JAN VAN DEN BRINK AND LEEN S T R E E F L A N D

YOUNG

CHILDREN

(6-8)-

RATIO

AND PROPORTION

0. I N T R O D U C T I O N In this report on young children's comprehension of ratio and proportion we start with two observations of one boy and with a number of observations in a classroom situation. This is why in the title we first mention the children and then the subject matter. The observational matter is described with a view to the didactical-phenomenological analysis of ratio and proportion, combined with a psychological analysis of the children's approach. This is followed by a description of methods of mathematical instruction in this field developed for children of this age.

1. O B S E R V A T I O N S 1.1. The ship propeller

Coen (7; 4) wants to know how the propeller (Figure 1) of a ship works. We discuss it in much detail, viewing: the small electric engine of his boat; the picture of a sea-going tug on the wall of his room. At the end he asks how big the propeller of a large ship is. His father tells him it would not fit into his room (which is about 3 by 4 meters). After a moment of silence he jumps to his feet, saying: -

-

It is true. In m y book on energy is a propeller like this (a distance

of about 3 cm between his thumb and forefinger) with a little man like that (about 1 cm).

1.2. Orca, the killer whale

Coen (8; O) and his father pass by a cinema where according to the posters they expect the film 'Orca, the killer whale'. The spectacular picture (Figure 2) shows a little man who on the back of the terrifying, smashing monster, tries to harpoon the animal:

Educational Studies in Mathematics 10 (1979) 403-420. 0013-1954/79/0104-0403501.80 Copyright 9 1979 by D. Reidel Publishing Co., Dordreeht, Lrolland, and Boston, U.S.A.

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I;

i Fig. l.

Fig. 2.

For the sake of sensation the size of the killer whale as compared with that of the man is exaggerated. Coen and his father look at the picture: What is wrong with it?, the father asks. The boy replies: That the whale smashes the boat to pieces.

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His father does not go further into the matter, although Coen's remark might have aimed at some wrong proportion. The father says: I t has something to do with the size o f the objects.

Coen (after a moment of silence): I k n o w what y o u mean. That whale should be smaller. When we were in England we saw an orea and it was only as tall as three men.

(Here the boy is alluding to an experience of 21 months previously (cf. Figure 3 : Dolphinarium Windsor Park, England)):

Fig. 3.

Four days later, just after another visit to a dolphinarium with a killer whale (albeit of only a good three meters (see Figure 4)) he explained why he knew the orca in England was as tall as three men: I remembered the orca, jumping out o f the water and touching the orange ball (Figure 3).

(He had not seen the photograph for at least one year; by the way it is a misleading one, because of the different distances of orca and trainer). After

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Fig. 4. this experience Coen emphatically wished to change 'the three men' into 'three to four men', which indeed corresponds better with the length of both killer whales. "2. ANALYSIS OF THE OBSERVATIONS 2.1. Mathematical-didactical Both observations involved four magnitudes (or rather rough magnitude values), which were compared in pairs: (a) In the first example these were: (1)

P1 : M1 : P2 : 342 :

the the the the

(2)

The pairwise comparison can be described by: room PI : M I

propeller of a big ship (bigger than the boy's room); father; picture of the propeller of a big ship; picture of a man beside the propeller in P2-

b o o k on energy = P2:3/12

The comparison is expressed in a qualitative way. (3)

Qualitative comparison and statement of the equality of ratios led the boy to accept his father's suggestion about the size of the propeller: 'It is t r u e . . ?.

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(b) The second example also involved four magnitudes (or rather rough values of magnitude), compared in pairs: (1)

O1 : M1 : 02 : 3/2 :

the killer whale on the poster; the man on the back o f the animal; the orca in Windsor Park; a man.

(2)

Now the pairwise comparison, on a qualitative level, may be described by a disproportion:

poster

Windsorpark

01 :M1 sa 02 :M2 (3)

There is, however, a new element involved, since the boy defines a numerical ratio for one of the pairs in order to make the comparison of the pairs easier: 'That orca was only as tall as three men'.

