( 7 ) . equilibrium in the hydraulic press . 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 . alongside was an exact duplicate of the apparatus so the subject could work with two liquids of different densities at the same time . one liquid was tap water tinged very slightly red , and the other was concentrated salt solution tinged very slightly blue . ( 8 ) . equilibrium in the balance . the balance arm ( and the associated supporting framework ) was made from Meccano strips . by this means the distance of the weights from the fulcrum could be quickly obtained . the weights were cut so that the weight plus attached hook weighed 2 , 5 , 10 or 20 g . ( 9 ) . projection of shadows . ( 10 ) . correlations . each of forty postcards had the head of a girl drawn on it . 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 . Inhelder and Piaget give no stages earlier than 3A , but the writer laid down criteria for 1 , 2A and 2B stages . subjects . 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 . in the comprehensive and secondary modern schools approximately equal numbers were drawn from the top and bottom streams of each year group . general technique . each subject was examined , individually , on four experiments , with everyone taking the experiment involving the combinations of colourless chemical liquids ( no 5 ) . 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 . the subject &apos;s actions were noted and his replies recorded verbatim . details of the exact procedure used in each experiment may be obtained from the writer . 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 . 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 . the subjects were asked to think aloud as much as possible . usually Inhelder and Piaget give details of three stages of thinking ; stages 2 and 3 usually being subdivided further into A and B stages . after examining our protocols it was thought better to subdivide the Inhelder and Piaget stages still further , and we usually used nine stages , viz : 1 ; 1-2A ; 2A ; 2A-2B ; 2B ; 2B-3A ; 3A ; 3A-3B ; 3B . in this way we were , in our opinion , able to classify our protocols within the framework provided by the authors . each protocol was studied by the writer and by the experimenter independently , and given a rating on the scale of stages . the results were compared and after discussion a final rating was given to each protocol . the assessment of some of the protocols was not an easy matter , and we can not 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 . in the experiment involving invisible magnetization the authors give a stage 3 only , not stages 3A and 3B , and we have kept to this . 3 . results . a number of tables are now given showing how the different groups performed on the various experiments . all our results are included . 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 . to determine this we used Kendall &apos;s coefficient of concordance W , which specifies the degree of association between a number of sets of rankings . first , the rank of each subject was calculated , separately for each of the four experiments . w was then calculated from formula 9.16 given by Siegel ( 1956 ) , p 234 ; this allows for tied observations . furthermore , if the total number of cases concerned is N , and N &gt; 7 , we may find the probability of any value as large as an observed W , by calculating xe2 = k(N-1)W , with d.f = N-1 , where k is the number of sets of rankings ( Siegel , 1956 , p 236 , formula 9.18 ) . accordingly xe2 was calculated for each W and the probability associated with so large a value of xe2 was found by referring to Siegel ( 1956 ) , table C , p 249 . table 11 shows the values of W , and the probability of finding an associated xe2 as large , Pxe2 , for the differing groups of experiments and subjects . 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 . for example , it could be that a particular experiment was rather easier or more difficult for one reason or another . 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 . the test assumes only that the variables studied have an underlying continuous distribution , and that ordinal measurement is possible for each variable . these conditions are fulfilled in the case of our data . 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 . 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 . 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 xe2 with d.f = k-1 , where k is the number of samples . once again the probability of finding a xe2 as large as H was found by referring to Siegel , table C , page 249 . hence table 11 shows also the probability of finding a xe2 as large as H , ph , for the differing groups of experiments and subjects . 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 . the four experiments which they undertook were : chemical combinations , pendulum , invisible magnetization , and equilibrium in the balance . reference to the values of ph 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 . in the third and fourth groups , however , ph &lt; 0.01 indicating that one or more experiments in each group can not be so regarded . 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 . 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 ph &gt; 0.05 . 4 . discussion . 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 . ( 1 ) . the main stages in the development of logical thinking proposed by Inhelder and Piaget have been confirmed . 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 . 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 . 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 . 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 . she had obtained a pass in art at G.C.E A level . 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 . 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 . ( 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 . moreover , the value of the coefficient of concordance W declines as the population becomes more homogeneous with respect to mental age . naturally there is no exact correspondence since the experiments and intelligence tests do not measure exactly the same thinking skills . among the preparatory and grammar school pupils , W = 0.89 , and among the primary and grammar school pupils W = 0.81 ( table 11 ) . 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 . 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 . 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 . 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 . ( 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 . 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 . below is a copy of part of the protocol of a boy aged 13 years 3 months . what happens to the shadow as you move the ring up and down the scale ? nearer the wall smaller , further away bigger . use two rings of different size , and move them until their shadows are exactly the same size , that is , they cover each other exactly . places the 5 cm diameter ring at 20 cm from light , and 10 cm diameter ring at 40 cm . why do the rings have to go in these positions ? well 10 is twice 5 , and 40 is twice 20 . 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 . he places 5 cm ring at 10 cm from light , 10 cm ring at 20 cm , 15 cm ring at 30 cm , and 20 cm ring at 40 cm from light . tell me exactly what you have done about the position of the rings . well 5 is 10 cm from torch , 10 is twice as big so it goes at 20 cm , 15 is half as big again so it goes at 30 , and 20 is twice 10 so it goes here at 40 . 