THE TRANSITION FROM CONCRETE
TO FORMAL THINKING
A thesis submitted in partial
fulfilment of the requirements
for the degree of Doc tor of
Philosophy.
by
Susan Clare Page
Australian National University
December, 1970,
181
CHAPTEB_I
SCORING OF PERFORMANCE ON THE FOUR GROUP TASK
Scoring of performance on the Four Group Task divides into two
distinct sections - firstly the learning of the sixteen combinations to
criterion and secondly the leve 1 of understanding displayed in the
subsequent questioningg
7.1
Measures of Efficiency in Learning the Combinations
The two basic indices of learning efficiency available are the
time taken and the number of combinations tried,
Time and number of
combinations to criterion wi 11 be the principal measures, although the
time to administration of the first "test" (T ) of the combinations may
1
also be informative.
It will be remembered, however, that a restriction
of the latter (to a maximum of 15 minutes) was imposed in most cases by
the experimenter.
Several points must be borne in mind in an examination of the
learning~
number of combinations tried during
TI1e first concerns the
fact that in some phases the subject is freely choosing which combination
to try, and at other times, during
11
tests", he is being asked by the
experimenter to predict the outcome of combinations in a systematic way
(refer Table
4,
Section
4.4.2).
Both of these activities give the subject
information about the outcomes of combinations, so that there seems to be
no objection to their being put together to give an overall index of the
number of combinations tried,
This index will be referred to as the "total
182
number of questions asked" (Tot. Qns.).
It is also possible to find on
what proportion, of the total number of questions asked, the correct prediction is made by the subject.
Measures of the numbe.r of correct answers
given during tests are also available, but of limited use, since the point
at which each test was given was not determined in an objective way$
The second point concerns the inadequacy of such measures as
the number of combinations tried, or the proportion of them on which a
correct prediction was made, as indices of the nstrategy", or method of
learning, adopted by the subject. No easy way of measuring such strategies
is available, although, as mentioned in Chapter 3, Dienes and Jeeves
(1965)
were able to use sequential patterns of combinations tried to derive
aoperator" and "patternu strategy
scores~
For reasons discussed earlier, the form of the task used in the
present investigation might have been expected to lead to even more obvious
adoption of an noperator 11 or "patternu strategy by the subjects than
occurred 1n Dienes and Jeeves'
(1965) study.
However, on completion of the
present study, it was remarkable to find that only two or three of the two
hundred and thirty six subjects tested had investigated the combinations by
means of a clear "operator" strategy.
For the most part, the investigation
of the combinations was initially haphazard, some times developing into an
"opera tor" or npa tternn strategy towards
the end.
Occasionally 1 after the
first or second test on all sixteen, a subject would go over the combinations in the same systematic fashion as the experimenter had done (in the
next ulearningn phase) but this, too, was a surprisingly rare occurrence
a
Because the existence of such strategies of investigation seemed
183
so hard to establish an attempt to do so was abandoned.
One overall index
of the sequential structure of combinations tried by the subject (in the
learning phases only) was derived as follows.
A tally was made, for each
coloured light (and irrespective of which panel -subject's or experimenter's - i t was on) of the number of times it appeared 1, 2,
tried~
times in a row in the combinations
3, 4 and "'5
A trial of a "doublen combinat-
ion, such as B and B was counted as an appearance of B twice in a row,.
Then, for all colours combined, the total number of times a run of
3 or
more (of the same colour tried) occurred was compared with the number of
runs of less than
3.
It was hoped that this would give some indication
of whether the subject had any tendency at all to look at "what Y does",
nwhat G doesu etc .. , in a systematic
way~
However it must be stressed
that the absence of such a strategy does not preclude the possibility that
the subject was thinking about the problem in this way.
It did seem, from
observation of subjects' behaviour, that they tended to see the possibility
of structuring the combinations in rows Cor by colours as uoperators") only
after a period of rather haphazard learning of them.
Unless it is poss-
ible to define such "turning points" where the method of approach changes,
indices of the sequential structure of the combinations tried during learning will not be very meaningful.
An analogy can perhaps be made with the
"Shift-point", from a primarily analytic to a primarily synthetic approach,
identified by John (1957) as occurring during the solution of problems on
the "P. S. I." apparatus.
TI1e pos si bi li ty of making an adequate study of this
aspect of performance on the Four Group Task is an interesting, but very
difficult, one and is beyond the scope of the present thesis.
184
Hence the measures of efficiency of learning the combinations
tabulated for each subject were those shown in Table 12 below.
A copy of
the scoringsheets from which they were derived can be seen in Appendix
I.
Of the tabulated measures shown in Table 12, only four were found to
be of use in the analysis of results.
These were the time taken to Test 1,
the time taken to reach Criterion, the total number of combinations tried
(in learning and in tests), and finally the proportion, of the total number
of combinations tried, on which the correct prediction was made by the
subject.
These wi 11 be discus sed in Chapter
9.
The examples presented in Table 12 are not necessarily typical
of the age groups from which they are drawn, nor do they exhaust the types
of performance found in the study.
They do serve to illustrate a number
of approaches to the problem which are rather clearly different, although
the elementary measures
complete records.
entered in the table convey far less than the
From the complete records (see Appendix I) it can be
speculated that these subjects have approached the task in the following
ways:-
J .A. (14-2) From the sequence and rather steady rate of trying the combinations it seems that the subject is simply adding new combinations, at a
steady rate, to a rote memory
store~
A few sequences show a tendency to
put several combinations involving the same colour together, but the idea
of trying each colour with itself does not occur.
these
11
Despite the fact that
doubles" have never been tried, the correct prediction is made for
each of them (in fact all 16 are correct) in the first test, which would
suggest that some pattern or system has been seen.
TIME
SUBJECT
COMBINATIONS TRIED
No. Correct/Total
Cexcl. tests) to:-
TIME TAKEN TO:-
Age
Initials
Sex
Group
Test 1
(mins)
Test 2
(mins)
Test 3
{mins)
CR11N.
Test 1
Test 2
Runs of same
No.~J/No.<J
Test 3
No./16 Correct in:Test 1 Test 2
Test 3
(mins)
Total No. correct/total No. tried in:Learning
Tests
Both(Total Qns.)
-
-
-
10.0
11/21
-
-
4/23
16
29.5
-
31.5
11/25
12/16
-
7/30
-
-
15.0
18/28
-
-
2.0
-
-
5.0
5/10
-
14
2.0
6.0
-
6.5
6/10
F
13
6.0
17.5
30.5
L.E.
F
11
8.0
19.0
-
24.0
I .Bl.
M
11
s.o
-
10.5
j.A.
F
14
7.0
M.B.
F
14
10,0
R.P.
M
14
8.0
D.M,
M
14
I .Br.
M
V.H.
- - - - - - - ---------- ---------
-
27.0
-
-
11/21
16/16
27/37
8
16
-
23/41
24/32
47/73
2/27
13
-
-
18/28
16/19
34147
-
1/12
13
-
-
5/10
16/19
21/29
13/16
-
6/21
11
14
-
19/26
27/34
46!6o
9/18
19/29
13/16
12/58
7
8
13
41163
31/51
72/114
12/23
24/26
-
10/44
11
15
-
36/49
27133
63/82
10/15
-
-
1/22
12
-
-
10/15
'-
10/20
26/35
'---
Table 12 : Measures of learning performance for a sample of eight subjects, The table shows measures of the time taken, and the number of
combinations tried, during the learning and testing phases on the Four Group Task. The time elapsed to the beginning of Tests 1,
2 and 3 (the latter two are not always applicable) and to Criterion (the end of the last test) is shown. The combinations tried
are separated into those during learning phases (chosen by the subject) and those during systematic tests conducted by the
experimenter. The order in which combinations were tested is shown in Table 4 of Chapter 4. The ratio of the number of runs ~ 3
of the same element tried, to the number of runs < 3 is also shown.
186
M.B. (1Lt-2) The sequence of combinations tried, and the fact that the subject
pauses and remarks on ideas being tried out in the first learning phase,
show that there is a clear tendency to look for a nrole" for each light.
Some of the difficulty encountered, with the Blue and Green lights particularly, is probably due to the fact that the "double ones" are not tried
(although Red and Red is tried),
Test one shows a lot of errors, except
with row one (Yellow plus X) and the Commutative Law does not seem to provide
the corresponding answers for column one.
In the second learning phase row
one combinations are omitted and the others tried systematically -the subject seems intent on discovering what each light
11
doesu ..
R.P. (14-6) From the start this subject asks questions which show a structured
approach to the problem:same?u
"Will the result of a combination always be the
nThere will be 16 combinations, won't there?u~
This seems to lead to
a search for rules(e.g. the Commutative Law) and the question "Can you have
two the same?".
However the trying out of combinations, in practice, seems
to be organised more into types of combination ("the same one back to front";
"two the same") than into rows of the matrix Call the combinations with
Yellow, then all with Green, etc.).
D.M. (14-2) After two minutes this subject has tried eight different combinations (two of them twice) and thinks that he knows them all.
Amongst those
not tried are the "double ones" and three of these are the only ones wrong
in Test one.
rule for
11
It seems likely that at least the Commutative Law and some
What Yellow doesu must be
understood~
I.Br. (14-2) The initial learning phase is extremely similar to that of the
previous subject, D.M., but in Test one two additional combinations are
wrong, necessitating a second learning phase.
In the second learning
phase, this subject tries each of the 16 combinations only once and in the
same order as the test ..
Test one may have
helped to structure the problem
for him.
(13-Sl The combinations are tried in no particular sequence and at a
V.H.
fairly steady rate which, as in the case of ].A., may indicate a rote
memory strategy for the first learning phase.
No double ones are tried,
but, in test one, two of these are correctly predicted even though overall
performance is poor.
During phase two learning, the Commutative Law is
expressed, but the only further structuring of the problem is the grouping
together of the ndouble
onesu~
In Test
two~
performance is little better
than it was in Test one, and then, interestingly, the final phase (three)
of learning shows the combinations tried in the same order as the
tests~
It seems possible that, for this subject, the test organisation of the combinations pres en ted real difficulties, interfering with her own method of
remembering
them~
L.E. (11-4) In eight minutes ten different combinations are tried, a few
favourites being returned to several
times~
The ndouble onesu are amongst
those omitted and three of these, plus two others, are wrong in the first
test.
The second learning phase concentrates very much on the double ones,
which are grouped together, although there is also rome slight evidence
of
grouping together uthose with YellowH and perhaps nthose with Blue'\
In the second learning phase, the rate of trying combina ti011s becomes rapid,
and in the second Test only one of the sixteen is wrong.
188
I.Bl~
(11-0) In no special sequence? eigl1t different combinat.ions are t:ried
(the ndouble ones 0 all omitted} and after five minutes the subject ls able
to get twelve correct in Test
one~
Of the four wrong? three are
~'/<double
one su ~
Comments and impressions of strategies such as those given
briefly above, for a small sample, can be gathered togetl1er for subjects
who succeed and fail in understanding various aspects of the s t:ructure of
the
problem~
Ideally, as mentioned
before~
some objective indices of such
aspects of the strategy of investigation should be derived,
however~
no clearly satisfactory way of achieving this is
At present,
apparent~
As
mentioned previously, the only objective scores used in analysis of the
results are four of those shown in Table 12, which at best serve as rather
gross indices of the speed, and perhaps efficiency, with which the problem
was mastered, at whatever level of understanding,.
analyses will be presented in Chapter
The results of these
9.
'l.:l:.__ Ratings of the Level of Understanding of the Four Group Task and of
the Axioms,
Scoring of the level of understandi.ng of the Four Group Task,
and of its axioms, as revealed by answers to the questioning that followed
attainment of criterion, remains to be described.
This questioning was
divided into the three sections already outlined in sections
and
4.4.3.3.
4.4.3.1, 4.4.3.2
Appendix II, which shows the details of the questioning, also
shows the level of understanding ascribed to each type of response to the
questions.
A summary of these ratings of level of understanding and the
189
rationale behind them, will be given here for each of the three sections.
Here again, the question of the reliability of such ratings
arises and the position with .respect to the .Four Group Task is identical
to that described in Chapter
6 for the Pendulum Problem.
As with the
Pendulum Problem, an attempt was made to rate only those aspects of
performance on the Four Group Task which could be considered objectively,
requiring a minimum of interpretation.,
Again, the only indication of
reliability available is the extent to which re-ratings made by the same
experimenter, after an interval of six
original ratings.
months~
are consistent with the
As before, whereas the original ratings were made by
listening to the taperecorded interview, in conjunction with answers
recorded on the questioning protocol (see section
the re-ratings were done without
~1earing
4.4.3.2
in particular),
the complete interview
again~
but using annotations made on a specially designed interview record.
Parts I and III of the questioning led to ratings of the overall level of
understanding of the Four Group Task.
The categories used were IIA and IIB
(Concrete Operational), ?III (Transi tionall, and IIIA and IIIB (Formal
Operational) and the number of subjects for whom the re-rating was different
from the original rating made six months earlier, was
6
(2,
14
year-olds;
3, 13 year-olds and 1, 11 year-old) out of a total of 236 (2.5%),
th" six ratings changed was of the performance of a subject
Each of
borderline
between the later Concrete Operational Stage (IIB) and tl1e Transitional
Stage (?III).
Such subjects were placed in Stage ?III only if they readily
grasped a formal operational system at the end of the interview, having
shown a concrete operational level of understanding up to that point. More
190
stringent criteria for distinguishing those who l'readily graspedn a system,
from those who did so only with considerable help, led to the unreliability
of ratings of these particular cases.
Fortunately, as will be pointed out
later, there were only 26 cases placed in the Transitional Stage in the
total sample, so that, for the majority of subjects, the stage rating on
the Four Group Task (by the one experimenter) can be regarded as extremely
reliable over time.
Part II of the questioning concerned four of the group
axioms and a property of the structure referred to as nconcrete reversibi Jityn ~
There were thus five separate ratings of the level of explanation given (one
for each "item") for each of 236 subjects.
33
Of the total of 1180 such ratings,
were altered in the subsequent re-ratings (2,8%),
Thus for the Four Group
Task, as for the Pendulum Problem, it can be concluded, with considerable
caution, that the ratings made are reliable indices of the level of perform-
ance.
Details of these ratings wi 11 now be given.
7.2.1
The derivation of stage categories from ratings of the overall level
of understanding of the Four Group
Tas~.
Rating of the overall level of understanding draws on answers in
both the Initial and Final Questioning Phases.
These will be discussed
separately and t1Te1T combined,
7.2.1.1.Ra.ting of answers to initial general questioning
after the learnil}_g_
criterion had been reached.
There are three parts to the initial questioning - a first general
question giving no direction to the subject;
a second more probing question,
dependent on the type of response made to the first;
and a final section
191
with standardised questions designed to test the knowledge and use of any
rules or ideas stated in the first two parts.
Details of this questioning
are in Appendix II, Part I.
On the basis of answers to all three parts of this questioning,
it was possible to assess a subject's overall approach to learning the
combinations, using the categories described be low~
The combinations could be learned:(A) Separately and by rote memory alone.
(B) With some grouping together of similar pairs as a mnemonic device.
(The most commonly occurring ngroups 11 were (i) Yellow with other
colours but not with itself (ii) the three combinations of Red,
Blue and Green (or perhaps two only, with RG omitted) and (iii) the
ndoublen ones, YY ~ GG, BB and RR..
These are exactly the same
'
sections of the matrix which Dienes and Jeeves (1965) state are
grouped together by the "pattern" strategist.
Other types of
grouping of pairs obtained were (iv) the pairing of
~
(rarely
all) combinations(s) with "the same one(s) back to front" (i.e. the
commutative property was used) and (v) a variety of idiosyncratic
selections of ua dark and a light colourn or nthe ones I like bestuc)
Those combinations not amongst the ones grouped together are remembered by rote memory. 1'
(C) With all sixteen combinations included in mnemonic groupings (such as
those described in Type (B) above) - none being remembered purely by
rote.
192
(D) By the discovery of an operational role for one (or possibly two) of
the elements, but with the remaining combinations remembered purely
by rote memory (as in Type (A) above) or by mnemonic structuring (as
in Type (C) above) or by a mixture of both (as in Type (B) above).
Most commonly, when an operational role was stated for only one
element, it was for the Yellow light.
(E) With the combinations structured by rows or columns of the matrix,
but without a role for each of the lights being understood.
most common version was nrt goes
around the panel.
at Y, with G it starts at G, and so on°.
The
With Y it starts
If such a structuring was
achieved at all, it seemed always to extend to include all sixteen
combinations.
(F) With the combinations structured by rows or columns of the matrix (as
in Type (E) above) but with a statement of an operational role for at
least one, but less than four, of the elements.
Most commonly, an
operational role was stated just for Yellow, less often for Yellow
and one or both of Green and Red.
A role for Blue was never given
without roles for the other three as well (as in Type (G) below).
(G) By means of finding some operational "role" for each of the four
lights in the system.
higher than
(For example;
Y=O, G=1, B=2, R=J and numbers
3 are equal to their remainder after division by 4;
a ncyclicu system of moves - in moves
or
to the right (for example)
Yellow moves any light 0, Green moves it 1, Blue moves it 2, and Red
moves it J).
193
The most difficult distinction to make was between a grouping
together of combinations, involving the Yellow
mnemonic device;
light~
~~role
and one which indica ted that some
light was understood.
which was merely a
11
for the Yellow
The main distinguishing feature seemed to be that
YB~
children who were grouping combinations such as YG,
YR (often one was
omitted) as simply "easy to remernbern, almost invariably failed to add YY
to the list.
The combination YY was more likely to be said to belong with
the ndouble ones'\ which were a different
matter~
On the other
hand~
a
child who saw these combinations in terms of a role played by Yellow (e.g.
"the Yellow light makes no difference to the light it goes with"), applied
this role to the situation ueven when it goes with itselfni and seemed
to see that this was a necessary part of the system.
The verbal
expressions used by the children were a less reliable indication of the
same thing.
Typically, a child who saw
merely that YG=G and YB=B are
similar combinations, and easy to remember, tended to recite them in full,
whereas the child who understood a urolen for Yellow was much more likely
to summarise and speak of ,any of the ones with Yellow 0 or
any of the colours stays the same".
11
With
Yellow~
Sometimes further questioning by the
experimenter enabled a subject who, at first, had merely said that YG=G
and YB=B were uthe easy ones n, to progress to the point of stating what it
was that made them so, and thence to some thing like a "role" for the Yellow
light.
In rating the subject's general level of approach to the learning
of combinations then, his response was classed as one of the types (A) to
(G) described above.
A record was also kept of whether the level indicated
194
by the first spontaneous answer was different from that reached after
further probing questions.
It should be noted that the types (A) to (G)
correspond fairly closely with those described by Dienes and Jeeves (1965),
except that the ones which involve structuring the matrix by rows or
columns (Types (E) and (F)) apparently did not occur in their study.
This
can probably be explained by differences in the form in which the task was
presented.
Lights which are switched on, arranged permanently on a panel
(in the appropriate order), are much more conducive to this structuring
than is a single card appearing in a
window~
which is replaced, after a
change in the combination, by a new single card.
The classification system described above formed the basis for
assigning subjects to the Stages of Concrete and Formal Thinking on the
Four Group Task.