When comparing the picture on the poster with his experience of two years previously the boy accepted, as it were axiomatically, a certain ratio orca: man, to wit, an orca = three men. This explains the emphasis with which, after his next experience (Harderwijk), he changed the ratio orca: man into three to four men. There is a striking similarity in the boy's approach to both situations. In both cases ratio comes up in an equivalence relation. In both examples the equivalence or non-equivalence of the underlying pairs of rough magnitude values is determined at a qualitative level, though in the case of the orca a numerical ratio plays a mediating role.

2.2. Psychological (ratios in cases of similarity) Both observations prove that the boy knows about ratios. The way of expressing, why in the one case there is equivalence of ratios and in the other there is not, shows that similarity as operational equivalence* is the background pattern. In the observation(s) concerning the killer whale the equivalence, or better non-equivalence o f three distinct situations is involved: the dolphinarium in Windsor Park; - the picture of the film; - the situation at Harderwijk. -

* Terminology of H. Freudenthal in the latest version of his Didactical phenomenology of ratio and proportion (Internal IOWO publication).

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Within the mental activity of the boy in comparing those three situations the application of transitivity is involved in spite of the fact he might not be conscious of it. That is what is meant by 'operational'. In this approach of the situations mentioned, ratio has been applied as an equivalence relation. In general the young child is able to recognize similar objects or situations notwithstanding scaling and/or picturing on different sizes. With respect to the situations described the boy's (operational but maybe not-conscious) criterion was similarity or dissimilarity, which means being able to operate with similarities. Freudenthal says about this ability: I go even as far as saying that congruences and similarities are built into the part of our central nervous system that processes our visual perceptions. The speed of identification of an object after the object itself or the observer has been rotated, or after its distance from the observer has been changed, presupposes, as it were, a computer programme in the brain which eliminates this kind of transformation. While I do not understand at all what such a programme looks like, its mere existence - which I do not doubt - is an enigma to me. Other authors make similar statements: The main point is, that we have here an elaborate and refined system for coding contour elements which is present in its main essential at birth and must therefore be 'built in' as a major feature of the visual system [3, p. 182]. . . . it seems very likely, that young children can take in and remember size ratios... [1,p. 96]. Nevertheless, utterances of 6 - 8 year-olds seem to be aimed at dealing with similarity as an operative equivalence, which is shown by their understanding of, and reasoning about, ratios. These ratios are always object-related within a situation or a system of situations. The (mental) availability of two or more situations of this kind is the precondition for comparing ratios. Various kinds of objects may serve as measures of comparison. Primarily the child bases his judgement on the ratios under consideration and their equivalence, n o t on comparing sizes of the same object on different scales. The child seems to need situations each of which is characterised by the presence of well distinguished objects, in order to come to terms with ratios. Both in the case of the propeller and the killer whale this phenomenon can be observed: Ca)

b o o k on energy

P1 : M1

room

= P2 (= room) : M2 (= father)

RATIO

(b)

poster

AND P R O P O R T I O N

409

Windsor park

Ol(rca) :ml(an) 4= Oz(rca) :M2(an) In his didactical phenomenology of the concept of ratio Freudenthal [4(a), p. 164] distinguished internal and external ratios: ratios within a magnitude (or the set of values assumed by this magnitude) on the one hand and between magnitudes (or the associated set of values) on the other. Our examples seem to indicate that the external ratios possess a psychological counterpart. Let us study this idea more closely. The distinction between internal and external ratio has a particular meaning in physics where ratios of magnitudes written as quotients or fractions result in numbers on the one hand and - very often - in new composite magnitudes on the other. For instance the ratio path : time defines the concept of velocity, and the ratio weight : volume determines specific weight. Psychologically this distinction is also relevant to the process of making conscious the idea of ratio, where internal ratio seems to be preceded by the external one. Here, in the examples given, external ratio should be understood as follows: ratio between two or more well distinguished objects in a specified situation which itself is part of a set of such situations. From a mathematical point of view such ratios can be considered as internal ones because in almost all cases only one kind of magnitude is involved - in the present case, length. However, on viewing magnitudes as a physical object or as a set of objects of the same kind (including its pictures), then the labelling of these ratios as 'external' can easily be justified, because the objects involved play the role of different magnitudes, and their ratios that of compound ones, which have to be compared. In particular, mapping by similarity appears to be a forceful means of producing and comparing this kind of ratio. Here we mean similarity in a broad sense: producing object-related similar situations where the objects themselves might even be displaced with respect to each other - a broader meaning than the usual one of similarities as mappings of the plane or the space. A further psychological analysis of both observations shows the following details: (a) Propeller: the first pair in: room

