OB What is A when B is 3? What is B when A is 25? (iii) If P is directly proportional to Q and P is 8 when Q is 2 (a) What is P when Q is 3? (b) What is Q when P is 16?

Comment: The warning at the end of section 3 is repeated, in particular items (i) and (ii).

Conclusion Many of the individual items described in this article will not be new to readers. However, it is hoped that the article will encourage teachers to introduce new ideas in a sequence best suited to their pupils' learning process.

References Dienes, Z. P. (1960) Building up Mathematics, Hutchinson. Dienes, Z. P. (1964) The Power of Mathematics, Hutchinson. Dienes, Z. P. (1965). Thinking in Structures, Hutchinson. Dienes, Z. P. (1973) The Six Stages in the Process of Learning Mathematics,

NFER.

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