The level of understanding shown in this initial question-
ing phase (Part I) could be altered, however, by new insights gained during
the next phase of questioning (Part II) about the axioms of the group.
To
discover any such new ideas, the general questioning (Part III) at the end
was included.
This will be discussed next, so that the overaLl method of
assignment to stages can be described.
The questioning about group axioms~
the scoring of which is independent of the overall assignment to stages,
will be discussed last.
7.2.1 .2
Rating of answers to the general questioning at the end of the
interview.
As a. final attempt to discover any ideas which the subject may
have omitted to mention
earlier~
or which may have occurred to him during
195
the course of the questioning about group Axioms 7 a very broad uanything
else you can tell me ?n question was introduced..
Any ideas volunteered
were explored to see if they had relevance to an understanding of the
task on a formal operational level.
If the suggestion that it was a bit
like numbers was NOT made at all by the subject, then he was asked directly
whether there was any resemblance.
in numerical terms he was
If he felt that the task could be seen
questioned to see how much progress he could
make toward finding what the numbers would have to be.
Some subjects left
it at nthey could really be any numbers you like" or nyou couldn't find out,
you would have to be told";
others went as far as suggesting that Yellow
would be 0 or 1 or ua small number";
others were able to add some values
for Green, Blue and Reel (e.g. a 0, 1, 2,
to db for a
4 or 5 or 6;
3 system) but did not know what
and others were able to arrive at the idea of
beginning again, so that Yellow was also
4, Green also 5 and Blue also 6
etc.,
TI1e details of the questioning procedure in this final stage are
shown in Appendix II, Part III, and it is apparent that the spirit in
which it is conducted is one of seeing whether the subject is able to go
further than he or she has before, with considerable npushingn ..
It is
therefore not legitimate to regard the formal operational systems (usually
numerical) arrived at in this way, as part of the child's own spontaneous
way of thinking about the problem.
However, if at this point, a child can
readily produce and grasp a number system which fits the structure of the
combinations (with some guidance), it does suggest that he has the necessary
operational structure to take that appro0ch himself.
The fact that he did
196
not do so spontaneously could be purely accidental, or it could be due to
a lack of confidence in his ability to work something out without instruction, or it could be because the structures are just emerging and hardly
ready for use,.
From the point of view of stage allocations, it seemed advisable
to be cautious about accepting an operational system derived, with
help~
in
It was therefore decided that the allocat-
the Final General Questioning.
ion of subjects to the Stages of Concrete and Formal Thinking should be
clone
principally in terms of their responses to the Initial General
Questioning.
One special category was aclclecl, for those who eli d not qualify
as formal thinkers in the initial questioning, but who, in the final question~·
ing, grasped an operational number system very readily, showed understanding
of its "cyclic" nature, and perhaps also realised that many of the rules they
had talked about before could now be explained in a new way (the Commutative
Law, Associative Law, and in some cases the Law of Inverses).
was labelled as a transitional stage, n7 Formal
This category
Operationaln~
The complete method of allocation of subjects to developmental
stages can now be described.
7.2.1
.3
It is in Section 7.2.1
.3
below.
Stage allocations based on the ratings of overall level of under-
standing of the Four Group Task.
Sections 7.2.1 .1 and 7.2,1.2 have described the way in which
responses in the Initial and Final Questioning Phases were used to make
ratings of the level of understanding of the Four Group Task.
These ratings
form the basis for the allocation of subjects to Five Stage categories;
197
Stages IIA and IIB (within Stage II of Concrete Operations), Stage ?III
(Transitional), and Stages IliA and IIIB (within Stage III of Formal
Operations).
The method of allocation to these stages, using the ratings
described above, is shown in Table
13 below,
these allocations was laid in Chapter
TI1e theoretical basis for
3.
I
TYPE OF APPROACH AND
UNDERSTANDING SHOWN
In Initial Questioning
(i)
(ii)
Spontaneous
Level
answer
reached
after
'\,probing
Type (A)
In
I
Final~stioning
I
I
I
Type (A)
Something less than a
readily understood
number (or cyclic)
system.
.....
Type
(A), (B), (C)
(D) or (E)
Type
(B),(C)
(D) or (E)
Type
(A), (B), (C)
(D) or (E)
Type
(A),(B),(C)
(D) or (E)
A complete number (or
cyclic) system readily
understood.
Type
(A), (B), (C)
(D) or (E)
Type
(F) or (G)
With or without a
complete system understood fully.
Type (F)
Type (F)
.
Type (F)
Type (G)
.
.
Type (G)
I
.... ..
I
I
Table 13:
I
I
STAGE
-DEVELOPMENT
OF
..
Concrete Operational
(I:) .. .pr_ob;>b.ly . IIA.
. . .
Concrete Operational
(II), probably IIB.
?Formal Operational
(?II I) (Transitional)
I
Formal Operational
(III), probably IliA,
I
.. ...
Anything further unnecessary, (although a
cyclic system may here
be turned into numbers)
I
.......
Formal Operational
(III),, probably IIIB.
Allocation of subjects to developmental stages on the basis
of res.ponses to questioning, in the Initial and Final Phases,
on the Four Group Task.
198
It should be noted again
that the number of subjects classified
as Transitional (? Formal Operational) was quite small (26 out of the 2.36
subjects);
and also that subjects who arrived spontaneously at an operat-
ional system during questions about the group axioms, and before the help
given in the final questioning, were categorised as transitional or- as
formal operational on their merits .. The number in this la tte:r category was
very small indeed C4 out of the total of 2.36).
These four subjects are
marked with an asterisk in Table 14A of Appendix V, because they are
exceptions to the rules (in Table 13 above) which determine tile overall
stage of development on the basis of responses in the Initial and Final
Questioning phases,
11 year-olcls,
Two of the four are 12 year-olds and the other two are
There are two further exceptions to these rules, also marked
with asterisks in the same table,
The first is an 11 year-old subject who
was rated as ?III overall, despite the fact that he reached Type (G) in the
Initial Questioning, because his grasp of the system was extremely shaky.
The second is a 12 year-old rated as IIIB rather than IIIA, with an
initial response of Type F, which remains Type F throughout.
TI1is was done
because
the questioning rather than her understanding was at fault,
7.2. 2
Ratings of the level of understanding of "concrete reversibility"
and of four of the group axioms.
Following the initial general questioning, a number of questions
designed to test for understanding of concrete reversibility and four of the
group axiomswas asked,
These are shown in detail in Appendix II, Part II,
with indications of the types of response obtained and of the level of
understanding inferred in each case.
In making ratings of the answers to these questions 1 there are
two additional aspects to be considered, in conjunction with the basic
fact of whether or not the axiom was understood.
The first aspect concerns
the degree of confidence or ease with which the child understands the
axiom~
To give an indication of this, a response was rated
n+n,
for any
axiom, when it (the axiom) was expressed by the child with little or no
help from the experimenter.
If, on the other hand, it was expressed only
after the probing and guiding questions shown in Appendix II, Part II, the
rating was "0" (understood with difficulty).
Thirdly, if not understood
at all, the rating of the child's response for that axiom was "-"
TI1e second aspect is a more important one.
It relates to tloe
subject's overall level of understanding of the structure, and concerns
the nature of the elements in terms of which he tries to understand and
explain the axioms.
He has a choice between talking about "Switching on
a Green light and then a Blue one 11
-
essentially concrete, real actions;
and talking about "A move of one to the right, then a move of two to the
rightn -which are actions, abstracted from the situation;
to "explain" the behaviour of the lights.
in his attempts
It was argued, in Chapter 3, that
the first is a concrete operational approach, where the elements of the
structure are representations of actual physical actions;
whereas the
second approach is one involving formal operations, which represent (by
some imaginary action) the roles of the lights.
It will be seen that three
of the group axioms (the Commutative Law, the Unit Element and the Associative Law) can be approached differently, but quite adequately, using
200
either concrete or formal operations as
elements~
·where the Law of
Inverse Elements is concerned, however, no account can be given, except
in terms of some system of formal operations which represent the roles of
the lights.
Thus the uinverse Element 11 , as a form of .reversibility, does
not exist in a concrete operational system.
The form of reversibility
which is present in the system of concrete operations 1s a much simpler
one, involving just the nundoing" of a concrete action itself,.
Specifically~
in the concrete operational system, turning on a Green light is ureversedu
by turning it off, but for a formal operational system it may also be
reversed by turning on the inverse element,
Red~
nTurning on Gu and
nTurning on Ru can only be seen as inverses if the subject thinks of them in
terms of nactionsn or urolesu such as nmoving one to the rightu and nmoving
one to the left",
Thus, in rating the responses of subjects to this section of the
questioning, a decision was made as to whether each axiom was understood
(if at all) in terms of a system comprising concrete operations (turning
lights on and off) or formal ore rations (numbers, or moves around the
panel) as elements,
This was indicated by the letters C and F respectively,
placed after the rating of understanding as
~~.t•
was necessary, of course, if the rating was
u-")~
or ''0' 1 (no such indication
The axioms wi 11 now be discussecl, one by one, to show the types
of response which were obtained, and the corresponding ratings made~
Com~
plete details of the analysis of responses, forming the basis for these
ratings, are to be found in Appendix II, Part II.
As mentioned previously,
no questioning about the Axiom of Closure was possible, since this was
201
explained to subjects in the instructions..
The questiuning about concrete
reversibility is, however» included here, and will be treated
Rating
fi_rsT~
Abbreviation
(i) Con ere te Rever si bi l i ty
Persistent suggestion that other
lights would have to be turned on
in place of one (or both) of those
already there.
0; Concrete
oc
Suggestion that one of the lights
be turned off, leaving the other
one, arrived at orily after considerable help from E - with no
expression for what has been done
except nB has been turned offn
(i~e
.. no ideas such as nsubtracted"
or nmoved back twon) ~
+; Concrete
+C
As for OC above, but answer given
spontaneously, with little or no
help.
0; Formal
OF
An answer stating that turning a
light off plays the opposite role
in a system to turning it on (e.g.
Turning B off moves the other light
uback two 11 since turning it on
moves it
11
forward twon)- bu. t
202
Rating
Abbreviation
].'ype of Re'U'_ons".
arrived at only after a good deal
(i) Con td.
of encouragement f:rom E •
.j,.; Formal
+F
As for OF
above~
but answer given
spontaneously, with 1i ttle or no
help.
(ii) Commutative Law
Even after considerable help,
cannot make any comment about the
same pair in the opposite order;
perhaps maintains that the answer
is different.
0; Concrete
oc
Expresses a commutative understanding (perhaps only for some pairs,
not all) after a considerable amount
of help from E.
The law is expressed
in terms such as nswi tching on B and
then R is the same as switching on R
and then B11
*; Concrete
+C
i
~eq
concrete actions.
As for OC above, but with little or
no help from E ..
0; Formal
OF
Expresses the fact that the order of
performing two formal operations
(e.g. +2+3;
or moving 2 and then
3
20.)
Rating
Abbreviation
Type of
Respon?_~,
to the right) does not affect the
(ii) Contd,
result - but needs considerable help
from E to do so.
+; Formal
+F
As for OF above, but with little or
no help from E.
(iii) Unit Element
Shows no understanding that combinations involving the Yellow light are
different from others, or can be
grouped and described in a particular
way,.
0; Concrete
oc
Expresses the idea that combinations
such as YG"'G, YB=B, YR=R (and perhaps
YY=Y) are neasyn and
usimilae~,
and
with some help from E. can express a
rule in a form such as "they stay the
same with yn or ny doesn't show up
very much in these
mixturesa~
The
response does got, however, indicate
any nroleu for Yellow in a formal
sys tern.
+; Con ere te
+C
As for OC above, but with little or
no help from F
to express the rule.
204
Rating
Abbreviation
:I:~f
Response
(iiD Contd.
0; Formal
OF
Accounts for the results of combina tions with the Yellow light by saying
uWell 1 Yellow is zeron or uYellow
doesn't move the light it goes with
at all" i.e" Yellow has some meaning
as a ttni t element in a system of
formal operations - however some help
from E is needed to arrive at the
formulation of the role.
+; Formal
+F
As for OF above, but with 1i t tle or
no help from E.
(iv) Inverse Element
No reason can be given for the results
of the combinations GR, BB and YY all
being Y.
These are perhaps labelled
the nhard onesn or "silly ones" 1 for
such reasons as ndark colours can 1 t
make a light colouru ..
0; Concrete
oc
Responses on a concrete level do not
+; Concrete
+C
occur, since no statement of inverse
roles of elements can be made when the
elements are seen purely as the concrete actions of nTurning on a Green
205
Rating
Abbreviation
light" etc.
(iv) Contd.
0; Formal
Type of Response
OF
Is able to see that lights which, in
combination, make Y are ugetting back
to the beginning again" and, with
help from E 1 arrives at some ninverseH
notion:- that they must nneutralise
each otheru, "cancel outn or nbe
opposi tesn.
From this there may or
may not be a progression to seeing
exactly why the formal nrolesn,
representing members of the pairs
(R and G) (B and B) and (Y and Y), are
uopposi ten. (R and G can be seen as
-1 and +1 or as +J and
-3
respectively;
B can be seen either as +2 or -2;
Y
is a trivial case).
+; Formal
+F
Is able to state why the pairs of
inverse elements make Y, in terms of
their formal roles in a system, without
very much help from E.
206
Rating
Abbreviation
(v) Associative Law
Answe1~
stating that the result would
be "Yellow, because that is the only
one leftn, or using some other ad hoc
rule;
o:r alternatively a statement
that "you couldn't know because for
three panels the rules would be
different."
0; Concrete
oc
With considerable help from E, is able
to work out, at least for one specific
set of three lights, that the results
of at least two of the three possible
methods of combination are the same and to say that this is to be expected.
There may be no reason offered as to
why it is expected, or perhaps the
Commutative Law is wrongly invoked as
a reason.
The elements being combined
are the concrete actions of "turning
on G11
+; Concrete
+C
etc~
As for OC above, but with little or no
help from E.
+; Formal
OF
With help from E, is able to see that
the same three combined in each possible
207
Rating
Abbreviation
-------
(v) Contd.
Type of Response
way will give the same result, and then
realises that this must be
so~
in view
Of the Hroles" the lightS have;-
these are just numbers 1, 2,
nrf
3 to add,
then of course it doesn't matter how
you add them upn etc
+; Formal
+F
I;
As for OF above, but with little or
no help from E.
A number of extracts from interviews with s'ubjects, on the Four
Group Task, are to be found in Appendix IV.
These have been edited some-
what in the interests of brevity and grammatical accuracy, but are
otherwise verbatim.
For each extract, the overall stage allocation, and/
or the ratings of level of understanding of the group axioms and of
concrete reversibility, are shown, as appropriateQ
For individual subjects tested, ratings of performance on the
Four Group Task, together with the stage allocations, are in the tables
of raw data in Appendix VII,
An indirect confirmation of the validity of ratings made is the
fact that, although two different experimenters participated in the testing,
the criteria fot
making ratings of performance were found to be equally
applicable to the interviews conducted by each,
Thus the idiosyncracies
in the questioning techniques of the two experimenters did not have an
effect on the levels of understanding displayed by subjects.
208
CHAPTER 8
TREATMENT OF DATA USING ONLY CATEGORISATION OF PERFORMANCE ON THE TASKS
8.1
General Comments
With data of the kind obtained in this study, reservations are
held about the application of quantitative statistical techniques, not
only for statistical reasons, but also because Piaget's account of the
development of thought distinguishes qualitatively different stages,
Although it has proved possible to derive quantitative scores on the two
tasks~
which reflect performance in the same way as do categorical stage
allocations~
analysis of the data using non-quantitative techniques is
preferred and will be discussed first.
TI1Us in this chapter performance
(in terms of stage categories) on the tasks is related in the first
instance to age, considering each task separately.
The relationship of
the two tasks to each other is then examined by similar techniques.
Next,
the relationships of sex, and of school attended, to performance on each
of the tasks are examined.
Finally, the level of understanding of each
of the group axioms and of "concrete reversibility" is related to the
overall level of understanding of the Four Group Task.
of the group axioms as ai temsn is also investigated..
The scalability
In the next chapter
quantitative scores on the tasks are introduced, and consideration of the
effects of I.Q. and mathematics ability is left until then.
Measures of
the efficiency of learning the Four Group Task will also be examined in a
quantitative way in the next chapter.
209
8"2
Levels of Understanding of the Four Group Task
7 described the way in which subjects were ,;ategorised,
Chapter
firstly on their initial response to questions about nhow the game works u
(categories A to G) and secondly taking into account the subsequent development of their ideas during initial questioning (categories IIA, IIB,
IIIA and IIIB).
The second categorisation also allowed rating of a child's
performance as characteristic of Stage II (IIA or IIB) on the initial
questioning, to be changed to ?III, because of his response in the final
questioning period,
The next two sections will show the relationship of the initial
response categorisation to the overall categorisation (described above)
into Stages IIA, IIB ?III, IliA or IIIB;
and then the distributions of
subjects in each of the age groups over these two sets of categories"
8.2.1
Relationship of initial response to overall stage allocation
Table
14
below shows, for all subjects combined, the number whose
initial response was of each of the types A to G.
Within the table these
subjects are broken down into those who stay at the level of thei.r initial
response and those who change during the initial questioning(,
210
Initial Response Category
A
B
c
D
E
F
G
Number that change
8
5
5
3
7
23
-
Number that stay
40
114
4
9
3
3
12
185
Total Number
48
119
9
12
10
26
12
236
• 17
.04
.56
.25
.70
.88
-
.22
Proportion changing
Table 1!J.:
Total
II
51-i
I
Numbers and proportions of subjects changing from each
of the initial response levels on the Four Group Task.
It is clear that the tendency to change is much greater, the
higher the level of the initial response.
This is demonstrated by a Chi-
square test of association performed on the data in Table 15 below.
Categories C, D and E have been grouped together to avoid small marginal
totals.
Category G has been excluded, since no change from it is possible.
F
Total
Initial Response Category
A
B
C,D,E
Number that change
8
5
15
23
51
Number that stay
40
114
16
3
173
To tal Number
48
119
31
26
224
Table 15:
The relationship of initial response ca tego£1: to
tendency to change,. The value of Chi-square for the
table is 99. 72, elf = 3, P< .001. The contingency
coefficient of association is 0.555.
211
Details of the levels to which subjects moved, from the various
initial levels, and of the age groups to which they belonged can be found
in Table 14A, Appendix V,
to move J.s
greater~
A summary of the trends is that the tendency
the older the subject;
greater, the higher the initial leveL
and the movement is also
To illustrate the second point,
seven of the eight subjects who move from level A move only to level B
(the other moves to G), whereas most of those who move from B, C, D or
E do so to G.
Those at F can only move to G,
It can be argued that the subsequent probing which occurred in
the initial questioning acted primarily to complete a formal structure
where it was incomplete (as in a move from F to G), but not radically to
alter the level of approach to the problem.
T11e fact that the two bases
of categorisation yield comparable results cannot be demonstrated statistically, since the classification into stages depends partly on the initial
response level and is therefore automatically related to it.
However Table
16 below shows the degree to which a division of subjects into those at
levels A? B and C ("Concretett) and those at levels D, E~ F and G (n? Formal
and Formal") on initial response would predict their classification later:;
as at Stages IIA, IIB ("Concrete") or at Stages ?III, IIIA, IIIB ("? Formal
and Formal").