b o o k on energy

Pa : M1 = 1'2 :Ms

was not explicitly given. The boy had to construct it from the context. Probably this happened after he had produced from memory the second pair, which

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was neither physically nor visually present. It appears from the boy's utterance that the mental construction of the similarity between both situations then led him to state a balanced proportion. The boy's mental activity was aiming at embedding the size of the ship's propeller in the field of his experiences of ships. It was not that he doubted the truth of his father's statement. (b) In the second situation the boy could use the height of a man as a (rough) standard to measure the length of an orca. Again he had to draw upon his memory to compare the available pair (orca and man in the poster) with the mental pair (orca and man in Windsor park). Here the mental activity was initiated through the problem posed by the poster. This led him to state inequivalence, in other words the conclusion that the orca on the poster was too large in proportion: 'The whale should be smaller'.

3. CONCLUSION Within the domain of congruence and similarity, as described in the preceding sections, 6-8 year-olds understand the meaning of ratio and proportion. This is witnessed by the way they deal with problems in this domain. As a consequence, teaching this aspect of mathematics should not be directed exclusively at formalising the concept (cf. Section 4). It is important to make the children conscious of the ways they reason about ratios and their characteristics. A method which can contribute to this aim consists of the teacher creating situations which are both surprising and conflict-provoking, while the children's initial interpretations and judgements are subjected to criticism and correction. In the next section a few examples from the classroom will be considered more closely, within and beyond the theoretical frame which has been set up above.

4. A C T I V I T I E S CONCERNING RATIO AND P R O P O R T I O N IN THE LOWER GRADES OF PRIMARY EDUCATION

4.1. Introduction In the following sections we shall discuss some classroom activities on ratio (and proportion) for 6-8 year-olds. In a way these classroom activities can be characterised by the term 'reappraisal of the direct perception' in order to focus on ratios in the perceptive field or 'perceptive accentuation of ratio' as Van Parreren calls it [6, p. 11 ].

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It is not only of psychological, but also of mathematical-didactical interest to observe the way in which this perceptive emphasis on ratio might take place and especially 'why' children are capable of this. Moreover it will be of interest to see whether, or in which aspect, the classroom experiences will show the role of internal and external ratios as described earlier. The examples will display the complicated learning process, which is, on the one hand 'steered by the field' and on the other hand 'steered intentionally', that is to say by the child's own plans or by those taken from other children [6, p. 12], thanks to the inquiring character of the instruction. The teacher tries to confront the children with surprising problem situations in which the children are motivated to feed back their reactions into their 'intentional steering'.

4.2. Liz Thumb (Maybe a relative of Lewis Caroll's Alice in Wonderland.) The teacher tells the story of Liz Thumb, the girl that once upon a time became as small as a thumb :

Fig, 5.

Liz Thumb is pictured on a worksheet. The pupils are asked to draw a flower, a stone and one of their own shoes at Liz Thumb's side. The results will show - according to the classroom experiences - quite a lot of wrong proportions, which lead to an interesting discussion: Liz Thumb is not as tall as m y thumb; she is smaller look here,

one o f the pupils remarks and he measures Liz Thumb on the sheet by means

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of his thumb. Apparently she has not been drawn full-sized so two scales are involved, namely the real Liz Thumb and her picture on the worksheet. This is the first spontaneous observation on ratios. But, there will be more problems demanding an explanation. Quite a number of children draw flowers or shoes, which are far too small. Note: they were supposed to draw one of their own shoes. By means of a sequence of questions the teacher tries to lead the children: How long is the grass?

