Piaget stresses that children can not visualize the results of the simplest actions until they have seen them performed , so that a child can not imagine the section of a cylinder as a circle , until he has cut through , say , a cylinder of plasticine . as always for Piaget , thought can only take the place of action on the basis of the data that action itself provides . while experience and general cultural opportunities are of great importance in helping the child to develop his concepts of space , it must not be forgotten that genetic causes , and temperament , play important roles too , especially the former . it has long been known that ability to manipulate shapes in the mind is present by 10-12 years of age , independent of measured intelligence . further , girls possess this ability to a lesser degree than boys , and it is likely that their inferiority in this respect is in part due to the differing kinds of activities in which they engage . it was suggested , too , by El Koussy in 1935 , that the ability depended on the capacity of the individual to obtain , and the facility to utilize , visual spatial imagery . El Koussy &apos;s point of view has recently received a little support from the work of Stewart and Macfarlane Smith ( 1959 ) using the electroencephalograph . Piaget would certainly admit that imagery supports spatial reasoning and geometrical thought , but is not in itself sufficient . chapter nine . concepts of length and measurement . before children come to school they are likely to hear many expressions used by adults and older children in relation to length and measurement . for example , most children hear their mothers speak of yards of material , or - less often - their fathers speak of feet of timber , or of the distance to the station or nearby town . more frequently , however , they hear of comparisons rather than the names of actual lengths , such as this is longer than that , or that is higher than this . these expressions are associated with many experiences ranging , maybe , from the length of nails to the height of mountains . likewise a child hears terms like near and far in relation to nearby or distant towns . again , from his play , or through watching the activities of grown-ups , he learns that a piece of string may be made shorter by cutting a piece off , or a stick made shorter by breaking it . likewise he learns that sticks and ropes may be joined to other sticks and ropes and so made longer . later we shall say a great deal about the view of the Geneva school regarding conceptual development in relation to length and measurement . it is sufficient to say here that it is out of these pre-school and out-of-school experiences , and out of infant school activities such as take place in the free choice period , that the child comes to understand the quality of longness or length - that is , the extent from beginning to end in the spatial field . during these experiences the child moves from visual , auditory and kinaesthetic perceptions , and actions to concepts . in activities involving counting a child may be asked to count the number of steps he has to take to cross the classroom . another child will be found to take a different number of steps . or , the lengths of short objects may be measured by the foot - the distance from heel to toe - or by the span from little finger to thumb when the hand is stretched as far as possible . from a variety of similar exercises the teacher can help her children to understand the need for a fixed unit of length for measuring purposes . of course , mankind has had exactly this problem of establishing fixed units , and a little history of measurement is an enjoyable and stimulating piece of work for older junior pupils . by the upper end of the infant &apos;s school the faster learners will be ready to be introduced to one of the agreed units of measurement , viz the foot . lengths of wood or hardboard , or plain foot rulers without end pieces or sub-divisions - which can be purchased - are given to the children , and they are instructed to measure various lengths and record their answers in a notebook . in the early stages they should be set to measure the lengths of lines drawn on the blackboard or floor , or to measure the length of pieces of string , paper , etc , all of which are cut to an exact number of feet in length . later , they can be set to measure the length of other objects in the environment to the nearest foot , so that if an object is nearly 3 feet long it is recorded as a full 3 feet . it is good , too , to let children estimate lengths before they measure , in the hope that it will lead to estimation with increased accuracy . with experience and maturity the pupils naturally become dissatisfied with a ruler that permits measurement to a foot only , for there are so many bits and pieces left over . this is the moment to introduce the inch , and a foot stick or foot ruler with inch marks on it . at the same time have work cards available on which there are lines drawn to an exact number of inches , or lengths of string and paper similarly cut for the pupils to measure . the next step is the measurement , to the nearest inch , of objects in the environment ; the children ought frequently to express their answer as , say , 1 foot 3 inches and as 15 inches , for this will help them to understand the relationship between two units used in the measurement of length . soon they will be found to be ready for a wall scale by means of which they can measure each other&apos;s height . this is an activity that creates great interest , since personal dimensions and growth are of great consequence to most children . next we come to the