It can be seen that a total of 30 out of 236 subjects (6
ten year-olds, 6 eleven year-olds, 2 twelve year-olds, 8 thirteen year-olds
and
8 fourteen year-olds) would have been differently classified, had only
the initial level of response been taken into account,
Thus i t appears
that a subject 7 s initial response to the questioning can be used as an
indicator of the stage of concrete or formal thinking which he has attained,
~
l);erall Classification
~
Classification
Ove~a.ll
IIA,IIB
!!:!l...lli.!
k'!U
?III,lliA,IIIB
A, B,C
]8
6
44
D,E,F,G
0
4
4
J8
10
48
Total
liA, IIB
Total
!!!..UiU
~
JJ
6
J9
D,E,F ,G
0
9
0
JJ
IS
48
Total
l 1 Y&Ali.S
12 YllARS
~
~
A,s,c
?III,IIIA,tllll
Overall Classification
Total
40
39
D,B,F,G
Total
4
40
IIA, IIS
!!!.!.!W.
~
44
A,B,C
Initial
~
D,I!,F,G
Total
19
7
]6
Total
"
"
30
"
48
TOTAL SAMPU!
?III,IIIA,IUB
14
5
Total
29
14 YEARS
A,B,C
?IH,lllA,IIIB
D,B, F,G
Overall Classification
IlA,I!B
Total
A,B,C
Overall Classification
IIA, liB
l!Il,IIIA,lllll
Overall Classification
Total
IIA,IIB
?lit,lliA,!llB
Total
153
,,
176
17
,,
'9
!.!.!i.ti!!
31
~
A,B,C
D,E,F,G
53
""
76
'"
48
Total
Table 16. Performance on the Four Group Task. The tables show
the relationship of initial response level to the final
~lassifica.tion for eacll age group and the total sample.
Since the two classification bases are not independent
no statistical tests of association are possible,
160
213
A more detailed breakdown of the figures presented in Table 16 can be
found in Table 14A of Appendix V"
8,2,2
Distributiom of age groups over categories on the Four Gro'!l2_Task
The fact that the initial response level and the final stage
allocation of subjects give much the same classifications can be seen in
the similarity of the distributions of age groups over the two sets of
categories.
8,2.2,1
TI1ese distributions are treated in Sections 8.2.2,1 and 8.2.2.2.
Relationship of initial response category to age.
Table 17 below shows the number of subjects of each age giving
initial responses at two levels (A. B or C) and (D, E, For G), and the
degree of association is tested by Chi-square.
Initial
Response
Category
10 yrs,
11 yr s.
Age Group
12 yrs.
13 yrs,
14 yr s.
44
39
40
36
17
176
4
9
4
12
31
6o
48
48
44
48
48
236
Total
r-A, B, c
D, E, F, G
Total
Table 17:
The relationship of initial response category to age.
The Chi-square value for the table is 53.539, df~14,
p<.001 and the contingency coefficient of association
is 0.430.
214
In order to decide which age groups differ from one another,
Chi-square tests were done in 2 x 2 contingency tables for each pair of
age groups.
The results of these appear in Table 18 belowo
reversal of the decision marked by an
elusions can be drawn as follows:-
asterisk~
With the
a consistent set of con-
that the distributions of 12, 11 and
10 year-olds do not differ amongst themselves, but do differ from those
of the 13 and 14 year-olds, which f!lso differ from each othec
expression "(10
= 11 = 12)
The
f. 13 f. 14" describes the conclusions, if "10"
can be interpreted to mean the distribution of the 10 year-ole! age group
over categories of initial response level
and similarly for
"11u~
n12n,
AGE GROUP
14
14
13
AGE
GROUP
12
13
12
11
10
15.197
pC001
29.983
p< 001
20.736
p<.001
32.774
p<. 001
4.039
p<.05
0.547
NS*
4.790
p<.05
1. 757
0,009
0
NS
11
NS
2.218
NS
10
Table 18:
Detailed comparisons of age group distributions on
initial response. Entries show the value of Chisquare and its significance level for comparisons
of each pair of age groups in 2 x 2 contingency
tables. If the difference between 13 and 11 years,
shown as nonsignificant*~ is taken to be significant,
then a set of consistent conclusions is (10 = 11 = 12)
f.13i14.
215
8.2,2,2
Relationship of overall stage allocation to age,
A comparable analysis of the age distributions across overall
stage allocation gives results congruent with those of the above section.
It is convenient to distinguish just three stages in overall performance
for the purposes of this analysis.
These wi 11 be a. Formal Operational
stage (III) comprising IIIA and IIIB, a Transitional Stage (?III) and a
Stage of Concrete Operational Thought (II), comprising I!A and IIB,
The
frequency distributions for the five age groups, and for the total sample,
over these stage categories are shown in Table 19 below, together with the
values obtained in a Chi-square test of
Overall
Stage
N
10 YRS.
%
N
III
3
6
8
7
15
II
38
TOTAL
48
?III
I
Table 19:
11 YRS.
association~
12 YRS.
13 YRS.
%
'OT~~
%
%
16
3
7
15
31
29
62
58
25
7
15
1
2
3
6
8
16
26
11
79
33
69
40
91
30
63
11
152
64
100
48
100
44
100
48
100
48
100 1236
100
%
N
%
14 YRS.
N
N
221
N
The relationship of overall performance on the Four Group
Task to age, The Chi-square value for the table is 65.346,
d, f ,-8, p < ,001 and the value of the contingency coefficient
of association is 0.466. Graphs of the percentage of
subjects in each category over age are shown in Figure 3(a),
page 220.
I
216
In order to decide which particular age groups have different
distributions over the stages, Chi-square tests were performed in all
possible 2 x 3 contingency tables and the results of these are presented
in Table 20 below.
Overruling the result
of the comparison marked with
an asterisk leads to a set of cons is tent conclusions identical to those
in section 8.2,2.1,
These are that the distributions of the 10, 11 and
12 year-old age groups over the stage categories do not differ from one
another, but do differ from those of both the 13 and 14 year-olds, which
also differ from each
other~
These conclusions may be expressed, as
before, by "(10 = 11 = 12) cl 13 I 14."
AGE GROUP
14
13
~
12
GROUP
11
13
12
11
10
15.532
p <. 001
42.967
P<.001
22.985
p<.001
36.069
p<.001
10.274
p<.01
3.873
NS*
10.5410
p<.01
7.283
NS
4.385
NS
2,625
NS
10
Table 20:
Detailed comparisons of age group distributions across
Overall Stages on the Four Group Task. Entries in the
table are Chi-square and probability values for each
of the possible 2 (age) x 3 (stage) contingency tables.
The comparison of 13 and 11 marked with an asterisk
has to be regarded as yielding a significant difference
to arrive at the set of consistent conclusions
C1o = 11 = 12) t 13 I 14.
217
It is clear from these results, and those of the previous section,
that the performance of the eleven year-olcls is more comparable with that
of thirteen year-olcls than the stated conclusions indicate.
In fact the
eleven year-olds appear to perform at a higher level than the twelve year·olcls.
This fact will be discussed further in a later chapter.
At this
point it is sufficient to note that, to arrive at this conclusion
statistic~
ally, amongst a set of compatible conclusions about the remaining age
groups, would necessitate the overruling of more than one of the results of
the tests of significance between pairs of age groups.
The set of con-
clusions proposed above (see Tables 18 and 20) overrules only one such
result (the same one) in the analyses of initial responses and overall
stage allocations.
For the remainder of this chapter, the set of categories of
performance on the Four Group Task to be used will be the overall stage
allocation, discussed in this section.
The results of section 8.2.2.1.
on initial response levels will be important for the discussion in Chapter
10, but not used any further in analysis of the data.
8.3
Levels of Performance on the Pendulum Problem.
Chapter
6
described the way in which a subject was assigned to
one of nine categories of performance on the pendulum problem.
All nine
of the categories were tentatively labelled as substages and transitional
stages within the period covering Concrete and Formal Thought, and age
distributions over the nine categories can be found in Table 21A of
Appendix V"
However, to demonstrate the relations hip of performance on
218
this problem to age
here~
three stage categories will suffice.
These will
be Stage IIIB (comprising IIIB and IIIB?A), Stage IIIA (comprising IIIA?B,
IliA and IIIA?IIjB) and Stage II (comprising IIB?IIIA, II!l, IIB?A and
IIA?B).
The frequency distributions of each of the five age groups, and
of the total sample, over these three categories are shown in Table 21
below, together with the value obtained in a Chi-square test of association,
10 YRS.
Stage
11 YRS.
%
12 YRS.
I
%
N
IIIB
4
8
8
17
11
23
20
42
27
56
70
30
IIIA
22
46
27
56
23
46
16
33
17
36
105
44
II
22
46
13
27
10
21
12
25
4
8
61
26
N
%
I
TOTAL
%
N
%
14 YRS.
N
N
%
13 YRS.
N
I
__j
TOTAL
48
Table 21:
100
48
100
44
100
48
100
48
100
2~:~
The relationship of performance on the Pendulum Problem to
age. The Chi-square value for the table is 41.852, d,f .=8,
P<.001 and the value of the contingency coefficient of
association is 0,388, Graphs of the percentage of subjects
in each category over age are shown in Figure J(b) page 220 ..
The set of conclusions about differences between particular age
group distributions, justified by individual comparisons, is slightly
different from that reached for the Four Group Task in sections 8.2,2.1
and 8,2,2,2.
Table 22 below shows the Chi-square and probability values
for the individual 2(age) x 3Cstage) comparisons on the Pendulum Problem,
One of the differences found to be nonsignificant, marked with an asterisk,
has to be regarded as significant to arrive at a set of consistent
conclusions ..
219
In the terminology used
"(10 = 11 = 12)
i (13
earlier~
these conclusions can be expressed by
= 14l".
AGE GROUP
------
14
13
12
11
5.073
NS
10.053
p<.01
17.3514
p<,01
30. 167
p<.01
7.997
p<.05
14.559
p(.001
1 . 013
NS
7.629
NS
13
AGE
3.884
NS*
12
GROUP
10
4.158
NS
11
10
Table 22:
8.4
Detailed comparisons of age grou12._distributions over
stages on the Pendulum Problem. Entries in the table
are Chi-square and probability values for each of the
possible 2(age) x 3Cstage) contingency tables. The
comparison between 12 and 13~ marked with an asterisk,
has to be regarded as yielding a significant result to
arrive at the set of consistent conclusions
(10 = 11 = 12) i (13 = 14).
Comparison of Performances on the Four Group Task and Pendulum Problem.
£JL.l
Detailed comparison of the separate relationships of the tasks to
age~
2
From the previous two sections,
8.j
3
and 8.J, it is clear that the
relationship of age to performance on the Four Group Task is quite similar
to that of age to performance on the Pendulum Problem.
The similarity can
be seen by a comparison of the graphs in Figures 3(a) and 3(b), page 220,
The curves which correspond most closely are that for the percentage
(a)
100
E_Q_l;.:f GROUP TASK
(b) PENDULUM PROBLEM
r
--e>--o
00
f
so>
A
70
\
/
~
~
Stage II
Stage ?I I I
Stage III
0---o
---..
90
Stagc I I
Stage IIIA
Stage lllB
80
70
6o
60
%
of
,______..
100 [
%
5o
of
S's.
so
s• s.
40
40
30
30
20
20
10
~
10
11
/
12
AGE GROUP
Figures }(a) and 3(b):
13
10
-
14
YRS.
10
11
12
AGE GROUP
Graphs of the percentage of subjects in each of three stage categories over Eve diff•··rent
age groups,
On the Four Group Task, the stages are Stage II (comprising IIA and IIB),
Stage ?III, and Stage III (comprising IliA and IIIB). On the Pendulum ProblP-rn, the sta;::es
are Stage II (comp.rising IIA?B, IIB?A, IIB, and IIB?IIIA), Stage IIIA (cornprising IIIA?IIB,
IliA and IIIA?B), and Stage IIIB (comprising IIIB?A and 1118).
13
14
YRS.
221
at Stage III (comprising IIIA and IIIB) on the Four Group Task and that for
the percentage at Stage IIIB on the Pendulum Problem,
To explore the
relationship between the tasks more fully, it is worthwhile to consider
more detailed breakdowns into stages.
The total number of stages identif-
ied on the Four Group Task is five, and the total numbers of subjects (all
ages) falling in these categories are 360IIB), 22(IIIA), 26(?III),
113(IIB) and 39(IIA).
If the distribution of the total sample over the
nine categories on the Pendulum Problem is examined (see Table 21A of
Appendix V) it is found that a breakdown into five categories with very
similar frequencies to those in the five on the Four Group Task can be
achieved,
I
Stage
The data are presented in this form in Tables 23 and 24 below.
10 YRS.
11 YRS.
N
%
N
IIIB
3
6
7
IIIA
0
0
?III
7
IIB
12 YRS.
13 YRS.
14 YRS.
%
36
36
15
12
25
22
9
6
8
16
26
11
21
44
9
19
113
48
JO
.9
19
2
4
39
17
100
48
100
48
100
236
100
%
N
15
1
2
8
16
17
1
2
2
5
7
15
15
7
15
1
2
3
28
58
28
58
27
61
IIA
10
21
5
10
13
TOTAL
48
100
48
100
44
IableB:
'51
N
N
%
%
N
%
Distributions of age groups and the total sample over five
stages on the Four Group Task, The Chi-square value for
the table is 72.410, d,f.=16, p<,001 and the contingency
coefficient has the value 0.484. For each of the five
stages a curve relating frequency to age group is shown
in Figure 4Ca), page 223,
AGE GROUP
I
I
10 YRS.
%
11
N
12 YRS.
%
N
2
4
6
12
6
14
9
19
12
25
35
15:
IIIB?A
2
4
2
4
5
11
11
23
15
321 35
151
IIIA?B
I 5
10
10
21
9
20
3
6
0
II I A, II IA? IIB' II B?I IIA
I 24
50
19
40
17
39
20
42
21
IIB, IIB?A, IIA?B
__j_ 15
32
11
23
7
16
5
10
0
0
nmu
143
•oo
48
•oo - 44
•oo
48
•oo
48
we
STAGE
N
IIIB
I
Table
2~:
YRS.
%
N
13 YRS.
%
14 YRS.
%
TOTA;:---1
N
%1
N
i
I 27
11
I
4311 01
43
I
0
38
16 i
'~6 '~
Distributions of age groups and the total sample over five st~
categories on the Pendulum Problem. The 01i-square value for the
table is 57o612, d.f.-16, p<.001 and the contingency coefficient
has the value 0.443. For each of the five stage categories a
curve relating frequency to age group is shown in Figure 4(b),
page 223.
[\)
[\)
[\)
(a) FOUR _GROUP TASK
(b) PEI'VpULUM PROBLEM
,..._._.,
100
Stage
Stage
St<1ge
Stage
Stage
)(·----K
t:r---<)
,.___..
oo
·---..e
--·----'»----""
<>---<>
90
80
70
70
6o
,.___,.
100
8o
·-----~---if"---------"'-,,
%
of
IIA
I IB
?Ill
IIIA
I II B
IIA?B, IIB?A, IIB
IIB?IliA, IIIA? IIB,
II IA ?8
IIIB?A
II IB
IllA
6o
%
................
so
.......... ,
S's.
of
'' ',
.40
30
/
""--
..................')f-_______ •
___ ... ___ ......
-------«
40
' ',
' '<''
20
So
S 's.
')(
/
/'
,Jib
30
'
__ ....... _.....
20
10
~.:------
10
10
11
12
AGE GROUP
Figures 4Ca) and
illJ2l:
13
14
YRS.
10
12
11
AGE GROUP
Graphs of tlle percentage of subjects in each of five stages over five age groups.
On the Four Group Task, the stages are Stage IIA, Stage IIB, Stage ?III, Stage IIIA
and Stage IIIB
On the Pendulum Proble·n, stages are grouped as follows:- OIA?B,
IIB?A and liB), (IIB?IIIA, IIIA?IIB and IliA), (!IIA?B), (liiB?A) and (IIIB).
YRS.
224
From Tables 23 and 24, and the graphs in Figures 4Ca) and 4Cb),
quite a close correspondence can be made between the trends over age on
the two tasks.
This involves equations of stages, or groups of stages,
labelled slightly differently in the two tasks;
for example the curve
representing the percentage at Stage IIIA on the Four Group Task is
apparently comparable to that of the percentage at a stag\' labelled
IIIB?A on the Pendulum Problem,
In general, performance at a given
level on the Four Group Task corresponds to performance at a slightly
higher level on the Pendulum Problem.
This may inclic ate either a mis-
labelling of stages on one or both of the tasks, or else a real differ-
ence in the level of thinking at which the tasks
discriminate~
question is taken up in the discussion in Chapter 11,
ing aspects of that
discussion~
This
Without anticipat-
it can be said that reasons were given
in Chapter 5 to expect that the exclusion of irrelevant factors in the
Pendulum Problem might be a necessary prerequisite for performance at
the formal level on the Four Group Tasko
Pendulum performances of a
sufficient standard are those at IIIA?B and above.
It is therefore
interesting to find that the clearest cor respondence between graphs of
stages on the two
tasks~
over
age~
is found when subjects in categories
IIIB, IliA and ?III on the Four Group Task are equated with those IIIB,
IIIB?A and IIIA?B on the Pendulum Problem;
and, the remainder, those
IIB and IIA on the Four Group Task equated with those IliA, IIIA?IIB,
IIB?IIIA, IIB, IIB?A and IIA?B on the Pendulum Problem.
26, and graphs (Figures 5(a) and 5Cb), page
Tables 25 and
226) are presented now,
225
showing this division into just two stage categories on each
task~
AGE GROUP
STAGE
CATEGORY
I
I
I
~
IIIB, IIIA,
?III
'
I' IIB,
IIA
YRS.
%
11 YRS.
N
%
12 YRS.
N
%
8
3
7
15
13 YRS.
N
%
14 YRS.
N
%
TOTAL
N
%
3
6
45
94
40
84
41
93
33
69
19
40 178
75[
48
100
48
100
44
100
48
100
48
100,236
~
16
31
29
6o
58
I
I'
'
TOTAL
Table 25:
25
1
I''
Dis tri buti ons of age groups and the total sample over two
categories of performance on the Four Group Tash_for
comparison with the Pendulu!IJ_!!."oblem, The value of Chisquare for the table is 52.218, d.f.=4, p<.001 and the
value of the contingency coefficient is 0.426. Graphs of
the percentages in these two stage categories over age
are in Figure 5(a), page 226.
,-------,-------------------------------------.-----l
STAGE
CATEGORY
10 YRS.
N
%
11 YRS.
N
%
12 YRS.
N
%
13 YRS.
N
%
14 YRS.
N
%
TOTAL
N
%1
~--------+------------------
IIIA?B and
above
IIIA and
~
4
8
8
16
11
25
20
42
27
56
70
30
44
92
40
84
33
75
28
58
21
44
166
70
I
--48
Table 26:
100
48
100
44
100
48
100
48
1 oo
236
1
ool
___
I __!