(Figure 6)

How tail will the flower be? Will the mouse fit into the shoe you have drawn?

By comparing the drawn objects pair-wise (grass and flower; shoe and mouse) the children are surprised to discover that their previous ideas were not correct. Sometimes they maintain their opinion concerning the size of objects they have drawn:

Fig. 6.

That sun o f yours is too small, the teacher objects. Uhh . . . that is why it is far away anyhow, is an answer.

Later on in the schoolyear (cf. Section 4.5) the influence of perspective on ratios will be taken into account.

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4.3. Overhead-projector After Liz Thumb an overhead-projector trick is played. The teacher asks the children to close their eyes. She puts a Cuisenaire rod (number two) on the projector table. The children are asked to open their eyes. They can only observe the black shadow on the screen. Then the teacher raises an impossible question: What is the eolour o f this rod?

One pupil reacts: I don't know. You will have to measure it.

The image (not the bar itself) is now measured by means of a greed rod: It fits, so the rod on the projector table is green, isn't it?

But the children demand: That green rod should be put on the projeetor table,

to enable them to compare both images and after that to make a better choice. In the same lesson the teacher places a coin on the projector table. The screen shows a black circle: A little round, a counter, the pupils mean. This is a coin, the teacher tells, which one is it? One guilder, because it is so big. No, the teacher says, this is one guilder,

and she puts one on the projector table. The pupils then make another guess about the first coin. It is wrong again. The teacher puts another coin on the projector table, namely the one of the pupils' guess and it goes on. By comparing the images the pupils determine what kind the first coin was. So it is obvious that the children only used external ratio in the previous examples, in the sense that rather than comparing objects with images they compared objects with each other and images with each other. Moreover, one can recognize a certain shift of interpretation and a trend towards abstraction: the children had first to abandon the natural criteria of monetary value and colour of sticks, which are just the characteristics asked for by the teacher. The children were compelled to take another criterion - ratio into consideration in order to solve the problem (shift of interpretation). On the other hand replacing monetary value and colour by ratio is a special case of abstracting. -

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The mathematical gist of the previous classroom situation is a mental construction based on visual data. In the course of instruction you see the children develop a strategy by trial and error which leads to a solution. Probably the children are unable to use the scale factor of enlargement of the overhead-projector. Though they knew the images were bigger than the real objects, they only compared the images. Ratios have been treated by the children in the external way. But how should one organize experiences concerning the internal ratio of an object and its image. To do this we have the children looking at photographs.

4.4. Reality-turn The teacher shows a part of a photograph on the screen of an overheadprojector:

Fig. 7.

The children are asked to indicate their own height on the picture. First they compare themselves with the classroom-door in order to relate their apparent size to the door on the picture. Notice that the children based their solution of the problem on external ratios. It is a big surprise when the remainder of the picture is uncovered (Figure 8) to see a little girl standing next to the house as tall as the height of the house (cf. [7, pp. 2 5 9 - 2 6 6 ] ) :

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Fig. 8.

I got it, it's a doll's house.t, one of the pupils shouts. First it was a real house with a real door. This view defined the choice of the ratios. But when the true proportion became clear, the pupils switched to the opinion that the house was a doll's house with a correspondingly little door. So the children discovered that the internal ratio between the classroomdoor and the door on the picture was not a given constant. They discovered it by what we call a 'reality-turn'. Reality-turns are a good preparation for the concept o f scale. They can occur in many contexts.