yard and yard stick ; a necessary unit when measuring longer distances . it is helpful to have some rulers divided into 3 feet with alternate sections , say , red and white , and a second set divided into 36 inches , with alternate inches of different colours . after comparing these with the whole foot , and with the 12-inch ruler previously used , the teacher should show that the yard ruler or stick is comparable with the length of her stride . by means of graded exercises similar in type to those described for feet , and feet and inches , we hope to get the child to the stage where he can measure a length as , for example 2 yards 1 foot 9 inches . the ordinary foot ruler with end pieces , and fractions of an inch up to 1/10 or even 1/16 inch , can be introduced when pupils are ready for it , but with the very slow learners simplified rulers may have to be used throughout the junior school . so far , activities and experiences that presuppose that the concepts of length and measurement are possible for children have been dealt with . have we , however , any clues as to the first beginnings of these concepts ? are there any conditions which are necessary before understanding of length can take place at all ? the Geneva school led by Piaget has carried out many interesting experiments in this field to which we now turn . the views of the Geneva school on the development of concepts relating to length and measurement . Piaget , Inhelder , and Szeminska ( 1960 ) have outlined the views on the way in which the child comes to understand length and measurement . in one of the experiments reported early in their book they study his spontaneous measurement . the experimenter showed the child a tower made of twelve blocks and a little over 2 feet 6 inches high - the tower being constructed on a table . the experimenter told the child to make another tower the same as mine on another table about 6 feet away , the table top being some 3 feet lower than that of the first table . there was a large screen between the model and the copy but the child was encouraged to go and see the model as often as he liked . he was also given strips of paper , sticks , rulers , etc , and he was told to use them if his spontaneous efforts ceased , but he was not told how to use them . the following stages were observed : ( a ) . up to about 4 1/2 years of age there was visual comparison only . the child judged the second tower to be the same height as the first by stepping back and estimating height . this was done regardless of the difference in heights of the table tops ; ( b ) . this lasted from 4 1/2 - 7 years of age roughly . at first the child might lay a long rod across the tops of the towers to make sure they were level . when he realized that the base of the towers were not at the same height , he sometimes attempted to place his tower on the same table as the model . naturally , that was not permitted . later , the children began to look for a measuring instrument , and some of them began using their own bodies for this purpose . for example , the span of the hands might be used , or the arms , by placing one hand on top of the model tower and the other at the base and moving over from the model to the copy , meanwhile trying to keep the hands the same distance apart . when they discovered that this procedure was unreliable , some would place , say , their shoulder against the top of the tower ( a chair or stool might be used ) and would mark a spot on their leg opposite the base . they would then move to the second tower to see if the heights were the same . the authors point out that in their view this use of the body is an important step forward , for coming to regard the body as a common measure must have its origin in visual perception when the child sees the objects , and in motor acts as when he walks from the model to its copy . these perceptions and motor acts give rise to images which in turn confer a symbolic value first on the child &apos;s own body as a measuring instrument , and later on a neutral object , e.g a ruler . ( c ) . from 7 years of age onwards there was an increasing tendency to use some symbolic object ( e.g a rod ) to imitate size . very occasionally a child built a third tower by the first and carried it over to the second : this was permitted . more frequently , though , he used a rod that was exactly the same length as the model tower was high . next , the child came to use an intermediate term in an operational way ( i.e in the mind ) , this , of course , being an expression of the general logical principle that if a=b , and b=c , a=c . children were found to take a longer rod than necessary and mark off the height of the model tower on it with a finger or by other means , so as to maintain a constant length when transposing to the copy . but , this transference is only one aspect of measurement ; the other aspect which must be understood is sub-division ; for only when this , too , has been grasped can a particular length of the measuring rod be given a definite value , and repeated again and again ( iteration ) . in the final stage it was found that children could also use a rod shorter than the tower , and it was applied as often as was necessary ; so that the height of the model tower was found by applying a shorter rod a number of times up the side . for the authors , then , the concept of measurement depends upon logical thinking . the child must first grasp that the whole is composed of a number of parts added together . second , he must understand the principles of substitution and iteration , that is the transport of the applied measure to another length , and its repeated application to this other . 