Piaget stresses that children cannot visualize the results of the simplest actions until they have seen them performed , so that a child cannot imagine the section of a cylinder as a circle , until he has cut through , say , a cylinder of plasticine .sx As always for Piaget , thought can only take the place of action on the basis of the data that action itself provides .sx 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 .sx 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 .sx 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 .sx 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 .sx El Koussy's point of view has recently received a little support from the work of Stewart and Macfarlane Smith ( 1959 ) using the electroencephalograph .sx Piaget would certainly admit that imagery supports spatial reasoning and geometrical thought , but is not in itself sufficient .sx CHAPTER NINE .sx Concepts of Length and Measurement .sx BEFORE children come to school they are likely to hear many expressions used by adults and older children in relation to length and measurement .sx 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 .sx 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' .sx These expressions are associated with many experiences ranging , maybe , from the length of nails to the height of mountains .sx Likewise a child hears terms like 'near' and 'far' in relation to nearby or distant towns .sx 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 .sx Likewise he learns that sticks and ropes may be joined to other sticks and ropes and so made longer .sx Later we shall say a great deal about the view of the Geneva school regarding conceptual development in relation to length and measurement .sx 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 .sx During these experiences the child moves from visual , auditory and kinaesthetic perceptions , and actions to concepts .sx In activities involving counting a child may be asked to count the number of steps he has to take to cross the classroom .sx Another child will be found to take a different number of steps .sx 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 .sx 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 .sx 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 .sx By the upper end of the infant's school the faster learners will be ready to be introduced to one of the agreed units of measurement , viz the foot .sx 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 .sx 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 .sx 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 .sx It is good , too , to let children estimate lengths before they measure , in the hope that it will lead to estimation with increased accuracy .sx 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 .sx This is the moment to introduce the inch , and a foot stick or foot ruler with inch marks on it .sx 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 .sx 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 .sx Soon they will be found to be ready for a wall scale by means of which they can measure each other's height .sx This is an activity that creates great interest , since personal dimensions and growth are of great consequence to most children .sx Next we come to the yard and yard stick ; a necessary unit when measuring longer distances .sx 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 .sx 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 .sx 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 .sx 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 .sx So far , activities and experiences that presuppose that the concepts of length and measurement are possible for children have been dealt with .sx Have we , however , any clues as to the first beginnings of these concepts ?sx Are there any conditions which are necessary before understanding of length can take place at all ?sx The Geneva school led by Piaget has carried out many interesting experiments in this field to which we now turn .sx THE VIEWS OF THE GENEVA SCHOOL ON THE DEVELOPMENT OF CONCEPTS RELATING TO LENGTH AND MEASUREMENT .sx Piaget , Inhelder , and Szeminska ( 1960 ) have outlined the views on the way in which the child comes to understand length and measurement .sx In one of the experiments reported early in their book they study his spontaneous measurement .sx 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 .sx 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 .sx 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 .sx 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 .sx The following stages were observed :sx ( a ) up to about 4 1/2 years of age there was visual comparison only .sx The child judged the second tower to be the same height as the first by stepping back and estimating height .sx 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 .sx At first the child might lay a long rod across the tops of the towers to make sure they were level .sx 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 .sx Naturally , that was not permitted .sx Later , the children began to look for a measuring instrument , and some of them began using their own bodies for this purpose .sx 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 .sx 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 .sx They would then move to the second tower to see if the heights were the same .sx 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 .sx These perceptions and motor acts give rise to images which in turn confer a symbolic value first on the child's own body as a measuring instrument , and later on a neutral object , e g a ruler .sx ( c ) from 7 years of age onwards there was an increasing tendency to use some symbolic object ( e g a rod ) to imitate size .sx Very occasionally a child built a third tower by the first and carried it over to the second :sx this was permitted .sx More frequently , though , he used a rod that was exactly the same length as the model tower was high .sx 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 .sx 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 .sx 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) .sx 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 .sx For the authors , then , the concept of measurement depends upon logical thinking .sx The child must first grasp that the whole is composed of a number of parts added together .sx 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 .sx