( 7 ) Equilibrium in the Hydraulic Press .sx An all-glass 50 c.c. hypodermic syringe , the piston of which could be loaded with different weights , was connected to a length of narrow glass tubing .sx Alongside was an exact duplicate of the apparatus so the subject could work with two liquids of different densities at the same time .sx One liquid was tap water tinged very slightly red , and the other was concentrated salt solution tinged very slightly blue .sx ( 8 ) Equilibrium in the Balance .sx The balance arm ( and the associated supporting framework ) was made from Meccano strips .sx By this means the distance of the weights from the fulcrum could be quickly obtained .sx The weights were cut so that the weight plus attached hook weighed 2 , 5 , 10 or 20 g. ( 9 ) Projection of Shadows .sx ( 10 ) Correlations .sx Each of forty postcards had the head of a girl drawn on it .sx The shape of the face , hair style and colouring differed for each girl , but the hair and eyes were coloured as indicated in the book .sx Inhelder and Piaget give no stages earlier than =3A , but the writer laid down criteria for =1 , =2A and =2B stages .sx Subjects .sx Our population consisted of 34 average and bright primary school pupils ; 14 average and bright preparatory school pupils ( aged 8-11 years ) ; 39 grammar school pupils ; 50 secondary modern school pupils ; 50 comprehensive school pupils ; 10 training college students ; 3 able adults whose ages ranged from 25 to 32 years of age ; thus making 200 subjects in all .sx In the comprehensive and secondary modern schools approximately equal numbers were drawn from the top and bottom streams of each year group .sx General technique .sx Each subject was examined , individually , on four experiments , with everyone taking the experiment involving the combinations of colourless chemical liquids ( no .sx 5) .sx After the subject had been introduced to the materials , and after some general discussions and sometimes free experimentation , he was asked to perform certain standard tasks and asked certain standard questions .sx The subject's actions were noted and his replies recorded verbatim .sx Details of the exact procedure used in each experiment may be obtained from the writer .sx It must be stressed , however , that the experimenter was quite free to vary the procedure by asking supplementary questions , or by prompting , or by experimenting slightly differently , if he thought it would be helpful .sx In brief our procedure was semi-structured and this is the best that one can do if the clinical approach is to be combined with some degree of standardization of procedure .sx The subjects were asked 'to think aloud' as much as possible .sx Usually Inhelder and Piaget give details of three stages of thinking ; stages =2 and =3 usually being subdivided further into 'A' and 'B' stages .sx After examining our protocols it was thought better to subdivide the Inhelder and Piaget stages still further , and we usually used nine stages , viz :sx =1 ; =1-=2A ; =2A ; =2A-=2B ; =2B ; =2B-=3A ; =3A ; =3A-=3B ; =3B .sx In this way we were , in our opinion , able to classify our protocols within the framework provided by the authors .sx Each protocol was studied by the writer and by the experimenter independently , and given a rating on the scale of stages .sx The results were compared and after discussion a final rating was given to each protocol .sx The assessment of some of the protocols was not an easy matter , and we cannot be sure that the more difficult ones were always rated correctly , although the ratings of these are not likely to be more than one stage out in the nine-stage scale that was usually used .sx In the experiment involving invisible magnetization the authors give a stage =3 only , not stages =3A and =3B , and we have kept to this .sx =3 .sx RESULTS .sx A number of tables are now given showing how the different groups performed on the various experiments .sx All our results are included .sx It is important to know to what extent the level of thinking of our subjects remained the same throughout the four experiments that each one undertook .sx To determine this we used Kendall's coefficient of concordance W , which specifies the degree of association between a number of sets of rankings .sx First , the rank of each subject was calculated , separately for each of the four experiments .sx W was then calculated from formula 9.16 given by Siegel ( 1956 ) , p. 234 ; this allows for tied observations .sx Furthermore , if the total number of cases concerned is N , and N 7 , we may find the probability of any value as large as an observed W , by calculating 5xe :sx 2: = k(N-1)W , with d.f. = N-1 , where k is the number of sets of rankings ( Siegel , 1956 , p. 236 , formula ) .sx Accordingly 5xe :sx 2: was calculated for each W and the probability associated with so large a value of 5xe :sx 2: was found by referring to Siegel ( 1956 ) , Table C , p. 249 .sx Table 11 shows the values of W , and the probability of finding an associated 5xe :sx 2: as large , P;5xe :sx 2: ; , for the differing groups of experiments and subjects .sx Even if there is a substantial degree of association between the level of thinking displayed by our subjects on each of the four experiments , it is necessary to determine if the experiments ( coupled with the manner in which the protocols were assigned to stages by Inhelder and Piaget ) were in fact drawn from the same population of experiments .sx For example , it could be that a particular experiment was rather easier or more difficult for one reason or another .sx Accordingly the Kruskal-Wallis one-way analysis of variance by ranks was used , as this test will decide if a number of different samples are drawn from the same population .sx The test assumes only that the variables studied have an underlying continuous distribution , and that ordinal measurement is possible for each variable .sx These conditions are fulfilled in the case of our data .sx First , the total number of subjects at each stage on each of the four tests was calculated , and the rank of each subject