Dis tri bu tions of age grOU['S and the total s am['le over two
categories of 2erformance on the Pendulum Problem 1 for
comJ2arison with the Four Grou[' Task. The value of Chi-square
for the table is 34.389, d.f.=4, p <. 001 and the contingency
coefficient has the value 0.357. Graphs of the percentages in
these two stage categories over age are in Figure 5(b), page
226.
(b) PENDULUM PROBLEM
(a) FOUR GROUP TASK
100
,..____...
r
"'
90
"'
·---·
Stages II and ?III
8o
so
70
70
\/
6o
S's.
40
I
f
I
f
.-
.........
10
......................................
......
/
I
/
I
/
I
I
of
I
11
/
So
... ........./
Jll
AGE GROUP
Figures 5Ca) and 5Cb):
////
JO
20
10
lJ
14
YRS.
.........
I
4o
I
12
IT lA
................
S's.
,//
....
Stage IIIB
%
,/
20 f
10
/
Stages II and
6o
/
So
30
~
8-----e
90
%
of
100
Stage III
........... -10
/
--
...................
11
/
,A
12
AGE GROUP
Graphs of the percentage of subjects in two stage groupings over five age groups.
On the Four GJ:o'..lp Task, the two stage groupings are Stages II and ?III (comprising
IlA, IIB and ?III) and Stage III (co~lprising IliA and IIIB). On the Pendulum
Problem the two stage groupings are Stages II and lilA (comprising IIA?B, IIB?A,
liB, IIB?IIIA, IIIA?IIB, IIIA and IIIA?B) and Stage IIIB(comprising IIIB?A and IIIB).
13
14
YRS,
227
8.4,2
Direct examination of the relationship of performance on the Four
Gr£.!:!J2..Jask to that on the Pendulum Problem.
The preceding section,
If
8.';{.
1,
does no more than demonstrate that
performance on the two tasks improves in a comparable way with
age~
Unless
it can be shown that it is the same children who succeed or fail on both
within any age group, no direct relationship between the tasks themselves
can be said to exist.
Unfortunately, with categorical treatment of data,
and unfavourable marginal distributions of subjects within any age group,
it is very difficult to demonstrate such relationships in a convincing
way~
The present data is no exception, although there is one age group, the
thirteen year-olds, where the distributions of subjects over categories on
both tasks is reasonably even,
A comprehensive contingency table, relating
the two tasks, for each of the age groups, is in Table 27A of Appendix V,
There are two ways to attempt to demonstrate the relationship
between the tasks in contingency tables,
TI1e first is to make the same
divisions into categories of performance on the two tasks for each age
group and to examine whether the sum of the five independent Chi-square
values (each with one degree of freedom, if the splits are dichotomous) is
significant on five degrees of freedom,
This would provide evidence that
the two tasks were related over the five age groupsj even if most or all
of the individual tests for separate ages failed to reach significance.
For the purpose of such tests, performance on the Four Group Task is
categorised as either Stage II (IIA or liB) or Stage III (?III, IIIA or
IIIB) and performance on the Pendulum Problem as either "better than Stage
IliA" (IIIA?B, IIIB?A or IIIB) or as "Stage IIIA or below" (IIIA, IIIA?IIB,
IIB?IIIA, IIB, IIB?A, IIA?B),
These are the only breakdowns on the two
228
tasks which result in marginal totals such that Chi-square may be calculated
for the two by two tables.
It should be noted that with this Cas with any
other possible breakdown) the calculation is questionable for the 12 yearold sample, since two of the expected frequencies fall below five.
The
contingency tables and associated Chi-square values are presented in Table
27 below.
Yates' correction for continuity is applied to the tables for
10, 12 and 14 year-olds,
While there is some tendency, in the tables for the 11, 12, 13
and 14 year-olds, for performance on the two tasks to be related, the
association is significant only for the 13 year-olds.
The sum of Chi-square
values for the five tables just fails to reach significance.
If the 10
year-olds are excluded (the grounds for this being tlmt they have extremely
few subjects in the "high" category on both tasks) then the sum of the four
Chi-square values for the remaining age groups is 9,6285, which, on four
degrees of freedom, reaches the
5% level of significance,
While a division of all age groups into the same categories on
each of the tasks enables the convenient summing of
Chi~squares
over tables
as has just been done, other considerations may justify a rather different
approach,
From the results of sections 8 ./Zand 8 .j3(of this chapter) it is
reasonable to assume that the two tasks discriminate best at different
levels of ability for different age groups.
Overall distributions of per-
formance on the tasks are different for the 14 year-old group from those
for the
13 year-old group, and in both cases differ from the distributions
for the 12, 11 and 10 year-old age groups.
While it is recognised that
1
~
Four Grou12 Task
~
Four Grou12 Task
II
>lilA
8
~IIIA
30
Pendulum
Problem
Total
38
Chi-square
?III,l!I
Total
II
9
9
39
10
48
0.1317, df
Pendulum
10
8
18
~IIIA
23
7
30
Total
33
15
48
Chi-square
Pendulum
?III,III
2.3855, df
1, N.S.
13 YEArtS
~
Four Grou12 Task
II
Total
>IIIA
Problem
1, N.S.
?III,Ill
Four Grou12 Task
II
Total
)IliA
17
3
20
~IliA
24
0
24
Pendulum
?III,III
Total
.> I !I A
11
12
23
<.IIIA
19
6
25
Total
30
18
48
Problem
Problem
Total
41
Chi-square
44
3
1.7397, df
1,
Chi-square
N.S.
4.1178, df
1, P< .05
14 YEARS
Four Grou12 Task
II
Pendulum
?III,IJI
Total
>IIIA
4
23
27
~IliA
7
14
21
Total
11
37
48
Problem
Chi-square
1,3855, df
1, N.S.
For the five age groups combined
df "' 5:Chi -square = 9, 7602
,05<p<.10
Contingency tables and Chi-square values for the relationship of Four Group Task performance
to Pendull.llll Problem performance in each age group. The Chi-square, with 5 df, obtained by
summing those for the five age groups is also shown.
230
tables may not be adjusted so as to obtain the most favourable set of
cell frequencies, it can be decided !'__££iori to divide performance into
categories which best suit the performance level of the age group concerned.
There is also justification for combining the 10, 11 and 12
year-old subjects into one group.
Thus~'
in the tables which follow,
new categories of performance have been introduced which discriminate
at the high levels for the
14
year-old group and at the low levels for
the group of 10, 11 and 12 year-olds combined,
The two tables will be
referred to as Table 28,
14
YEARS
Four Group Task
II
?III,IIIA
0
5
7
12
II IB ?A
and
belo w
11
15
10
36
TOTAL
11
20
17
IIIB
Pendulum
Problem
IIIB
TOTAL
--
I
Chi-square = 6.0391, d,f. = 2, p<
contingency coefficient = 0.334.
48
.05.
Value of
231
10' 11 ' 12 YEARS COMBINED
Four Group Task
IIA
I
IIB?IIIA
and abovej
Pendulum
----Problem
IIB
?III,III
TOTAL
17
64
26
107
IIB an~ 11
below
19
3
33
83
29
140
'
TOTAL
I
28
Chi-square = 6.6752, d.f. = 2, p < .05.
of contingency coefficient ~ 0.2133.
Table 28:
i
I
-------4
Value
Contingency tables and Chi-square values for the
relationship of performance on the Four Group
Task to that on the Pendulum Problem, with
categories adjusted for high performance levels
in the 14 year-old group and low levels in the 10,
11 and 12 year-olds combined.
From the preceding analysis it can be concluded, with some
caution, that performance on the Four Group and Pendulum tasks is
related, over and above the relationship due to their separate improvement with age.
With performance measured simply by categorisation on each
task, it is not surprising that the relationship demonstrated is small and
that it appears only when the divisions into categories are such as to
discriminate appropriate performance levels at each age.
Attempts to
partial out other variables such as men tal age, wi 11 not be made until
the quantitative treatment of data is introduced, in Chapter
9.
232
8.S
Relationships of Sex and School to Performances on the Pendulu~
and Four Group Task.
Since each age group divides into almost equal numbers of mitle
and female subjects, it is possible to look at the relationship of this
variable to performance on the Pendulum Problem and Four Group Task,
despite the difficulties in achieving a sui table breakdown applicable
to each age on either of the latter tasks.
D1e same is true of school
attended as a variable, although the breakdown of subjects into schools
is nearer to 60 : 40 than to
So : So.
Associations of these two variables with
p2
rformance on the
Pendulum Problem will be considered first.
8.S.1
Relationship of sex and school to performance on the Pendulum
Problem.
If the same breakdown of performance on the Pendulum
Problem is
required at each age, so that Chi-square values for the association with
sex and school can be summed over age groups? then the most suitable is
into the following three categories:-
Stage IIIB( !IIB and IIIB?A),
Stage IIIA( IIIA?B, IIIA and IIIA?IIB) and Stage Il(IIB?IIIA, liB, IIB?A
and IIA?B).
The resulting 3 x 2 contingency tables, for each age group,
relating Pendulum performance to sex are presented in Table 29 below.
~0.~
Pendul:>m Problem
lQ_YEARS
Pendulum
H
'"
"
Total
Proble~
lilA
lllB
Total
"
24
u
10
24
"
"
Q
Chi-square
,,
"
lilA
0
16
s,,
Total
1]
'"
'
Total
H!A
10
Itl6
H
Total
22
15
5
"
2]
"
44
~
M
9
""
Total
2, N_ S.
Chi -square
Chi-square
~
2,
"'< ,01
12
Total
!IJA
IIlB
6
9
24
10
"
24
16
20
48
4.2000, M
0.8626, df "'2, N.S.
For the five age groups combined,
df "10-
Table 29
4'
Pendulum Problem
6
Chi-square " 5.8212, M
n
11 YEARS
12 YEARS
Pendulum Problem
8
25
10.7!'40, M
Chi-square
1,()0\)0, M " 2, N.S.
H
.-n
"
-------
48
4
"<otal
lUll
Chi-square" 2].5768, p< 01
Contingency tables and Chi-square values for the relationship of sex to
performance on the Pendulum Problem at each of the five age groups. The
sum of Chi-square values for the five age groups is also shown,
2' N. S.
234
From the preceding tables and significance tests it can be
seen that there is a somewhat complicated interaction of age and sex
effects on the Pendulum Problem performance"
The sum of Chi-square
values for the five age groups 1 with 10 degrees of
at the 1% level.
freedom·~
is significant
Within age groups, there is a superiority of boys to
girls at age 11, of girls to boys at ages 12 and
13 (although these latter
effects are not significant, tested in isolation),
It is interesting to
note that whereas boys are more likely to achieve a very high level of
performance CIIIB) than girls, at the early age (11 years), at 12 and 13
years the superior performance of girls does not arise from a comparatively
large proportion in the very high category, but rather from a large proportion reaching IIIA rather than II Cas compared with the boys).
If the ages
11, 12 and 13 can be taken as the "transitional" period, then the interaction described above may indicate different charac te ris tic rates of
development for boys and girls,
Another possibility would be a greater
variation amongst boys than amongst girls in their rates of
development~
Further discussion of this will be left to Chapter 11"
To examine the effect of school attended, the same breakdown of
performance on the Pendulum Problem can be used.
values in 2 x
3
Table 30 be low.
The data and Chi-square
contingency tables for each age group are presented in
~
10 YllARS
Pendulum Problem
II
IIIA
IIIB
A(P)
13
12
2
School
Attended
Pendulum Problem
IIiB
IliA
II
Total
27
A(P)
8
17
2
27
School
Attended
B(P)
9
10
2
21
BCP)
5
10
6
21
Total
22
22
4
48
Total
1)
27
8
48
Chi-square
0.161], df = 2, N.S.
3.8165, df
Chi-square
~
Pendulum Problem
13 YEARS
lilA
IIIB
Total
7
11
9
27
B(P)
3
12
2
17
Total
10
2)
11
44
Chi-square
IliA
IIIB
Total
7
1)
9
29
B(S)
5
3
11
19
Total
12
16
20
48
II
School
Attended
4.0]34, df = 2, N.S.
A(S)
Chi-square
4.9130, df "'2, N.S,
14 YEARS
Pendulum Problem
II
A(S)
3
B(S)
Total
Chi-square
4
IliA
IIIB
Total
10
15
28
7
12
20
17
27
48
0.5441, df
2, N.S.
For the five age groues combined
df"' 10;-
Table 30
2, N.S.
Pendulum Problem
II
A(P)
School
Attended
Total
Chi-square =
13.4683, .10<p<.20
Contingency tables and Chi-square values for the relationship of school
attended to performance on the Pendulum Problem wi thiri each age group.
The sum of Chi-square values for the five age groups is also shown.
236
From the tables and significance tests
above~
no relationship
exists between school attended and performance on the Pendulum Problem.
It is interesting to note that there are greater tendencies toward a
relationship in the 11 transitionalu age groups, 11, 12 and 13 yearsl as
was the case for the relationship of sex to this task,
discussed further.
This will be
It should be noted that the summing of results across
primary and secondary schools, for the overall Chi-square, mO;y not be
justified,
However a comparable analysis keeping primary and secondary
schools separate does not alter the results.
While no attempt is made to test interaction effects of age,
sex and school on the Pendulum Problem, a breakdown of numbers in the
performance categories in terms of these three variables simultaneously
is provided for inspection in Table JOA of Appendix V,
There do not
appear to be interaction effects present which would interfere with interpretation of the two separate analyses just presented.
of summing across values of the other
variable~
Tims, the method
when examining either one
of these, seems justifiable.
8.5.2
Relationship of sex and school to performance on the Four Grouj2
Iask.
The effects of school and sex on Four Group Task performance
s:
can be examined in a similar way to that employed in 8~f'~ 1 for the Pendulum
Problem.
For these purposes, performance on the Four Group Task is
categorised into Stage II (comprising IIA and IIB) and Stage III (comprising ?II!, IliA and IIIB).
The small numbers of subjects at Stage III in
237
the 10 and 12 year-old age groups creates some problems, and Yates'
correction for continuity has been applied where
necessary~
As in the
analysis of Pendulum Problem performance, there are two exrected
frequencies below five in the relationship of sex to task performance
at 12 years, and consequently some doubt about the validity of the test
applied to that table.
Table 31 below presents 2 x 2 contingency tables and the Chisquare values for each age group..
The individual Chi-square values are
then summed and the overall value tested for significance on
freedom.
5
degt'ees of
It is apparent that no relationship between sex and le\·el of
performance on the Four Group Task is
discernible~
either at any
individ·~
ual age, or for the five age groups taken as a whole.
The corresponding analysis of the relationship be tween school
attended and performance on this task is carried out by Chi-square tests
of association on the 2 x 2 contingency tables reported in Table 32, be low
(page 239).
Yates' correction for continuity is applied to the tables
for 10 and 12 year-olds,
lQ~~
Four Group
li
T.~.§.L
?III,II!
Total
!I
6
M
fot al
7
M
Sex
?II!,IIl
Sex
20
38
Total
Chi -square
10
0.6040, df
Total
48
JJ
Chi -square
1, N.S.
M
?lll,III
21
15
0.0096, ctf
1, N. S.
13 YEARS
~
t2_ur ..Q£?..1f.£____}'aslc
~;: Groc~~
!I
25
17
Total
22
M
Sex
II
?III,III
Total
14
10
24
16
8
Sex
F
19
3
22
24
-:=.:.=
Tot a 1
40
Chi-square
4
0.2750, df
44
Total
1, N_S.
30
Chi -square
1R
1,8
0 3501!, df "' 1, t\.S.
14 YEARS
Four Grou12 Task
?III,III
Total
9
15
24
F
10
14
24
Total
19
29
48
II
M
Sex
Chi -square
0.0864, df
1, N. S.
For the five tables cotnbined,
df = 5, Chi-square "'
1.3254, N S.
Contingency tables and Chi-square values for the relatioYJship of sex to perfonnance
on the Four Group Task in each age group. The sum of Cni-square values for tile five
age groups is also shown.
~
Four Grou2 Task
H
A(P}
School
Attended
?III,HI
20
11 YEARS
Four Grou12 Task
Total
H
27
7
20
6(P)
13
Total
33
18
3
21
Total
38
10
48
'
s.
Chi-square
.
~
School
Attended
H
?IU,UI
Total
2
27
B(P)
15
2
17
Total
40
4
44
o.oooo,
df
27
21
15
0.8112,
"
1, N. S.
II
6
29
B(S)
7
12
19
Total
30
18
48
Chi-square "' 8.8272, df = 1' p< ,01
14 YEARS
n
?I!! 1 Ill
A(S)
12
16
28
B(S)
7
13
20
Total
19
29
48
Total
~
Chi-square
= 0.2976, df = 1, N.S.
For the five age groups combined
df
Total
23
School
Attended
1 1 N.S.
?lii,III
A(S)
Four GrOUJ2 Task
~
48
Four Groul?: Task
25
Chi-square =
Total
7
13 YEARS
Four Grou12 Task
A(P)
?III,!II
School
Attended
B(P)
Chi-square " 0,3062, df "' 1,
A(P)
= 5:- Chi-square " 10.2422
,05('p<.10
Table 32 Contingency tables and Qd-square values for the relationship of school attended to
performance on the Four Group Task in each of five age groups. The sum of Chi-square
values over the five groups is also shown.
School attended is not .related to performance on the Four Group
Task, except at the age of 13 years.
Within that age group, attenders of
School B(S) tend to perform better on the task than attenclers of School
A(S).
Similar, though slight, tendencies are found in the other age
groups, but neither the individual tests, nor the Chi-square obtained by
summing the values for the five groups 1 reaches the
cance.
If the values for just the
5% le\'el of signifi-
13 and 14 year-old age groups are summed
the Chi-square value is 9.1248, which, on 2 degrees of freedom, is significant at the 2% level.
Since these two age groups are attending the same
secondary school B(S), whereas the 10, 11 and 12 year-olds are attending
one of its contributing primary schools, B(P), there may be justification
for excluding the 10, 11 and 12 year-olds and considering just the 13 and
14 year-olds in combination.
Further discussion of the finding of an
effect of school attended, for just one, possibly two, of the age groups
will be left until Chapter 11.
As was clone for the Pendulum Problem, a breakdown of performance
on the Four Group Task by Age, Sex and School simultaneously is presented
for inspection in Table 32A of Appendix V.
interaction effects are
attempted~
As before, no tests of the
but there are none immediately apparent
which would interfere with the interpretation of the separate analyses,
reported above, for effects of sex and school.
241
~,,§__ UnderstandilJJl: of the Group AKioms and C',oncrete Reversibili iJ'., Foy£
Group Task.
The remaining aspect of performance to be described in this
Cllapter is the le>/el of responses to questions about group a.xioms and
concrete re .versibili ty ~
This questioning took place between the i ni. t.ial
and final phases of questioning for level of understanding of the Four
Group Task.
Responses to this section of the questioning will be dis-
cussed now and related to the levels of understanding of the Four: Group
Task as a whole, as treated in section 8,j'Z.
Chapter
7 described the scoring of levels of explanation of
each of the axioms and of concrete reversibility.,
Except for the question,-
ing about Inverse elements, the responses can in each case be categorised
as follows:Description of Explanation
Overall Level
Given~
no explanation given
oc
an explanation given in terms of
concrete operations, with help.
+C
an explanation given readily in
terms of concrete operationse
OF
an explanation given in terms of
formal operations, with help.
an explanation given readily in
terms of formal operations.
