An Investigation of Methodological Problems in the Cognitive Developmental Diagnosis of Transitivity

Discrepancies in the age of emergence of transitivity have been found by various investigators. Piagetians argue that transitivity emerges at approximately age seven to eight years while other researchers claim that children as young as four to five years have transitivity skills. It can be argued that these differing conclusions about the age of emergence of transitivity are possibly due to differences in methodological variables. The major objectives of the present investigation were to determine whether differences in the age of emergence of transitivity are due to methodological differences in: 1) type of training (active training vs. passive training) b) order of presentation of training pairs during training (random vs. ordered), and/or c) type of response criterion demanded (judgment vs. explanation). Subjects age 4-5 to 5-0, 6-0 to 6-7, and 7-6 to 8-0 were administered one of four premise pair training tasks: passive ordered pair, passive random block, active ordered pair, or active random block. Subsequent to the training phase subjects were tested on inference pairs using one of two types of response criteria: judgment only or judgment plus explanation. Significant age and type of training effects were found for both the judgment and explanation criteria on inference pair errors. No stgnif.icant order of trainlng pairs effect was found for either criterion. These results lend partial support to the major hypotheses. It ca..~ thus be concluded

In any of these three cases, the tester will misinterpret the absence of the symptom response as the absence of transitivity.
The possible causes of false positive errors are: 1) the child might just guess, 2) the child might directly perceive that A was longer than C without using transitivity, or J) the child might use another (nontransitive inference) solution strategy. One such strategy (Smedslund, 1963) is that the child might code the initial premises (A)B, B)C) as "A is long but Bis not long, and Bis long but C is not long." Since A has been coded as being "long" and C "not long" the child will choose A when asked to solve the AC inference relationship.
In this way the resulting display formed two vertical columns running in ascending or descending order from card one to card five and card two to card six. In the random display the five premise pairs were presented in a random vertical arrangement (e.g. (5,6), (2,3), (1,2), (3,4), (4,5)). (1) Subjects must discover the initial premise relations for themselves in an active task. No attempt is made by the experimenter to directly ask the child to compare A with B or B with c.
The child must figure out for him or herself how to acquire the needed premise information.
Children who do not measure will not obtain this information. It is possible that this difference makes the task a more difficult one and thus hinders the expression of transitivity.
(2) Subjects are allowed to concretely manipulate the materials in the task.
They have the opportunity to interact with the initial premise stimuli. They are even required to actively construct one of the initial premise conditions (tower two or C  Youniss, 1968;Riley, 1975Riley, , 1976Riley & Trabasso, 1974;Trabasso, 1975;Trabasso, Riley, & Wilson, 1975;Youniss & Murray, 1970) or using both the judgment and the child's explanation of his/her judgment (McManis, 1969;Smedslund, 1960Smedslund, , 1963 1. Transitivity will significantly increase with age for both the judgment and judgment plus explanation criterion.
2. The judgment plus explanation response criterion will be more difficult for younger children than for older children.
3. Children who receive the active training conditions will evidence significantly more transitivity than children who receive the passive training conditions. 4. Children who receive randomized block training will evidence significantly less transitivity than children who receive ordered pairs training.

5.
The effect of the more difficult randomized block training will be more pronounced for younger children than for older children.
6. The facilitative effect of the active training condition will be more pronounced for younger children than for older children.

Subjects
Chapter II METHOD Subjects were 120 Caucasian children including 60 boys and 60 girls.
These age levels were selected so that they would span the controversial age of emergence phenomenon. The experimenter and subject manipulate one of the five sticks.
The subject is given the other four sticks and the experimenter continues saying ' Now here are the other four sticks.
I want you to make sure that these all work in the same way as the other one.' The subject then manipulates the other four sticks. The experimenter continues 'Now, if I put these sticks onto this rack you can only see the colored top of the sticks.
The rest of the stick is hidden behind this piece of black cloth.' The experimenter demonstrates the rack and screen apparatus with the first color order pair. 'You see -these two sticks look about the same length from the front but if I turn the rack around then you can see that one stick is longer than the other.' The experimenter turns rack around and lets the subject see the sticks of different lengths on the back side.
'Now I'm going to show you two sticks at a time and ask you to (Active conditions -'make one longer than the other one') (Passive conditions -'tell me which one is longer or which one is shorter'). The important thing is to try to remember which sticks are longer than other sticks and which sticks are shorter than other sticks.
I'm going to give you lots of practice with the sticks and try very hard to remember which is longer and which stick is shorter in every two sticks I show you. After you practice for awhile then we'll try to see how much you remember.
So be sure you try to remember which sticks are longer than others and which sticks are shorter than others e Now tell me what you are supposed to try to remember while you practice?' After the subject successfully repeated the task directions the training phase beg an. The factors in the design were age (4-5 to 5-0, 6-0 to 6-7, and 7-6 to 8-0), type of training (active and passive), and order of training pairs (randomized block and ordered pairs). The schematic representation of the experimental design is presented in Figure 2. (4-5 to 5-0, 6-0 to 6-7, 7-6 to 8-0), type of training (passive and active), and order of training pairs (random block and ordered pair).
Hartley's F max test (Winer, 1971) was applied to the data and no violation of the homogeneity of variance assumption was found [F max (12,9)= 6.302, n).05]. The summary of the analysis of variance of the judgment error scores on inference pairs is presented in Table 1.
As can be seen in Table 1 Table 2. This interaction is depicted in Figure J. Table 1 Summary