4.5. Perspective Another trick that upsets the internal ratio o f an object and its image is

perspective (Figures 9 and 10). At school all the pupils will receive a 'camera' (that is to say the cover of a matchbox) in order to make a picture o f the teacher. They move back, because they want a picture of the teacher from top to toe. By doing so they discover that the cover can also be turned. The pupils learn to interpret ratios from the point of view of a photograp h e r . . , a little bit to the left, going backwards etc. Ratio will then have been embedded in the setting of perception. The pupils then have a look at some photographs of the school and answer some questions:

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Where did the photographer stand when he took these pictures o f the school? Which classroom is this? I f you were on the picture, what would be your size then? Which o f the photographs has been taken from a short distance? How do you know that?

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One pupil reacted to the last question by pointing out the apartment-building, which rises high above the school. He says: The apartment-building will become smaller as you come closer to the school, because the school will get bigger then.

After that he counts the little balconies of the apartment-building to be sure. This early concept of perspective in young children can be used to upset internal ratio : besides the normal pictures of the school we look at some 'mad' photographs (Figures 11 and 12):

Fig. 11. Peter says: The fish I caught is twice as tall as me, isn't it?

A pupil replied: Peter is a liar.t He cannot carry a fish like this, or can he?

The child's own experience in fishing is important. Perspective rather than experience was the clue when the children looked at the picture of the elephant:

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AND LEEN

STREEFLAND

Fig. 12.

The elephant is that big, his keeper fits into his trunk. How is it possible? 4.6. In conclusion By means of perspective and reality-turn the children grasp the non-variance of the internal ratio. By an open approach, favouring intentional steering, we found a way to develop all instructional and research aspects of ratio in agreement with the child's mental development. This statement is emphasized by the seemingly wrong answers children give to the question on the next work-sheet:

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m !

Fig. 13. The shape has been enlarged linearly by a factor 2: What is the factor o f enlargement o f the area? Many pupils answered that the area has remained unchanged. 'The grid has grown', they reasoned, or 'we have come closer to the house'. In our opinion these 'wrong' answers prove the necessity to search for activities which embody all aspects of ratio and proportion and to make them conscious to the pupils. That is to say to create situations, within which their conceptions can be confronted with and (eventually) changed because of the 'official' ones. Whether or not, a question like the last one is intended to be ambiguous, children might react that way. Our concern is to take into account the ideas (on ratio and proportion) they already have. L O. W.O. Utrecht BIBLIOGRAPHY [0] [1] [2]

Brink, F. J. van den, Strokenaanpak I en H (An Approach to Ratios by Bars, I, II) (A Mathematical-Didactical Internal Study), IOWO, Utrecht, 1975. Bryant, P., Perception and Understanding in Young Children, Methuen & Co. Ltd., London, 1974. Desjardins,M. and H6tu, J.C., L'activitd mathdmatique darts l'enseignement des fractions, les Presses de l'Universit6 du Qubbec, 1974.

420 [3]

[4]

[5] [6] [7]

JAN VAN DEN BRINK AND LEEN STREEFLAND Dudwell, P., Children's Perception and Their Understanding o f Geometrical Ideas, in: 'Piagetian Cognitive Development Research and Mathematical Education', National Council of Teachers of Mathematics, U.S.A., 1971. For an extended and thorough didactical-phenomenological analysis of ratio and proportion, the reader is referred to the last sections of: (a) Freudenthanl, H., Weeding and Sowing - Preface to a Science o f Mathematical Education, Reidel, Dordrecht, Holland, 1978. (b) Freudenthal, H., Lernzielfindung im Mathematikun. terricht, in: 'Zeitschrift flir P~dagogik', jahrgang 20, heft 5, pp. 719-729, 1974. Piaget, J., Understanding Causality, (especially Ch. 12: 'Linearity, Proportionality and Distributivity'), W. W. Norton & Company Inc., New York, 1971. Parreren, C. F. van, Niveaus in ontwikkeling van het abstraheren, State University of Utrecht (internal), 1978. 'Five Years IOWO - on H. Freudenthal's Retirement from the Directorship of IOWO', Educational Studies in Mathematics 7 (1976), 3.