In the case of the axiom concerning inverse elements, no explanation in
terms of concrete operations is possible, hence the only categories used
are -
OF and +F.
§~,_2~~
Relationship of overall leyel ontll~~£2LC':_ Gr£flP Task to the~ level
explanation of axioms and cone!:.:: te ~::.,~r sibi li ty,,
Table JJA of Appendix V shows a detailed breakdown of tbe
frequency of response in each of the categories described in 8~~babove,
with a simultaneous classification according to overall le\ el of under7
standing of tbe Four Group Task,
three categories are used:-
In tbis latter classification only
Stage II (comprising IIA and IIB), Stage
?III and Stage III (comprising IliA and IIIB),
To relate performance on
the Four Group Task as a whole, to the level of explanation of each
axiom, in such a way as to be able to sum Chi-square values a,Gross age
g:roups, it was
necessary~
firstly, to combine the 10, 11 and 12
year~
olds into one gr-oup and, secondly, to split subjec. ts into just two ca tegaries on each task..
The two categories on the level of understanding of
the Four Group Task are Stage II and Stage III (?III and III combinedL
The two levels of explanation of each axiom and of
are
11
~oncrete
.reversibility
Concrete or no explanation" (-, OC, +C) and "Formal explanationn
(OF, +F),
Table 33 below gives the 2 x 2 contingency tables, for each
age group, and for each of the five aspects questioned,
The Chi-square
values are summed across age groups and tested for significance~
Yates
1
correction for continuity was applied in every table, since each has one
expected cell frequency at, or below, five.
(i) Concrete Reversibility
Level of
"
~
~
?III, III
Total
Chi-square
~
"
"
"
,,
14 1532, Of
z
)7
?Ill,Ill
48
Total
1, p < ,001
)0
JJ
Total
1P
P1
"
?Irt,lli
10
"'
20
48
Tot,J.l
1]0
JC
140
JO
"
"
37
20.4501' M
Chi-square
-,C
JJ
Total
-,C
Total
"
"
"
Levc 1 of EX£ !an,< t ion
Level d
h~lanation
-,C
Level on
Four
J\; YEARS
_l~ll
ll~
Ex)21ano.tion
14 YEARS
-
1,
P< ,DOl
Sum of Ci1i-sqware values over •<ge groups equals 70.71\1\S, df
J6
Chi-s<wan =
3,
1, p <.001
1 1'52, M
p<.001
(ii) Commutative Law
14 YEARS
Level
Level on
-,C
~
~
n
Task
?lll,III
Total
Level of
Total
JJ
12
13 YEARS
o~lanation
"
18
H)
]]
"
19
48
Chi-square " 7 ,J255, df = 1, p < 01
-,C
Tctal
-,C
]0
P1
n
]0
?III,IIJ
10
18
Total
40
48
Chi-square
= 12.96oO,
0
df
= 1,
" " "''""
Level of Ex121ana t ion
ExQlanatio~
"
?lii,Ill
Sum of Chi-square values over a~:e groups equals 4J.6}6o, df
P1
2')
7
"
140
133
Total
P< .001
Total
Ctoi-square = 23
3505,
"'
1, P<JlOl
j, P< .001
(iii) l!nit Element
14 YEARS
"
JJ
10
YEARS
Level of ExJ2lanation
Level of Explanation
-,C
-,C
Total
Total
Level on
~
~
Task
"
10
?III,II!
Total
J8
n
JJ
'9
J7
]9
48
Su<O of Chi-square values ov~r age groups equals 8},0154, df
?III,IH
"'
"
Total
J,
P< .001
1JJ
17
'9
Uv) •nve:rse: Element
14 YEARS
ll
10
?III,III
11
"
Total
Level of ExJ21anation
Total
-,C
Level on
Four
Group
Tas)S_
, ,
13 YEARS
Level of Explanation
Total
-,C
ll
30
0
30
ll
26
37
?lli,tU
6
12
18
?IU,HI
27
48
36
12
48
Chi-square" 10,5300, df" 1, P<.Ol
Sum of
Chi-sq~are
Chi-square " 23.2296, df " 1' P<.001
values over age groups equals 78,8552, df
Total
-,C
11
Total
10 YEARS
Level of Exelanation
Total
1"
0
"1
17
12
29
128
12
140
Chi-square" 115.0956, df" 1, p<,001
3, p<.001
(v) Associative Law
14 YEARS
12
Level of Ex[:!lanation
Level on
~
Gcoup
Task
Total
-,C
30
ll
"1
0
,1
10
18
?IH,!II
19
10
"
1]0
fO
140
11
0
11
ll
?III,III
13
24
37
?Ill,Ili
24
24
48
Total
" 1, p<.001
Chi-square"' 17.8189, df"' 1, p<.001
Total
Chi-square " 11.7936,
"
38
Total
Chi-square " 36.1852,
Sum of Chi-square values over age groups equals 65,7977, df "3, p<.OO\
Table 33
Total
0
If
30
10 YEARS
-,C
Total
-,C
11
Level of ExJ21anation
Level of Explanation
2 x 2 Contingency tables and Chi-square values relating the overall level of
understanding of the four Group Task to the level of explanation given of each
of the Group Axioms and of Concrete Reversibility. The association is tested
separately for three age groups, and the Chi-square val01es then summed and
tested on three degrees of freedom, in each case. Two levels of overall unde.·standing are used on each case·, these being Stage n {comprising substages IIA
and I!B) and Stages ?UI and III combined (the latter comprising substages IIIA
and UHI), Two levels of explanation of the axioms are used in each case, these
being -,C (comprising"-", "OC" and "+C") and F (comprising "OF" and "+F"), It
should be noted that, in the case of the Inverse Element, "C" explanations are
not possibie, so that the lower category reduces to one of "-"alone,
"
" 1, P< .001
From Table 33 it can be seen that the level of explanation of
concrete re\Tersibility and each of the four group axioms tested is closely
related to the overall le:vel of understanding of the Four Group Tasko
It
is not surprising that there is one almost empty, or empty, cell in each
of the contingency
tables~
since, where most of the axioms are concerned
it is not possible to give a formal explanation without having arrived at
some formal account of the roles of each of the elements,
It is by no
means guaranteed, on the other hand, that a person who has conceptualised
the role of elements formally will be able to give an account:, in these
terms~
of the axioms which the combinations of elements obey,
The sig-
nificant associations obtained indicate, empirically 7 that the second
ability does follow, given the first.
It is also interesting to examine interrelationships between
performance on individual axioms and concrete reversibility.
This will
be done in the next section.
806~2
Interrelationships between performance on concrete .reversibility
and the group axiom"'_.
Since performance on each individual i tern depends on overall
level of understanding of the Four Group Task, interrelationships among
items will be examined separately for those subjects categorised in
Stage II (IIA and IIB) and for those categorised in Stages ?III and III
(IIIA and IIIB) combined, on overall level of understanding.
8,6,2.1
Interrelationships between performance on concrete reversibili_t_y:
and axioms for subjects categorised in Sta~ II on the Four
Group Task.
34 below shows the number of subjects, of those whose
Table
performance was rated in Stage II on overall level of understanding of
the Four Group Task, giving different le\cels of explanation of the
items (i), (ii), (iii), (iv) and (v), concerned with concrete reversibil-
ity and the group axioms.
The distribtuion is shown for each age group
and for all age groups combined.
(i) Concrete Reversibility
AGE
(ii) Commutativitz.
Level of Explanation
GROUP
OC
-~>C
OF
"'F
I
Level of Explanation
Total
+C
OF
Ye,_ar-,s-+--0~-4--7,--0--0----1-l--+-0--3
8
0
16
o
o
30
37
0
o
4o I
Years
Years
3
Years
*F
To ta,l
OC
~
14
15
o
o
30
26
11
o
o
4o
2
20
12
0
0
33
0
7
26
0
0
33
I3
J
10 Years
0
22
16
0
0
38
0
4 34
0
0
38
Total
5 86
61
0
0
152
3
28 121
0
0
152
Table continued on page
247,
I
-------ITEM
AGE
GROUP
1
1
(iii) l!nit Elemen_t:
Inverse Element
I
~~~ Level of Explanation 'Level of Explanation
~---- ~
- OC +C OF +F
I14 YRS.
0
4
Total_L___-
OF
+F
Level of Explanation
7
0
1
1I
10
1
o
11
1
3 5
20
0
2
30
30
0
0
30
117
0 31
0
2
40
40
0
0
40
11 YRS.
7 24
0
33
33
o
o
10 YRS.
6 3 29
0
38
0
0
12
YRS.
Total
Tabl~:
7
17 18111
2
6 3 o o
11
5
I
I
8
0
0
30
I
' 17 1 0 13
0
0
40
33
13 1o 1o o o
38
10 17 11
0
33
38
I
o_6_-_'5_2_,__1_5_1__1___o__1_5_2j_59 48_4_5_o 0
!52
l
38
0
0
combined~
A summary of the distributions on items for all age groups
(combining the levels of explanation ( +C, OF, +F) since there are so few
35
of these subjects giving explanations at formal levels), is in Table
below.
Level of
Explanation
CD
-
5
oc
86
61
+C, OF~ .;.p
Total
35:
1
+--------4-----------i
Distributions, of those subjects categorised as Stage II (IIA
or IIB) on Overall Level of Understanding of the Four Group
Task, over Levels of Explanation of items (i )-(v) concerned
with Concrete Reversi bi li ty and Group Axioms., Dis tr ibu tions
are shown for each age group separately and for all ages
Table
I
I
- OC +C OF +F Total I
.~---~-1
Total
3
1 13 YRS.
1
I(iv)
I
---1
-.-------~-~~
Item
1152
J
cu:J_ Ci~Civ)
T:'
I
28
. 121
I
I 152
17
151
18
o
I
l~152_j
r--"117
1
152
( v)
I
'
59
48
b
'
45
I
I !52
Distributions, of those subjects at Stage II
on the Four Group Task, over levels of
explanation of items (i) to (vL Age groups
are combined~
The items may tentatively be ordered in difficulty as (ii)
(easiest), (iii), (i), (v) (iv).
In fact item (iv) is, for all practical
purposes, impossible for S_tage II subjects.
An attempt to test for a
Guttman Scale of items (ii), (iii), (i) and (v), witll tllree levels of
response on eacll, yields 76 scale and 76 nonscale response patterns for
the total of 152 subjects and is not worth pursuing further.
not the proportions of subjects in categories on the items
Whetber: or
differ~
consid-
ering the items a pair at a time, may be tested by Chi':"'square 1 as may the
association be tween levels of performance on each pair of items.
These
two types of tests are presented, with the 2 x 2 tables on whicll they are
based, in Table 36 below.
All age groups are combined for these tests.
In the Table, the i terns wi 11 be referred to by the initials
of their title, as
Ci) C.R.
follows:~
-
Concrete Reversibility
(ii) C. L.
- Commutative Law
(iii) U.E. - Unit Element
(iv) I.E. - Inverse Element
( v) A. L.
-
Associative La1M
/
IM
/(ett-f-'>
J
T. II.
ll
j
/;S'
3.
!
249
'
J
·
.
I Test of Association
___1__
i ITEM PAIR (and 2 x 2 Table)
1
1-c,--)-C-.R-.-a-n·d--(1-.l-.)-C-.-L.
I
+C
(iil-,OC
Total
I Test of Difference
J
b t
p
t'oro'
e ween ropor. ,l ::;
!
i
i
I
c~~;:;e-= .!(.9859 I ~bi-square ~
48.6400
O·o2'1-b ,
d-:<nroi
j
d.f.=1,p<,05A'-S.Id.f.=·l,pc.CI01
[Ci)
j:
I Total
I[Ci)
I
j
67
91
Contingency
7
54
61
Coefficient
31
121
0.1812
o. 011.1
152
C.R. and (iii) U.E.
I
+C
(iii) -,OC
len
1 -,
I ~~
Total
d. f. = 1, N.S.
22
69
91
Contingency
13
48
61
Coefficient =
j(i)
C.R. and (v) A.L.
Chi-square =
I
(v) -,OC
cl. f. = 1 , p
+C
'Ci)
Total
U,
j:
",om
I
i
~.6208 I Chi-square
3·/h.'f·'
<. O§. N- S.
I
I
d. f. = 1 , p
= 4.4138
¥· ¥713
< . o5
1
70
21
37
24
I Tota~;( ii) C • L.
91
Conting~y_
61
Coefficient =
-=-
45
+C
8.1774
D. I:;;{, 7
152
and (ii ·.) U .E.
Ciii)-,oc
Total
Chi-square =
5-.39~8
¥·tt690
d. f. = 1 , P <
. os
Joe
12
19
31
[ +C
23
98
121
35
117
152
1 Chi-sgu"c~e = .0.]80')
1
I
I
,Cii)
Total
'•
0.03~3
1
I -,
o
D./i/'2.
1 Tota=l=3=5======
117
152
[
I
I
Chi-s quare = 38. 21!39 [
if!..
4.1
Chi-square = 0.1672
0·3
gd.
d.f. = 1, N.S.
Contingency
Coefficient = ~L
<).
'??'t
'===--===------~j_====·-==-
------~
250
ITEM PAIR (and 2
X
2 Table)
Test of Association
(ii) C.L. and Cv) A.L.
Chi-sguare
I
I
I
d. f. = 1 ' #ri>.
Total
""C
F<;'. OS '
oc
+C
81
5
31
I
!
40
121
9\
--··---
I
I
Cii)
26
---:i
Chi-square = ~~~
I
8f. 6
d.f. = 1, p<.001
- 3.3g9(l
If-· ;J II
(v)-,OC
Test of Difference
between Proportions
Contingency
I
Coefficient =
G.-1~494-1
D.lo4-Cf~
(iii) u
I
I
Total 107
152
45
:g. and
(v)
A.L.
I
-~ Chi-sguare
I'
I
=
~2
s:
(v)-,OC
(iii)
26
'
+C
Total
9
35
81
36
117
Total 107
45
152
oc
+C
d. f. = 1 I
~-&.
I
~
Chi-square = 57.60ool
;o i d.f.
·cS!
I
Contingency
Il
OQ
I
J
= 1, p<.001
I
Coefficient = 0. 0466
I
I
I
Table 36:
For each possible pair of the items (i), (ii), (iii) and
(v),a 2 x 2 table is shown and two Chi-square tests
performed, TI1e first tests the association between le-vels
of explanation given on the two i terns, The second test
examines whether the proportions of subjects falling in
the categories are different for the two i terns.
The results of the tests reported in Table 36 indicate firstly
that the correlations be tween i terns, measured by the contingency coefficient,
Only three of the six intercorrelations reach the 5% leYel
are very smalL.
of
significance~
understanding.
Thus eacl1 item measures a relatively
ind~::pendent are~
It should be remembered that these conclusions are drawn
only for subjects at one level of overall understanding (Stage II) of the
of
251
Four Group Task.
The second finding is that the order of difficulty put
forward on inspection earlier must be modified by saying that the proportiomin category u.,.cn do not differ significantly for items (ii) and (iii),
but that all other such differences are significant.
The order thus
becomes ((ii) = (iii))< (i) <(v) < (iv), from easiest to most difficult.
As mentioned before, Civ) is, in fact, impossible for Stage II subjects,
and has therefore not been included
8.6.2.2
in the tests reported above.
Interrelationships between performance on concrete reversibili tv
and group axioms for subjects categorised in Stage ?III or Stage
III on the Four Group Task.
Table
37
below shows the number of subjed:s, of those whose
performance was rated as ?III or as III (including IIIA and IIIB) on o\·erall
understanding of the Four Group Task, giving eli f ferent levels of explanation
of the items (i), (ii), (iii) (iv) and (v), concerned with concrete revers-
ibili ty and the group axioms.
The distribution is shown for each age group,
and for all age groups combined.
I
AGE
I, GROUP
I
(i) Concrete Reversibilitx:T Cii) Commutativity
1
Level of Expplanation
1
0
1
10
7
19
37
113
YRS,
1
2
4
0
11
112
YRS.
0
1
3
0
I
0
5
I
0
3
3
10
1 0 YRS.
lrotal
J
Level of Explanation
1
0
-~-------o_c ___
+c
_______
+_F
___T_o_t__a_l__-4___._.__o_c____•_c___o_F____+_F___.1_'o__
ta__
l ______ '
11 4 YRS-
11 YRS.
~
I
ITEM
I
0
0
18
2
17
37
18
0
0
10
1
7
18
0
4
0
0
4
0
0
4
0
7
15
0
0
9
0
6
15
4
1
2
10
0
1
8
0
1
10
26
8
39
84
1
49
3 31
84
I
to
I
·--j
Table continued on page 252.
!
252
~--
-~---+L-----~----(iii) Unit Element
---...,,---ITEM
C:~~p
I
Level of Explanation
I -
1;
YRS.
113
YRS.
OC +C
?F ~F
---7
2 27
1ra-,
II
l
Level of Explanation
Level of Explanation
OF
:~Total___~oc
37
-3-7--+- ,, 14 12
0
7
1 10
18
6 5
7
18
112
YRS. 11
0
0
0
0
4
4
3
0
4
111
YRS.
0
0
5
0 10
15
4 4
15
o
2
5
o
3
10
7
7
0
3
24 3 54
84
j1o YRS.
~Total
II
;J
Table
J1:
---~--1
(v) Assoc_!o_~_!i_y_i__t_y_
Total
0
1
_
(iv) I12verse Element
2
10
34 25 25
84
I
0
7
6
4 20
37
2
2
4
0 10
18
16
14 20
5 39
84
A summary of the distributions over response categories on
items (i) to (v) with the categories combined into a dichotomy of (-, OC,
Level of
Explanation
I
-, oc, ..c
r j J6
38
below.
(i)
(ii)
1
,,
F=O=F,==+F====~•=~=4=+·
Total
TaJ:l.le 38:
!
Item
1
_Ls4
1
s4
(iv)
(iii)
27
57
1
[ LU :_ __~_:_
Distributions, of those subjects categorised as Stage ?III
and III (III& or IIIB) on Overall Level of Understanding
of the Four Group Task, over Levels of Explanation of
i terns (i)-(v) concerned with Concrete Reversibility and
Group Axiomso Distributions are shown for each age group
and for all ages combined.
+C) versus (OF, +F), is in Table
I
+C OF +F Total
I
(v)
5o=+;
34
1
I
41]
~E-~8~J
Distributions, of those subjects at
Stages ?III, III on the Four Group Task,
over levels of explanation of items (i)
to (v)" Age groups are combined.
253
A tentative ordering of difficulty of the i terns is (iii) (easiest),
(iv), Ci), (v) and (ii), although the differences in number of subjects
giving (OF or +F) explanations is very small in some caseso
An examination
of goodness of fit to a Guttman scale yields 58 scale and 26 nonscale
response patterns,
for the total of
84
subjects, and is not worth pursuing
Since an ordering of the i terns for difficulty cannot be achieved,
furtber.
the main question of interest becomes whether or not responses to pairs of
items are
correlated~
All possible associations between pairs of items
were tested for signifir.ance by Chi-square, and the contingency coefficient
present~
deriv·ed as a measure of the degree of association
Chi-squarf.,' was
also used to test the significance of the difference between the
in response categories, for each pair of i
the tests were
basecl~
~TEM
below,
All age groups are combined
--~--
OF,+F Total
3
Test of Association
Chi -square = 28, 72g
;t(,,
(1) C.R. and (ii) C,L,
34
39
Initials will be used to refer to items as in Table
PAIR (and 2 x 2 table)
-,OC,+C
The 2 x 2 tables on which.
and the Chi-square 1 contingency coefficient and prob-
ability values are presented in Table
for three tests,
terns~
d.L
= ·1,
p"(,001
37
16
31
47
5o
34
84
T
)6.