Explanatign Scores
The means and standard deviations for all explanation data are reported in Table 8 in Appendix B. The data collected for the explanation criterion were analyzed by a 2X2X2 analysis of variance. The analysis contained the following factors: age of subjects (6-0 to 6-7 and 7-6 to 8-0), type of training (passive and active), and order of training pairs (random block and ordered pair).
The 4-5 to 5-0 age group was excluded from this analysis due to the lack of variability in their explanation scores. The mean number of errors on inference pairs for this group was 11.950 (SD= .224).
Only one subject in all conditions gave one correct explanation out of a total of 240 possible correct explanations. The means for the subjects age 6-0 to 6-7 and 7-6 to 8-0 were 6.950 (SD= 4.249) and 4.900 (SD= 3.523) respectively.
The means for the three age groups are depicted in Figure 4.
Hartley's F max test (Winer, 1971) was performed and no violation of the assumption of homogeneity of variance was found [F max (8,4) = 7.426, n).05].
The summary of the analysis of variance of the explanation error scores on inference pairs is presented in Table J.
As can be seen in Table 3 Table 9 in Appendix B.
The data collected on training pair errors during testing were  Table 3 Summary age (4-5 to 5-0, 6-0 to 6-7, 7-6 to 8-0), type of training (passive and active), and order of training pairs (rc1...ndom block and ordered pairs). Hartley I s F max test (Winer, 1971) was used as a preliminary procedure and no violation of the homogeneity of variance assumption was found [F max (12,9) = 5.022, n).05]. The summary of the analysis of variance of the training pair errors during testing is presented in Table 4.  Table   5. This interaction is depicted in Figure 5.
Due to the significance of the age by type of training interaction it was necessary to analyze both the age effects within the two levels of type of training and the type of  Table 6. It can be seen in Table 6   Subjects in the youngest age group (4-5 to 5-0) demonstrated significantly less transitivity than subjects in the other two age groups (6-0 to 6-7 and 7-6 to 8-0).
Yet, the subjects in the latter two age groups did not differ significantly from each other on either criterion. Active training was found to be significantly more difficult than passive training and more specifically, active training was significantly more difficult than passive training for subjects in the middle (6-0 to 6-7) and older (7-6 to 8-0) age groups.
No difference in type of training was found for subjects in the youngest age groups.

Overall Implications and Methodological Issues
The evidence in support of the contention that type of response criteria demanded affects the age of emergence of transitivity is consistent with the findings of the few previous investigators who have addressed this question in their work (Brainerd, 1973;, 1977;Roodin & Gruen, 1970 tion data seem to support this conclusiono As can be seen in Table 6, subjects age 7-6 to 8-0 gave many more explanations in which only correct training pairs were mentioned (category 5) than did the subjects age 6-0 to 6-7. It is possible that a more stringent explanation criterion than the one used in this study and that of previous researchers (Smedslund, 1963;Roodin & Gruen, 1970) Riley's (1977) findings.
Given the setting in which this study was conducted and the time constraints imposed by the schools themselves, both visual and verbal feedback were used to facilitate training.
It is thus possible that subjects! judgments could have been based upon "nontransitiveN hypotheses (Smedslund, 1963 It can be seen that category 3 explanations were given by 37.92% of the subjects age 4-5 to 5-0, but only 6.67% by subjects age 6-0 to 6-7, and only 4.17% by subjects age 7-6 to 8-0. It    o.oo Table 9 Summar y of Mean