found from the single series that resulted .sx Thus H , the statistic used in the Kruskal-Wallis test , was calculated from formula 8.3 given by Siegel ( 1956 ) , page 192 , as this allows for tied observations .sx Since in our case there were four samples , and the number of subjects in each sample is greater than five , H is distributed approximately as 5xe :sx 2: with d.f. = k-1 , where k is the number of samples .sx Once again the probability of finding a 5xe :sx 2: as large as H was found by referring to Siegel , Table C , page 249 .sx Hence Table 11 shows also the probability of finding a 5xe :sx 2: as large as H , P;H ; , for the differing groups of experiments and subjects .sx The results of the remaining ten training college students were not analysed in this manner on account of the smallness and homogeneity of the sample .sx The four experiments which they undertook were :sx Chemical Combinations , Pendulum , Invisible Magnetization , and Equilibrium in the Balance .sx Reference to the values of P;H ; in Table 11 shows that the experiments in the first , second and fifth groups may be regarded as random samples drawn from the same population of experiments .sx In the third and fourth groups , however , P;H ; 0.01 indicating that one or more experiments in each group cannot be so regarded .sx Experience gained in examining the subjects indicated that the Projection of Shadows , and Correlations experiments found in the third and fourth groups , respectively , were likely to be responsible for this .sx Consequently the remaining three experiments in each of these groups were subjected to the Kruskal-Wallis test ; and for each of the two groups of three experiments the value of H so obtained was such that P;H ; 0.05. =4 .sx DISCUSSION .sx The following discussion deals principally with the educational implications of the study , and in order to be succinct the findings are grouped under a number of points .sx ( 1 ) The main stages in the development of logical thinking proposed by Inhelder and Piaget have been confirmed .sx It seems that the authors are correct in suggesting that it is only rarely that average to bright junior school children reach the stage of formal thinking .sx The ablest of the secondary modern and comprehensive school pupils certainly attain the stage of formal thought , but not all the older grammar school pupils always do so .sx There is a suggestion that ill-digested snippets of knowledge , mental set , and expectancy , are affecting thinking more in the students than among the school pupils .sx The student with the poorest performance was aged 19 years , and on the four experiments her replies were classified at the =2A , =2B , =2B and =2B-=3A stages .sx She had obtained a pass in Art at G.C.E. 'A' level .sx However , the least able of the secondary modern and comprehensive school pupils certainly remain at a low level of logical thought even at 15 years of age , and many of these do not seem to pass beyond the =2A-=2B stage of thinking .sx This is a finding the authors do not mention , and it leads one to suspect that the school population in Geneva which they examined consisted of able children .sx ( 2 ) By getting each subject to undergo four experiments and analysing the results by means of a non-parametric statistical technique , it has been possible to show that there is a considerable agreement between the levels of thinking that the subjects display in the four experiments .sx Moreover , the value of the coefficient of concordance W declines as the population becomes more homogeneous with respect to mental age .sx Naturally there is no exact correspondence since the experiments and 'intelligence' tests do not measure exactly the same thinking skills .sx Among the preparatory and grammar school pupils , W = 0.89 , and among the primary and grammar school pupils W = 0.81 ( Table 11) .sx In these groups the Mental Ages of the pupils ranged from 8 years to well above 15 years ( the M.A. usually accepted for average adults ) , whereas in the primary school group alone , for which W = 0.52 , the mental ages would range from 8 to 13 or 15 years .sx The authors give no evidence on this issue , but one would certainly expect some such stability of thinking skills if their general theory is correct .sx Again the Kruskal-Wallis test gave reasonable grounds for assuming that eight of the ten experiments may be regarded as samples drawn from the same population of experiments .sx The Correlations experiment is too easy for secondary , but not for primary pupils , compared with the other eight experiments ; while the Projection of Shadows test placed too many subjects at stage =2B .sx ( 3 ) The majority of our protocols show much the same kind of reasoning as those of Inhelder and Piaget , and support many of their statements .sx For example , the authors maintain that , at the level of formal thought , the child comes to the Projection of Shadows experiment assuming proportionality from the start .sx Below is a copy of part of the protocol of a boy aged 13 years 3 months .sx 'What happens to the shadow as you move the ring up and down the scale ?sx ' 'Nearer the wall smaller , further away bigger .sx ' 'Use two rings of different size , and move them until their shadows are exactly the same size , that is , they cover each other exactly .sx ' Places the 5cm .sx diameter ring at 20 cm .sx from light , and 10 cm .sx diameter ring at 40 cm .sx 'Why do the rings have to go in these positions ?sx ' 'Well 10 is twice 5 , and 40 is twice 20 .sx ' After placing three rings of different diameter correctly in position he is asked to place four rings of different diameters in position so that their shadows coincide .sx He places 5 cm .sx ring at 10 cm .sx from light , 10 cm .sx ring at 20 cm .sx , 15 cm .sx ring at 30 cm .sx , and 20 cm .sx ring at 40 cm .sx from light .sx 'Tell me exactly what you have done about the position of the rings .sx ' 'Well 5 is 10 cm .sx from torch , 10 is twice as big so it goes at 20 cm .sx , 15 is half as big again so it goes at 30 , and 20 is twice 10 so it goes here at 40 .sx '