Test of Difference!I
between Pro ortio~~
Chi "square =
"T;;"r~-1
Cf· 7bl.S'i
.
I
d.f. = 1, p·C01
I
•
foefficient = ~
Total
proportion~
(), /.fCJO/
254
i ITEM PAIR
(and 2 x 2 table)
Test of Association
I
-of
Test
5iffe~e-r1C:~
between Proportion~]
1---------+----~----ll.
Chi-square = zy,61Er I Cbi -square ...
!CD C.R. and (iii) U.E.
!
i Ciii)-,OC,+C
I(~)
OC
23
OF,+F
'<?· 760.;.1
d,f. = 1, p <,001
I
i
[ +C
I OF
Total
14
37
Contingency
Coefficient =
::,",
d.f. = 1' p<.OS
~
I
o.
I
43
47
1
,,
,,
"
1
3'31.;6-~
I
Chi-2guare
¥2.
(iv)-,OC,+C
1
! ( i)
I ! oc
I ,c
OF,+F
Total
29
8
37
5
+F
I
i Total 34
~-----·
So
I
84
A.L.
Chi-square. =
29.568
3!· 3
v) -,OC,+C
30
I
----1------·- - - ··--1
(i) C.R, and (v)
ijci) (
'I
/').$7'/l.l
47
!
I -
N.S.
I
Contingency
Coefficien!_ = ~
l oc
= 1,
,-:~1
o.b?ebl
•
OF
J
cl.f.
d.L = 1, p<.001
I
I
I
4
01i-sguare =
lfc-5Bfr I
.r. fill 'fi I
OF ,+F
Total
7
37
37
47
44
84
d,f,=1,p<.001
Contingency
Coefficient =
Cbi~c:_quar~ = &;Z-}:7'11
o·S'32S
d.f.=l,N.S.
!
255
. .
~·-·--·-f.A----.---.--fTestc·fDiff;;~enze:-·!
and 2 x 2 table
.· Test o
ssoc~at.1on '~b
.t
. p.ropor·1ons
t.··. . i~
e--ween
--"~
i
!Cii) C.L. and (iii) U.E.
Chi-square " 36.98.?! 1 Chi-square" 3J,OOI'l I
1
---------Ill .14-oJ 1
.n. !Nt~q 1
:ITEM PAIR (
11
I·
.
Jc:.Biil-,OC,+C
I OC
27
OF,+F
23
l+c
I
I::
1
cl,f. = 1, p<.001
Total
50
-.--
I
I
·. ·
1
Contingency_
1
I
Coefficient -
Total
I
0
34
34
27
57
84
l(ii) C.L. and (iv) I.E.
15.624
Chi-square=
~~-square-·
(iv) -,OC,+C
OF,+F
Total
cl.f. =
1, p<: .001
iCHJ
-~
cl,f,
= 1, P< ,01
.
I -
I
I OC
29
21
50
~
· OF
I +F
i Total
5
29
34
34
5o
84
Contingency
1
Coefficient - B-;;-#3-2'
Ii
' 'J
i
i
l(ii) C.T. and (~·) A.L.
I ..
:1
9.846
16· :>8 68
I:Z·:lo;g;,_ I
I
I
!
cl,L '" 'i, p<.001
(v) -,OC,+C
1('.:.1)
oc
1
I;;
Total
OF,+·F
Chi -square '' ·
Total
cl.f. =
I
1.!.·?6}'!; I S,:hi-sguare
1•·"'•" 1
1, p<.001
I
cl.L--
.l,
=
4.167
I
;;;a67i
"c/ '
p<.O.
!i
I
33
17
50
Coefficient
27
7
34
=====---·-
44
84
-~I
o. 3'fU> I
I
I
I
J
256
!-ITE;-PAIR
( and 2 x 2. table)
1
·~----
l(iii)
Tes t...ot~ A. ssoc.1a
. t'.1on
·----·------+------
:.
. Test of Diffe:rence
P
.
Between
Chi-square = 14.700
U.E. and (iv) I.E.
I
roporbOl:!:"j
I~,;:-~
13· ql.3lf
(iv)-,OC,+C
(iii)
~~
OF,+F
19
8
Total
27
d,f. = 1, p < .001
i' d.
Contingency
I
I
z. ¥-1:
= 2.130
f. ·· 1, !l E.05N.s.[.
1
Coefficient = ~
OF
+F
15
Total
34
o.
57
5o
I
84
r-----------+----b
.
I
J
iCiii) U.F. and (v) A.L.
Chi-square = 27.B48
Chi-square- r-§:19-
I
, (v)-,OC,+C
(iii)
II-
2q..
d.f. = 1, P<.OO'
q. 1£s 5
d.f. = 1, p<.01
I
OF,+F
24
I OC
1;
Total
27
i
Contingenrr
Coefficient = B.§8g
,,===4=1==!:=}7
I Total
40
44
84
i
l(iv) I.E..
I
and__0::2__~~·
(v)-,OC,+C
OF,+F
Total
~(~v)
24
I oc
10
34
I+C
d.f. = 1, N.S.
Contingency
Coefficient - &;-379
OF
16
+F
I Total
I
Table
d. f. = 1 ' p < • 001
39:
34
5o
44
84
o. '31
For each possible pair of the items (i), (ii), (iii), (iv), (v)
a 2 x 2 table is shown and two Chi-square tests performed, The
first tests the association between levels of explanation given
on the two items,
The second tests whether the proportions of
subjects falling in the categories are different for the two
i terns$
257
From the results of the tests reported in Table 39 l t can be
seen that the levels of explanation given are significantly related for
e\~ery
possible pair of items 9 and that the size of the contingency
coefficient of association is from about
0.4
to about
o.o.
correlation of performance on these i terns is much higher
P
Thus the
for subjects
in Stages ?III and III of overall level of understanding of the task,
than it was for subjects at Stage II.
A consideration of the results of the significance tests of
differences between proportions makes it very difficult to reach any
conclusion about an ordering of difficulty of the items.
below, Table
40,
In the matrix
pairs of items for which the proportions differ at the
5% leyel of significance are marked with an asterisk, pairs which differ
at the 1% level with two asterisks and at the 0.1% level with t:u:ee
asterisks~
ITEM
(i)
Concrete
Reversibility
(ii)
Commutativity
(i)
(iii)
Unit
Element
Civ)
Inverse
Element
*
***
(ii)
(v)
Associ a ti vi ty
*
**
(iii)
Civ)
(v)
No, in
48
(OF,+F)
Table
40:
34
57
So
4.3
The number of subjects, amongst those at Stages ?III and
III on the Four G1:oup Task, who give OF or +F levels of
explanation of each of items (i) to (v) is shown, Asterisks
in the matrix mark Chi-square tests of significance of the
difference between proportions reaching the 5% (''), 1% (''*)
and 0.1% C***) levels of significance.
258
Perhaps the most justifiable conclusion about the order of
difficulty is that (iii)< ((iv) = (i) - (v)) < (ii), from easiest to most
difficult~
where< is used to signify ~*is easier than"~
However, other
sets of conclusions are almost equally justifiable and the most obvious
general conclusion is that the differences in difficulty are not great.
The fact that one cell in each of the 2 x 2 tables is, almost invariably,
near zero gives encouragement to the notion that some meaningful ordering
of items, could be demonstrated
if techniques of measuring the explanat-
ions given were improved.
It should be noted here that there are aspects of the questioning
procedure which may give rise to doubts about the validity of the levels
of explanation to which subjects' .responses were assigned,.
This is
particularly true for subjects at Stage III on tbe task overall, and
pertains to the amount of encouragement given to subjects to give a 'formal'
account of eac11 axiom?
Little encouragement was given to improve the
explanation of some axioms (especially (ii), tbe Commutative Law) beyond
an initial concrete level, but on otber items ((iv) and (v) in particular)
a good deal more probing and
npushing 1 ~
was done by the experimenter_,
This
may well have obscured any real differences in the le,·el of difficulty of
the items,
259
CHAPTER
9
QUANTITATIVE MEASURES OF PERFORMANCE ON THE TASKS AND THEIR R):!LATIONSHI PS
TO OTHER VARIABLES.
9.1
The Derivation of Quantitative Measures of Performance on the
Pendulum Problem and the Four Group Task.
Whereas there are substantial statistical and theoretical
arguments for treating data, collected on Piagetian tasks, purely in terms
of qualitative
stage categories (as was done in Chapter 8), a number of
advantages accrue if a quantitative index of performance can be
derived~
These advantages pertain, in particular, to correlational analyses of the
relationship of Piagetian task performance to that on other tests, such
as intelligence and ability scales,
An examination of performance on both the Pendulum and Four
Group tasks, in the present study, shows that it is possible to devise
quantitative indices which nmeasuren developmental level in a way compar-
able to the stage categorisations.
TI1e derivation of these scores will
be described in sections 9.1.1 and 9.1.2 below,
Essentially they consist
0 and + on
in using the scores O, 1 and 2 in place of the ratings
rated aspects of the tasks described in Chapters
2.1.1
6
and
7.
Quantitative measures of performance on the Pendulum Problem
The fact that nine substages within the overall Stages II and
III are differentiated, in performance on the Pendulum Problem, suggests
that a sufficient range of performance exists to justify the assignment
260
of quantitative scores.
This impression is strengthened by an examination
of the rated aspects of performance on the task (some relating to the
subject's method of investigation of the p.roblem, others relating to the
content of the conclusions drawn) which determined allocation to stages.
It is clear that performance on all the rated aspects is improving through-
out the transition from early Stage II to late Stage III, and that some
aspects are more difficult than others in terms of the age at which con-
sistent success first
occurs~
It is possible that, with some refinement
of the questioning procedure, and standardised rules for rating the aspects
of performance, a set of items conforming to strict scale requirements
(e.g. those of a Guttman scale) could be devised.
not attempt to do this.
The present study does
However, it was shown in Chapter 6 that patterns
of ratings could be used to place subjects unequivocally in one of the
substages, and it therefore seems reasonable to regard the task as one in
which each aspect might be scored and the sum of scores on all aspects
taken as a quantitative index of performance.
The way in which this was
clone, to yield a Total Score, made up of two constituent scores, on the
Pendulum Problem is as follows.
I
Pendulum Co11tent Score.
It was indicated in Chapter 6 that there are six aspects of the
conclusions drawn by a subject which can be rated.
These consist of two
aspects of the conclusions about each of the three variables involved
(the length of the string, the size of the weight, and the amplitude of
oscillation).
The two aspects are firstly whether or not a correct
261
ordering of the effects of a variable is achieved, and secondly whether
or not the effect ascribed to a variable is the correct one.
6
In
a system of rating the performance of a subject as -, 0 or + on each
of these aspects was described,
If these ratings are now changed to
scores of 0, 1 and 2 respectively then the total score on the six aspects
(out of a possible 12) is the _Eendulum Content Score.
II Pendulum Method Score
The second as pee t of performance for which ratings are described
6 is
in Chapter
the subject's method of investigation of the problem,
Six
relatively independent items relating to the method used were rated-, 0,
or + as for the content aspects.
These ratings may be transformed into
scores of 0, 1 and 2 in a similar way to that for the Pendulum Content
Score.
Again, a total score (out of a possible 12) is attained, which
will be referred to as the Pendulum Method Score.
III Pendulum Total Score
The sum of the Pendulum Content and Pendulum Method Scores gives
the Pendulum Total Score, out of a possible
IV Pendulum
24.
Sta~Score
It is also possible to assign the scores 1 to
9 to the substages
of performance identified on the Pendulum Problem (score 1 for S.tage IIA?B
...
tryough to score 9 for Stage IIIB),
The distributions of subjects over the Scores I, II, III and IV
262
on the Pendulum Problem, described above, are given in Tables 41A, 41B,
41C and 41D of Appendix VI.
The means and standard deviations for each
age group separately, for two combined age groups (10, 11 and 12 years
combined;
and 13 and 14 combined) and for the total sample are shown in
Table 41 below,
Age Group ( Yrs.)
Scored
Combined Age Gps,
---
10
11
12
13
14
10,11,12
13' 14
Sample
X
8.15
8.92
9.30
9.46
9.83
8.77
9.65
9.13
s
1,56
1.91
1.56
1,90
1 .69
1. 74
1,80
1,82
X
6.46
7.23
7.70
6.90
8.44
7.11
7.67
7-34
s
1 .90
1,80
2.15
2,63
2.48
2.00
2.66
2.30
14.60 16.15 17 .oo 16.35 18.23
15.89
17.29
16.46
Aspect
Pendulum
Total
--
I
pontent
Pendulum
II
Method
Pendulum
X
III
Total
s
--
3.15
3.45
3.35
4.10
3.88
3.44
4.08
3.77
X
4.77
5.67
6. 14
6.29
7.08
5.51
6.69
5.99
Stage
s
2,00
2.18
2.01
2.12
1 • 69
2.13
1.95
2.13
t-
n
48
48
44
48
48
140
96
236
-Pendulum
IV
Table [!_:
The means and standard deviations of Scores I, II,
III, and IV on the Pendulum Problem for each of
the five age groups, for two combined age groups and
for the total sample.
The distribution of subjects over the four scores do not preclude
the ajJplication of one-way Analyses of Variance, to test the significance
of differences between the mean performances of the age groups.
The results
of such an analysis, for each of the four scores, are shown in Table 42
263
below.
For detailed results, see Tables 42A, 42B, 42C and 42D of Appendix
VI.
Score
F value obtained
p
I Conclusions justified
by t tests
I
p endulum Content
6.63 (d. f. 4. 235)
(.001
II
pendulum Method
5. 81 (d. f. 4. 235)
<.001
[m
P endulum Total
6.55 ( cl. f. 4. 235)
(",001
)1, o<
IV
Pendulum Stage
8.04 ( d, f. 4. 235)
<.001
)l,o<~,,;a,2=~13)~14
)l,o<)i<n'l<,27t13l0 14
yt, 0 =)<,, =)<,2 )<))112 =)1,4)
SA,, '/1,2=fl,3l')lt,4
--
Table 42:
The values of F, and their si gni fie ance levels
obtained in one-way Analyses of Variance to
compare the performance of five age groups on
each of the Pendulum Scores. Also shown, for
each score, is the set of conclusions drawn by
the application of t tests to pairs of means.
TI1ese t tests employed the within groups estimate
of variance, in each case.
With some reservations about the sui tabi 1i ty of the scores,
product-moment correlation coefficients were computed between two pairs
of scores on the Pendulum
Prob~em,
These were between Scores I and II,
the Content and the Method Scores respectively;
IV, the Total Score
and between Scores III and
(being the sum of the content and method scores), and
the Stage Score (stage categories scored 1 to 9, respectively),
These two
correlations were obtained for each age group separately, so that the
improvement of both scores with age would not result in a spuriously high
value.
The values of the coefficients (each of which differs from zero at
264
the 0.1% significance level), and 1% confidence limits for the population
values, are shown in Table 43 below.
--
I
AGE
GROUP
n
Values of Correlation Coefficients between:Content (I) & Method (II)
Scores
Coefficient
10 YRS. 48
1% limits for
Total (III) & Stage (IV)
Scores
f'
Coefficient
1% limits for f'
+.652
( + .355<f<+ .815)
+.900
C+. 795<f<+.950)
+.722
( +. 480<f< +. 860)
+.941
(+.875<f<".970)
12 YRS. 44
+.614
C+.3o5<f<+.8o5J
+.923
C+.840<f<+.965)
13 YRS. 48
... 632
( +. 345<f< +. 808)
+.950
( +.896<f< + .977)
114 YRS. 48
+.757
(+,530<f<+.876)
+.926
(+ .854 <f<
11 YRS. 48
I
Table 43:
+.967)
"The table contains the values of product-moment correlation
coefficients, between two pairs of scores on the Pendulum
Problem, for each of the five age groups. Fisher's z transformation is used to set 1% confidence limits for the
population value (f) of the correlation in each case. Each
of the correlations shown differs from zero at the 0.1%
level of significance.
Correlations of the order of +.6 to +,7 between the Content and
Method Scores justify the summing of these two to give a Total Score.
It
appears that the Content and Method Scores each measure some unique aspects
of performance, although there are other aspects which they measure in
commonQ
This conclusion is a tentative one, which would need to be verified
by more thorough methods of i tern analysis.
The very high correlations
between the Total Score and the Stage Score support the conclusion that they
are equivalent indices of the same developmental abilities.
265
9.1.2
Quantitative measures of performance on the Four Group Task.
While it is clear that the range of performance on the Four Group
Task is not as great as that on the Pendulum Problem, it seemed worthwhile
to attempt to derive quantitative scores on both.
The results described in
Section8.6.1 show that the level of explanation of group axioms and concrete
rever si bili ty is related to the overall stage allocation on the Four Group
Task.
It was therefore decided to score firstly the stage allocation and
secondly the levels of explanation of axioms and concrete reversibility,
and to sum these two scores to give a Total Score on the Four Group Task,
The way in which these scores were derived is as follows.
I Four Group Task Stage Score
Only five stages (IIA, IIB, ?III, IliA and IIIB) were distinguished on the Four Group Task.
These are scored 2,
4, 6,
8 and 10 respect-
ively, to give a comparable range to that on the Pendulum Content and
Method Scores.
This is the Four Group Task Stage Score.
II Four Group Task Axioms Score
Explanations of concrete reversibility and four of the group
axioms have been rated as
n
-
n
.
described in Section 7 .2.2. To obtain the Four Group Task Axiom Score, a
score of 0 is given to "-" explanations, a score of 1 to
uocn
explanations, and a score of 2 to "OF!t and n+pn explanations.
and
uq,cn
Summing over
the five items yields a score with a possible maximum of 10 identical to
that for the Four Group Task Stage Score,
266
III
Four Group Task J:otai Score,
The sum of the Fou.r Group Task Stage Score (out of 10) and the
Four Group Task A>d.om Score (out of 10) gives the Four Group Task Total
Score (out of a possible 20),
The distributions of subjects over the Scores I, II and III on
the Four Group Task, described above, are given in Tables 44A, 44B and
44C of Appendix VI.
The means and standard deviations for each age group
separately, for two combined age groups (10, 11 and 12 years combined;
and 13 and 14 years combined), and for the total sample are shown in
Table 44 below.
.
I
~roup
Scored
Aspect
(Yrs,)
Combined Age Gps.
Total
10
11
12
13
X
4.25
5.04
3.77
5.33
7.38
Stage
s
1.92
2.37
1,68
2.78
F.G.T.
X
3.96
4.65
3.50
s
1.41
2,20
0.93
F.G.T.
x
8.21
III
1Ax~4
9.69
7.27 10.20 14.52
8.42
12,36
10,02
4.40
2.25
3.53
5.40
4.79
I
lc:,G.T.
13,14
Sample
4.37
6.35
5.18
2,52
2,07
2,83
2.59
4.85
7.15
4.05
6,00
4.84
2,62
2.48
1,67
2.78
2.39
10,11,12
I
'
II
!Axiom
~
I
C~L~~
Tabl~:
48
-
-+---------+--~
5.18
4.75
~--~~~~8~~~-4_8~:~~~-~~1-4_0~~~~~-~9-6____....,~~_-:2_3~6~:
The means and standard deviations of Scores I, II and
III on the Four Group Task for each of five age groups,
for two combined age groups and for the total sample.
One-way analyses of variance were performed on the three scores
on the Four Group Task, to test the significance of differences between
the mean scores of the five age groups.
The results of these analyses are
shown in Table 45 below and the details are in Tables 45A, 45B and 45C of
Appendix VI.
r-
I
F value obtained
Score
I
I
I
III
I
p
Four Group Task
Stage
17.70 (d.f. 4, 235) < .001
Four Group Task
Axiom
23.09 (d.f. 4. 235)
I Four Group Task
Total
Conclusions justified
by t tests
<)1, 0 =)112 )<yi1117«13)<_;Jl14
.001
yt, 0 ~~ 2 )<>Jt,, ')l13)-014
22.57 (d.f. 4. 235) <.001
~u, o=il12 kyti11 =/'13l<)A-14
<
J
Table 45:
The values of F, and their significance levels, obtained
in one-way Analyses of Variance to compare the performance
of five age groups on the Four Group Task. Also shown,
for each score, is the set of conclusions drawn after the
application of t tests to pairs of means. These t tests
employed the within groups estimate of variance, in each
case ..
Because of the restricted range of performance on the Four Group
Task Stage (I) and Axiom (II) Scores in the 10, 11 and 12 year-old age
groups, product-moment correlation coefficients were not computed for these
age groups.
The range of scores on these two indices is better (although
still somewhat restricted) in the 13 and 14 year-old age groups and the
product-moment correlations, computed for these age groups were + .847 (1%
confidence limits:- +.695<{'<+.925) and +.815 (1% confidence limits:+.640
<I'< +.910) respectively.
The very high values of the correlation
I
268
coefficients obtained, despite the restricted ranges of scores, support
the summing of the Stage and Axiom Scores to give a Total Score on the
Four Group Task,
Possibly the correlations are higher than desirable,
since it could be argued that the scores are measuring exactly the same
aspects of understanding of the task,
However, if it is remembered that
the Axiom Score is the sum of scores on five separate aspects, it is
clear that each of these individual aspects would have a much lower
correlation with the total and with other scores,
The correlations of
above +,8 do not seem inordinately high if they are regarded as correlations between two halves of a test.
A detailed item ana lysis, treating
the group axioms and concrete reversibility as single items, is not done,
since no comparable way of breaking down the Stage Score is available,
9.2
Interrelationships of Quantitative Scores on the Pendu!.um Problem,
the Four Gro!l2 Task and otherVariables.
Because of the restricted ranges of scores on the Four Group
Task, and to some extent also on the Pendulum Problem, in the 1 o, 11 and
12 year-old age groups correlations between scores on these tasks and
other variables are not computed for these age groups,
For each of the
13 year-old and 14 year-old age groups, however, it is appropriate to
intercorrelate the following quantitative scores:-
Assessed I.Q., raw
score on the Standard Progressive Matrices, Mathematics Mark, Pendulum
Total Score (out of 24) and Four Group Task Total Score (out of 20).
Tables 46A, 46B and 46C below present the matrices of correlations for
the 13 year-old subjects in School A(S), in School B(S) and in both
269
schools combined, respectively,
Correlations with the Mathematics Marks
are not computed for the combined schools, since it cannot be assumed that
the two examinations are comparablee
A.I.Q.
S. P.M.
**
A.I.Q.
+.825
S.P.M.
Maths.
Pend./24
F.G.T./20
+ .671
**
**
+.544
**
+.535
**
+,699
**
+.550
*
+,452
*
+.466
*
+,428
Maths
**
+,538
Pend,/24
F.G.T./20
Table 46A:
Product-moment correlations between five variables
for the 13 year-old subjects in School A(S), The
number of scores is 29, (*indicates that the
correlation differs from zero at the 5%,
at the
1%, level on a two-tailed test).
"*
A. I. Q.
A. I.Q.
S.P.M.
S.P.M.
Ma ths.
Pend./24
*
+.475
*''
+,714
+. 147
+.515*
'll*
+,236
**
+.723
+,322
+ .571*
+,614
Maths
Pend./24
F.G. T./20
*
+,539
F.G.T./20
Table 46B:
Product-moment cor relations be tween five variables
for the 13 year-old subjects in School B(S). The
number of scores is 19. (* indicates that the
correlation differs from zero at the 5%,
at the
1%, level on a two-tailed test).
*"
270
A.I.Q.
S,P.M.
**
A. I.Q.
•.703
Pend./24
F.G.T./20
+,386
+,489
**
S.P.M.
+.443
**
*"If
+,533
**
Pencl./24
+,535
F.G.T./20
TableJI2g_:
Product-moment correlations between four
variables for the total sample of 13 yearold subjects. The number of scores is 48.
C* indicates that the correlation differs
from zero at the 5%, ** at the 1%, level
on a two-tailed test),
Tables 47A, 47B and 47C below present similar correlation
matrices for subjects in the 14 year-old age group.
A. I. Q.
A. I.Q.
S.P.M.
Maths
Pend./24
S.P.M.
Ma ths.
**
+,748
**
+,809
+,266
+,283
**
+.576
+.237
+. 184
**
+.503
*
+.447
Pencl./24
F.G. T./20
+.292
F. G. T ./20
Table 47A:
Product-moment correlations between five variables
for the 14 year-old subjects in School A(S), The
number of scores is 28. (* indicates that the
correlation differs from zero at the 5%, ** at the
1%, level on a two-tailed test),
271
A.:LQ.
A. I.Q.
S,P.M.
Maths.
Pend,/24
**
+,825
**
+,679
+ .304
-.036
*
+.435
+,318
.ooo
*
+,221
S.P.M.
Maths
F.G.T./20
+.496
Pend,/24
+. 175
F.G.T./20
Table 47B:
Product-moment correlations between five variables
for the 14 year-old subjects in School B(S). The
number of scores is 20, (* indicates that the
correlation differs from zero at the 5%,
at the
1%, level on a two-tailed test),
**
A. I.Q.
S.P.M.
**
A.I .Q.
+,770
S.P.M.
Pend. /24
• 159
F.G.T./20
*
.295
. 103
+
Pend. /24
.246
F.G. T. /20
Table 47C:
Product moment correlations between four variables
for the total sample of 14 year-old subjects. The
number of scores is 48. (* indicates that the
correlation differs from zero at the 5%, ** at the
1%, level on a two-tailed test. + indicates a
difference from zero, at the 5% level, on a onetailed test).
In other studies (Lovell 1961;
Pumfrey 1968;
Stones and
Heslop 1968) the practice has been to relate performance to measures of
Mental Age, rather than I.Q.
TI1is is principally because the studies
comprise small numbers of subjects at any given age, so that correlations
272
between tests must be computed for subjects varying widely in age, and
then age partialled out of these correlations.
It has been found that
it is mental age, rather than chronological age, which contributes markedly
to such correlations.
In the present study, with relatively large samples
of subjects in several narrow age ranges, a different approach to the
problem of general ability, and its relationship to performance on Piagetian tasks, may be adopted.
For subjects of the same age, an I.Q. score is an appropriate
measure of relative ability.
It can be seen from Tables 46C and 47C that
Assessed I.Q. correlates +,386 with the Pendulum Total Score, and +.489 with
the Four Group Task Total Score, in the 13 year-old sample;
and +.159 with
the Pendulum Total Score, + .295 with the Four Group Task Total Score, in the
14 year-old sample<
The correlations between Total Scores on the Pendulum
and Four Group Tasks, in these samples, are +,535 and +.246 respectively.
When Assessed I.Q. is partialled out of these last two correlations, they
become +,428 (significantly different from zero, at the 1% level, on a twotailed test) and +,201 (not significantly different from zero).
The raw score on the Standard Progressive Matrices can be considered as an alternative measure to the Assessed I. Q.
It is a little unclear,
however, exactly how differences in such a raw score should be interpreted,
For subjects of the same chronological age, differences in raw score on
S. P.M. appear to relate to differences in Assessed I. Q. (correlations are
+.497, +.601, "".629, +.703 and +,770 in the 10, 11, 12, 13 and 14 year-old
age groups, respectively),
Since the only available Australian norms for
273
the Standard Progre$sive Matrices give I .Q. ranges (of about 15 I .Q ..
points) it is not possible to give each subject an exact I.Q. using his
score on this test.
Further, it is not possible to us-e the raw score
on the Standard Progressive Matrices to obtain a mental age, firstly
since there is no sfl,tisfactory way to decide which raw score corresponds
to an I. Q. of 100 in any given age, and secondly because the mean raw
score changes extremely little, with age, after a certain point.
An
attempt to derive a Mental Age (in months) -corresponding to each raw
score, from the available norms of I.Q. ranges yielded the results shown
below.
For each raw score, an attempt was made to find the age Cthe mid-
point of a range) for which that raw score corresponded to the I .Q. range
whose mid-point was 100.
Raw Score
S. P.M.
Mental Age ·
(months)
Raw Score
S.P.M.
Mental Age
(months)
24
120.5
33
145.5
25
122.5
34
148.5
26
124.5
35
151 .5
27
126.5
36
155.5
28
129.5
37
158.5
29
132.5
38
161 .5
30
135.5
39
165.5
31
138.5
40
179.5
32
141.5
41
203.5
274
Over the raw score range from 24 to 38, it is clear that the
increase in months of mental
consistently 2 or 3;
age~
for unit increase in raw score, is
however above a score of 38 (and, presumably,
below 24, although no norms are available) the relationship changes
markedly.· Since many raw scores obtained in the present study were
beyond these limits, no attempt was made to use this test to derive
mental ages.
This being the case, it is not possible to combine subjects
from different age groups in order to intercorrelate scores on the Pendulum and Four Group Tasks, and then partial out the correlation clue to
mental age.
If the raw score on Standard Progressive Matrices is regarded,
then, as a rough measure of non-verbal I.Q. (for subjects of the same
age), it may be partialled out of the correlation between the Pendulum
and Four Group Task Total Scores within any age .group, as was done for
Assessed I.Q.
For the 13 year-old sample the correlation of Standard
Progressive Matrices raw score with the Pendulum Total Score is •.443,
and with the Four Group Task Total Score +,533 (from Table 46C);
for
the 14 year-old sample the two corresponding correlations are +.103 and
+ ,272 (from Table 47C).
Partialling out the correlations with this
variable from the correlations of the Pendulum and the Four Group Task
Total Scores (+.535 in the 13 year-old sample;
+.246 in the 14 year-old
sample) reduces them to + .402 (significantly different from zero at the
1% level) and +.228 (not significantly different from zero) respectively.
If it can be ar guecl that the difference in raw score on Standard
275
Progressive Matrices, between the 13 year-old and 14 year-old age groups,
is of a similar order to the difference between them on Pendulum Total
Score and on the Four Group Task Total Score, then a combination of the
two age groups in order to intercorrelate the tasks and partial out
general ability could be con template d.
The means and standard de via t-
ions of raw scores on the Standard Progressive Matrices, for the five
age groups used in the present study, are shown in Table 48 be low.
Standard
Progressive
Matrices
Age Group ( Yrs.)
10
11
12
13
14
Total
Sample
Mean
35.85
38.98
40.79
41.77
43.13
40.09
s. d.
7.13
6.6o
5.63
5.88
6.27
6.78
n
48
48
44
48
48
Table 48:
236
Means and Standard deviations of raw score on
the Standard Progressive Matrices for the five
age groups and for the total sample.
A one-way analysis of variance, details of which are in Table
48A of Appendix VI, yields a
significan~
F ratio (F = 9.64, p<:: .001,
d.f. 4, 235) and t tests between pairs of means justify the conclusion
thatji
10
<</1
=-)i ~/11J)<Ic
11
12
14 •
A comparison.of these results with the
figures in Table 41 and 42 relating to the Pendulum Total Score, and
the figures in Tables 44 and 45 relating to the Four Group Task Total
Score,
indicates that scores on the three variables do improve simil-
arly with age.
Thus the raw score on Standard Progressive Matrices may
be taken as a measure of general intellectual ability (non-verbal),
although not calibrated in terms of mental age, and may be used in a
similar way to mental age in correlations which combine different age
groups.
Table
49 below presents the intercorrelations of the three
variables under consideration for the
13 and 14 year-old age groups,
combined.
Raw Score
Pendulum
Four GrouE Task
S.P.M.
24
20
**
.289
S.P.M.
**
.411
**
.450
Pend.
F.G.T.
Tabl<:'.Jl2:
Product-moment correlations between three
variables for subjects in the 13 and 14
year-old age groups, combined. The nt@ber
of scores is 96. (* indicates that a
correlation differs from zero at the 5%,
** at the 1%, level, on a two-tailed test).
When the cor relations with raw score on the Standard Progressive Matrices
are par tialled out of the correlation between the two tasks ( + .450) it
reduces to
+.379, which remains significantly different from zero at the
1% level.
It 1s not advisable to use a similar correlational approach to
the above to investigate the relationship of Mathematics Mark to performance on the Pendulum and Four Group Tasks since, as was pointed out
earlier, it cannot be assumed that marks in the examinations of the two
different schools are comparable.
Analyses restricted to one school at a
time are not satisfactory because the numbers of subjects are so small
277
that correlations have a very large standard error.
One possible way to
combine results from the two schools is to use the categories of Low,
Medium and High mathematics marks by which the sample at each age is
stratified,
If the subjects in the top thirds of the two schools are
combined, and similarly for the middle and bottom thirds, it is reasonable to assume that differences in standard between the schools will
result in only a small number of misclassifications, in the boundary
regions.
If measures such as Total Score on the Pendulum and Four Group
Tasks are then made the dependent variables, with subjects classified as
above, Analysis of Variance techniques may be used to investigate the
relationship of Mathematics Mark to performance on the tasks.
This
approach has the added advantage of making it possible to study the
interaction effects of more than one variable on task performances.
It is
only in the 13 and 14 year-old age groups that Total Scores on the Pendulum and Four Group Tasks are of sufficient range foJ: any such effects
to be detectable.
In the 13 and 14 year-old age groups, then, 2 x 3 factorial
Analyses of Variance were performed, with subjects classified into two
levels of raw score on the Standard Progressive Matrices and three levels
of Mathe rna tics Mark in school.
two dependent variables;
In each age group analyses were done for
the Pendulum Total Score (out of 24) and the
Four Group Task Total Score (out of 20).
The relevant means and standard
deviations are reported in Tables SOA and 51A (for the 13 and 14 year-olds,
respectively) of Appendix VI.
The means and standard deviations for the
two schools separately, as well as combined, are shown in those tables,
although no attempt to analyse the effects of school attended was made
here.
Summary tables of the analyses of variance are presented in
Tables
50
and
51
below (for the
Dependent Variable:
Rows (S.P,M.)
and
14
year-olds, respectively).
Pendulum Total Score (out of 24)
Mean square
F
p
73.50
73.50
5.25
<.os
SLt:m of Squares
d,f.
Source
13
1
Cols. (Maths)
2
22.75
11.38
0.85
N.S.
~ows x Cols.
2
60.75
60.75
2.28
N.S.
Within Gps.
18
23().50
13.31
Total
23
Dependent Variable:
Source
I
3()6.50
Four Group Task Total Score (out of
d. f ..
Sum of Squares
20)
Mean square
F
p
<.01
~ows (S.P.M.)
1
181.50
Cols. (Maths)
2
101 .33
50.67
3. 16
N.S.
Rows x Cols.
2
36.oo
18.00
1 . 12
N.S.
18
288.50
16.03
23
607.33
~""'"
Gps.
Total
Table
50:
--
181 .50
11
.32
Summary tables for Analyses of Variance performed on the scores
of 13 year-old subjects classified into two levels of S.P.M.
raw score (row variable) and three levels of Mathematics Mark
(column variable). Analyses are reported for two dependent
variables; Pendulum Total Score (out of 24) and Four Group
Task Total Score (out of 20), Means and standard deviations
for the cells of the 2 x 3 table are reported in Table 50A
of Appendix VI, but subjects were randomly discarded to give
an equal number in each cell (namely 4) for the analyses.
The subjects used in the analyses are indicated in the table
of raw data for the 13 year-olcls in Appendix VII.
279
Dependent Variable:
Source
Pendulum Total Score (out of 24)
d. f.
Sum of Squares
Mean square
F
p
Rows (S.P.M.)
1
0,38
0,38
0.04
N .S.
Cols, (Maths)
2
125.08
62.54
6,00
<.OS
Rows x Cols.
2
9.75
4,88
o.67
N.S.
Within Gps.
18
187.75
10.43
Total
23
322.96
Mean square
F
p
Dependent Variable:
Source
Four Group Task Total Score (out of 20)
d~f.-
Sum of :Squares
Rows (S.P.M.)
1
1. 04
1. 04
o.os
N.S.
Cols. (Maths)
2
1 51 • 08
75.54
3.90
(.05
Rows x Cols,
2
81 ,08
4o.S4
2.09
N.S.
___j
Within Gps,
18
348.75
Total
23
581.95
Table 51:
19.38
--
Summary tables for Analyses of Variance performed on the
scores of 14 year-old subjects classified into two levels
of S. P.M. raw score Crow variable) and three levels of
Mathematics Mark (column variable). Analyses are reported
for two dependent variables; Pendulum Total Score (out of
24) and Four Group Task Total Sco10e (out of 20). Means and
standard deviations for the cells of the 2 x 3 table are
reported in Table 51A of Appendix VI, but subjects were discarded randomly to give an equal number in each cell (namely
4) for the analyses, The subjects used in the analyses are
indicated in the table of raw data for the 14 year-olds in
Appendix VII.
280
From Tables So and 51 above, it is clear that there are
different effects of the two variables, on scores on the tasks, in the
13 and 14 year-old groups.
It seems worthwhile, then, to combine the
age groups and perform two three-way Analyses of Variance, to discover
any significant interaction effects with age.
The combination of age
groups necessitates the assumption that, even though four different
examinations in mathematics are used to make classifications of subjects
into the Low, Medium and High groups, these nevertheless reflect comparable divisions in rna thema tics abi 1i ty.
In addition to this, the combin-
ation of subjects, across age groups, into Low and High categories of
raw score on the Standard Progressive Matrices may involve some errors
of classification, since the division is between 44 and 43 for the 14
year-olds and between 43 and 42 for the 13 year-olds.
It is assumed that
any such errors would have a negligible effect on the results.
Summary
tables for the two three-way Analyses of Variance are provided in Table
52A of Appendix VI.
The F ratios obtained and their significance levels,
are shown in Table 52 below.
Dependent Variable:
Source
Pendulum Total Score (out of 24)
F ratio
d,.. f.
p
A (Age)
4.21
1 ,36
'(.OS
B (S.P.M.)
3.55
1 ,36
c
5.25
2,36
AB
2.07
1 ,36
N.S.
AC
2,36
N.S.
BC
0,98
0,77
ABC
2.20
2,36
(Ma ths)
2,36
N.S.
'(,OS
-- --
N.S.
N.. S.
--
281
Dependent Variable:
F ratio
d, f.
p
13,98
1 ,36
<.001
5.93
1 ,36
<.05
7,06
2,36
(,05
4.38
1,36
N. S.
AC
0,07
2,36
N.S.
BC
0,26
2,36
N.S,
ABC
3.05
2,36
N.S.
Source
A (Age)
I
B (S.P.M.)
c
(Ma ths)
AB
-
I
Table 52:
9.3
Four Group Task Total Score (out of 20)
Summary tables for two three-way Analyses of
Variance combining the scores of the 13 and 14
year-old subjects used for the analyses
reported in Tables So and 51 above.
Measures of Efficiency in L.earning the Four Group Task r;nd their
Relationship to other Variables.
The available measures of efficiency in learning the sixteen
combinations of the Four Group Task have been described and discussed in
Chapter
7 •
Table 12 of section 7
.1
showed the measures tabula ted for
each subject, but indicated that only four of these were actually used
in analysis.
The tables of raw data in Appendix VII provide only the
four measures used in statistical analyses.
These are the Time to Test
and the Time to Criterion (both to the nearest half minute), the Total
Number of Questions asked (i.e. the number of combinations tried by the
subject during learning phases, plus those asked about by the experimenter
during tests and retests) and the Percentage, of the Total Number of
282
Questions, on which the correct outcome was predicted by the subject.
Section
9.3. 1
with age.
will describe changes in the four measures of efficiency
These efficiency measures will then be related to subjects'
mathematical and general ability in section
9.3.2
and to their levels of
performance on the Pendulum Problem and Four Group Task in section
9.3.1
9.3.3.
Changes in measures of efficiency in learning the Four Group
Task with age.
Table 53 below presents the means and standard deviations of
scores on the four efficiency measures, for each of the five age groups,
and for the total sample.
Age Group (Yrs,)
Efficiency
Measure
(i) Time
Test
10
-X
1
(ii) Time
Critn.
(iii) Tot.
Qns.
(i v) % Qns.
correct
s
X
s
X
s
X
s
n
Table
53:
11
12
13
14
Total
Sample
9.97 9.35 12.79 11 .o5 6.90
4.38 4.73 7.01 7.35 2,58
9.96
5.76
22.02 19.09 22.60 22.13 14.28
7. 72 8.47 9.88 9.67 5.44
19.98
8.87
68.54 58.69 81.14 75.04 49.40
19.91 24.47 27.70 29.99 13.92
66.31
26.20
65.17 66.23 64.57 61.90 66.79
7.33 9.00 8.54 10.81 8.50
64.94
9.01
48
48
44
48
48
236
Means and standard deviations of scores on four
measures of efficiency in learning the sixteen
combinations of the Four Group Task. Figures
are shown for each age group and for the total
sample.
283
The differences between the mean performances of the five age
groups were tested by means of one-way Analyses of Variance, details of
which appear in Tables 54A, 54B, 54C and 54D of Appendix VI.
The F
ratios obtained, their significance levels, and the conclusions drawn
about differences between particular means, are shown in Table 54 below.
''
I
F ratio and
d. f.
Efficiency ·
Measure
p
Conclusions justified
by t tests between pairs
of means
(i) Time T
1
7-74 (d. f. 4, 235)
"(,001
/« 14<)-1<1 o ~/!11 l <)A12if13 l
(ii) Time Critn.
8.53 (d. f. 4. 235)
"(. 001
}f.
13.87 (d.f. 4. 235)
< ,001
(JA-,4 =?11
2.34 (d,f. 4. 235)
N.S.
(iii) Tot. Qns.
(iv) % Qns, Corr.
Table 54:
9.3.2
14
<~~~
-JI -tl -/( )
10l1f/ 12? 13
J<c_A, of'<12'ft13l
No differences
The table shows the F ratios obtained, the degrees of
freedom, and the level of significance, in one-way
Analyses of Variance to test the differences in per formance of five age groups on four measures of efficiency
in learning the Four Group Task, Also shown is the set
of conclusions drawn about differences between the age
group means. These conclusions are reached by the
application of t tests using the within groups estimate
of variance in each case.
The reJ_ationship of measures of efficiency in learning the Four
Group Task to mathematical and general ability.
The four measures of efficiency in learning the Four Group Task
may be correlated, within each age group, with Assessed I.Q., with raw
score on the Standard Progressive Matrices and, for each school separately,
with Mathematics Mark in a school examination.
Since the intercorrelations
are uniformly low, except between the four efficiency measures themselves,
the complete set is presented only as Tables 55A, 55B, 55C, 55D and 55E
of Appendix VI,
The table of intercorrelations for the 13 year-old age
group is shown as Table 55 below.
It is sufficiently typical to give
an indication of trends in the tables for the other age groups as well.
Only in the 11 year-old sample do some correlations of Assessed I.Q.
with efficiency measures reach
A. I. Q.
significance~
S.P.M.
A(S)
Maths
B( S)
Time
T1
Time
Cr i tn.
Total
Qns.
% Qns.
cor r.
**
(+,671
**
+,714) +,017
-.142
-.204
+. 100
**
(+,6<)9
**
+. 614)
+,033
-.083
-.178
+. 181
Maths A(S)
+,069
-.005
+.037
-.003
B( S)
-.232
-.384
-.432
+ .274
**
+,678
**
+.452
**
-.448
**
**
-.632
**
+,703
A. I. Q.
S. P.M.
Time T1
Time Cri tn.
+. 762
**
-.658
Tot, Qns,
% Qns. Corr.
Table 55:
9.4
Product-moment correlation coefficients between four
measures of efficiency in learning the Four Group Task
and measures of mathematical and general ability, for
the 13 year-old age group. For School A(S), n = 29;
for School B(S), n = 19; and thus, for the total 13
year-old sample, n = 48. (*indicates that the correlation differs from zero at the 5%, ** at the 1%, level
on a two-tailed test),
The Relationship of Developmental Level to Intelligence and to
Efficiency in Learning the Four Group Task.
In periods of transition from one developmental stage to the
next, it is difficult to predict the changes which may occur in the
285
efficiency with which a task is learned,
Although new thought structures
may ultimately provide a more efficient framework for the memorising and
coding of information, while they are in a formative stage they may
actually confuse and impede the memory processes.
It is therefore inter-
esting to categorise subjects into early and advanced developmental stages
on both the Pendulum Problem and Four Group Task and to investigate whether
there are associated differences in their efficiency of learning the
latter.
Two-way factorial Analyses of Variance are used in preference to
correlational techniques for two reasons.
Firstly, they allow interaction
effects of the developmental levels on both tasks to be investigated.
Secondly, they are applicable to the 10, 11 and 12 year-old age groups,
where subjects may be categorised suitably into early and late developmental
levels, but where the quantitative scores on the Pendulum Problem and Four
Group Task do not cover sufficient range for correlational analyses.
This
type of analysis also offers a way of investigating the relationship of
measures of intelligence to developn1ental level on the Pendulum and Four
Group tasks, amongst the 10, 11 and 12 year-olds.
For the purposes of these analyses, subjects are dichotomised,
as closely as possible to the median, into early and late developmental
levels on both the Pendulum and Four Group tasks.
The analyses are per-
formed separately, and the points of dichotomisation different, for each
of the five age groups.
Tables 56A, 57A, 58A, 59A and 6oA of Appendix VI
provide the means and standard deviations of scores on two measures of
intelligence and four measures of efficiency in learning the Four Group
Task, for subjects in the cells of the relevant 2 x 2 tables.
These
286
tables are for the 10, 11, 12, 13 and 14 year-old age groups respectively,
Tables 56, 57, 58, 59 and 60 below report the significant
effects obtained
in Analyses of Variance for each of the six variables in each of the five
age groups.
Subjects were randomly discarded to give equal numbers in the
four cells, and those subjects actually used in the analyses are indicated
in the tables of raw data in Appendix VII.
Details of the analyses of
variance are in Tables 56B, 57B, 58B, 59B and 6oB of Appendix VI.
Dependent
Variable
Significant
Effects
F
ratio
d.f.
p
(i) Assessed I.Q.
Interaction (Pend.
X F.G.T.)
32.47
1 • 12
<.001
(ii) Raw score, S.P.M.
Interact ion (Pend.
X F.G.T.)
6.27
1 '12
<.05
(iii) Time to Test 1
None
Civ) Time to Criterion
None
(v) Total Questions
Interaction (Pend,
X F.G.T.)
4.74
1 '12
<.06
(vi) % Qns. Correct
None
Table 56:
I
Significant effects obtained in two-way Analyses of
Variance with 10 year-old subjects categorised into
two developmental levels on each of the Pendulum and
Four Group Tasks. Results are reported for six
dependent variables. Relevant means and standard
deviations are in Table 56A, and summary tables of
the Analyses of Variance in Table 56B of Appendix VI.
I
_j
287
r
I
I
Dependent
Variable
Signific;;tnt
Effects
F
ratio
d.f.
p
6.10
1,24
< .os
'
(i) Assessed I. Q.
Main effect, F. G. T.
Cii) Raw score, S.P.M.
None
I( iii) Time to Test
Main Effect, Pend.
10.28
1 ,24
<: ,01
Main Effect, Pend.
18.10
1,24
< ,001
Main Effect, F.G.T.
7.55
1 ,24
<
(v) Total Questions
Main Effect, Pend,
9.12
1,24
< ,01
(vi) % Qns Correct
Main Effect, Pend,
5.96
1 ,24
< .. os I
1
l(iv) Time to Criterion
Table 57:
I
I
Significant effects obtained in two-way Analyses of
Variance, with 11 year-old subjects categorised into
two developmental levels on each of the Pendulum and
Four Group tasks. Results are reported for six
dependent variables, Relevant means and standard
deviations are in Table 57A, and summary tables of
the Analyses of Variance in Table 57B of Appendix VI.
Dependent
Variable
F
ratio
d .. f"
p
(i) Assessed I.Q.
Main Effect, F.G. T.
6.67
1,20
<:.OS
'(ii) Raw score, S.P.M.
None
'<iii) Time to Test 1
None
l(iv) Time to Criterion
None
l(v) Total Questions
Main Effect, F.G.T.
(vi) % Qns. Correct
None
I
Table 58:
I
.os
Si gni fi cant
Effects
!
'
I
II
21.81
1,20
<.001
Significant effects obtained in two-way Analyses of
Variance, with 12 year-old subjects categorised into
two developmental levels on each of the Pendulum and
Four Group tasks. Results are reported for six
dependent variables. Relevant means and standard
deviations are in Table 58A, and summ<:~ry tables of
the Analyses of Variance in Table 58B of Appendix VI.
288
--
Dependent
Variable
,~,--
Significant
Effects
F
d. f.
p
-
ratio
(i) Assessed I.Q.
Main Effect, F.G.T.
7.88
1 ,24
(ii) Raw score, S.P.M.
Main Effect, F.G. T.
5.56
1,24
< .01
< .o5
(iii) Time to Test 1
None
(iv) Time to Criterion
None
(v) Total Questions
Main Effect, F. G. T.
6.20
1.24
<.o5
(vi)
% Qns. Correct
Table22_:
None
Significant effects obtained in two-way Analyses of
Variance with 13 year-old subjects categorised into
two developmental levels on each of the Pendulum
and Four Group tasks. Results are reported for six
dependent variables. Relevant means and standard
deviations are in Table 59A, and summary tables of
the k1alyses of Variance in Table 59B of Appendix VI •
.
Dependent
Variable
I
Significant
Effects
(i) Assessed I. Q.
None
(ii) Raw score, S.P.M.
None
(iii) Time to Test 1
None
(iv) Time to Criterion
None
(v) Total Questions
None
(vi)
% Qns. Correct
Table 60:
Interaction (Pend.
xF.G.T.)
F
ratio
d.f.
p
I
4.54
1,32
< .05
Significant effects obtained in two-way Analyses of
Variance with 14 year-old subjects categorised into
two developmental levels on each of the Pendulum and
Four Group tasks. Results are reported for six
dependent variables. Relevant means and standard
deviations are in Table 6oA, and summary tables of
the Analyses of Variance in Table 60B of Appendix VI.
289
9.5
Effects of Age, Sex and School Attended on Quantitative Scores on
the Pendulum and Four Group Tasks, and on Measures of Efficiency
in Learning the Latter Task,
The effects of sex and school attended, on the four measures
of efficiency in learning the Four Group Task, were examined by means
of two-way Analyses of Variance.
for each age group.
The analyses were performed separately
The means and standard deviations of scores on
efficiency measures, for subjects categorised as for these analyses,
appear in Tables 61A, 62A, 63A, 64A and 65A of Appendix VI.
These
tables are for the 10,11, 12, 13 and 14 year-old age groups, respectively.
For the analyses, subjects were randomly discarded to give equal numbers
in the cells and those subjects actually used are indicated in the tables
of raw data for the respective age groups, in Appendix VII.
Tables 61,
62, 63, 64 and 65 below report the results obtained in these ana lyses,
details of which appear in Tables 61B, 62B, 63B, 64B and 65B of Appendix
VI.
I
I
l(i)
I
Dependent
Variable
'
Significant
Effects
Time to Test 1
None
l(ii) Time to Criterion
None
(iii) Total Questions
None
l(i v) % Qns. Correct
Table 61:
I
F
d .. f ..
None
Significant effects obtained in two-way
Analyses of Variance, with 10 year-old
subjects classified by Sex and School
attended. Results are reported for four
dependent variables.. Relevant means and
standard deviations are in Table 61A, and
summary tables of the Analyses of Variance
in Table 61B, of Appendix VI.
p
I
290
I
~
I
(i )
--
Significant
Effects
None
(i i) Time to Criterion
None
1
ii) Total Questions
l(iv)
L
d.f.
p
1 ,36
< ,05
None
% Qns. Correct
I
Table 62:
F
ratio
r--~-
Time to Test 1
I (,
I
II
Dependent
Variable
Main Effect,
School
7.22
Significant effects obtained in two-way Analyses
of Variance, with 11 year-old subjects classified
by sex and school attended, Results are reported
for four dependent variables, Relevant means and
standard deviations are in Table 62A, and summary
tables of the Ana lyses of Variance in Table 62B,
of Appendix VI.
I
Dependent
Variable
Significa nt
Effects
(i) Time to Test 1
None
(ii) Time to Criterion
None
kiii) Total Questions
None
ICiv) % Qns, Correct
None
-------"--·
Table 6].:
F
ratio
d.f.
p
I
I
J
Significant effects obtained in two-way Analyses
of Variance, with 12 year-old subjects classified
by sex and school attended, Results are reported
for four dependent variables. Relevant means and
standard deviations are in Table 6JA, and summary
tables of the Analyses of Variance in Table 6JB,
of Appendix VI.
291
Dependent
Variable
Significant
Effects
(i) Time to Test 1
F
ratio
d. f.
p
None
(ii) Time to Criterion
Main Effect,
School
4.15
1 ,32
< .05
l(iii) Total Questions
Main Effect,
School
6.80
1,32
<
liv)% Qns, Correct
Table 64:
'
None
Significant effects obtained in two-way Analyses
of Variance, with 13 year-old subjects classified
by sex and school attended, Results are reported
for four dependent variables, Relevant means and
standard deviations are in Table 64A, and summary
tables of the Analyses of Variance in Table 64B,
of Appendix VI.
Dependent
Variable
Significant
Effects
(i) Time to Test 1
None
(ii) Time to Criterion
None
l(iiD Total Questions
None
( i v) % Qns. Correct
None
Table 65:
.05
F
ratio
I d.f.
p
Significant effects obtained in two-way Analyses
of Variance, with 14 year-old subjects classified
by sex and school attended. Results are reported
for four dependent variables. Relevant means and
standard deviations are in Table 65A, and summary
tables of the Analyses of Variance in Table 65B,
of Appendix VI.
292
Since there are clearly very few effects of sex and school
attended on the measures of efficiency in learning the Four Group Task,
it is not worthwhile to pursue changes of such effects with age, in a
three-way analysis.
In the case of the quantitative measures of devel-
opmental level on the two tasks, however, there are more consistent
effects of sex and school attended, some of which appear to change with
age.
To conclude the analysis of results, then, results of two three-
way Analyses of Variance (Age x Sex x School Attended) on the Pendulum
Total Score (out of 24) and the Four Group Task Total Score (out of 20)
will be presented.
The means and standard deviations of scores in the
twenty cells <5 x 2 x 2) may be found in Tables 61A, 62A, 63A, 64A and
65A of Appendix VI.
number
Subjects were randomly discarded to provide an equal
(8) in each cell, and those subjects whose scores are actually
used in the analyses are indicated in the tables of raw data, for the
res pee ti ve age groups, in Appendix VII.
The Ana lyses of Variance are
reported in Tables 66 and 67 below, for the Pendulum and Four Group Task
Total Scores respectively.
293
'
~
d. f.
Source
Sum of Squares
Mean square
F
p
(Sex)
1
23.25
23.25
1.95
N.S.
B (School)
1
20.31
20.31
1 • 71
N.S.
4
214.59
53.65
4.51
<,01
AB
1
0,05
0,05
o.oo
AC
4
185,84
46.46
3.90
BC
4
27.79
6.95
0,58
N .S.
4
73.54
18.38
1.54
N.S.
140
1665.87
11.90
159
2211,24
c
(Age)
I
-
ABC
Within Gps,
1.....Total
r
Table 66:
I
The table presents the results of a three-way
Analysis of Variance, with subjects classified
by Age Groups (5), Sexes (2) and Schools
Attended (2). The dependent variable is the
Pendulum Total Score, out of 24. Means and
standard deviations for each of the cells may
be found in Tables 61A, 62A, 63A, 64A and 65A
of Appendix VI, and those subjects used in the
analysis are indicated in the tables of raw
data in Appendix VII.
N. S.
< ,05
294
I
i'
\A
Source
d.f.
Sum of Squares
Mean square
I
F
p
(Sex)
1
5.26
5.26
0.30
N.S.
IB (School)
1
209.31
209.31
12.01
< ,001
fc
4
910,21
227.55
13.05
< .001
'
lAB
1
3.90
3.90
0,22
N,S.
I
AC
4
29.59
7.40
0,42
N.S.
BC
4
69.60
17.42
1.00
N.S.
ABC
4
53.94
13.48
0.77
N.S.
Within Gps.
140
2440.62
17.43
Total
159
3722.49
I
(Age)
L
I
Table 67:
--
The table presents the results of a three-way Analysis
of Variance, with subjects classified by Age Groups
(5), Sexes (2) and Schools Attended (2). The dependent
variable is the Four Group Task Total Score, out of 20.
Means and standard deviations for each of the cells may
be found in Tables 61A, 62A, 63A, 64A and 65A of
Appendix VI, and those subjects actually used in the
analysis are indicated in the tables of raw data in
Appendix VII.
This chapter concludes the description
of results obtained in the study,
of
the scoring and analysis
Chapter 10 puts forward an interpretation
of the concrete and formal operational levels of approach to the Four Group
Task, using some further detailed material from interviews.
The relation-
ships obtained between the levels of performance on the Four Group Task and
scores on other variables are then discussed, together with the methodological and theoretical implications of the study, in Chapter 11,