Representation

Overall, 68 percent (69/102) of children who received the video analogue described both the problem and solution to the problem which included the appropriate criteria to warrant being scored on representational ability. This is a 27 per cent increase from session one. A chi square analysis revealed significant differences in children’s ability to represent the base problem across sessions χ2

(1, N = 17) = 5.781; p<.05. This suggests that children’s representational ability improved over the 30 week period.

As in session one, the second most frequent description was the solution only (15 per cent). A reliance on solution only may affect children’s ability to solve the problem in that they do not see that the target problem is related to the base problem (because they do not recognise that they are structurally similar problems). Table 11 shows that solution only was more prevalent in the problems needing an auxiliary task where children found the relations harder to represent. Results revealed that 3-year-olds represented 58 per cent of the base problems and four-year-olds represented 76 per cent of problems. A chi square analysis revealed significant differences between age group χ2

(1, N = 17) = 3.594; p<.05. This indicates a developmental difference in representational ability between 3- and 4-year-olds in session two in that 4-year-olds better able to represent than 3-year-olds.

Representation also significantly increased for both 3- and 4-year-olds across session (46 per cent and 58 per cent; χ2

(1, N = 17) = 2.885; p<.05, and 37 per cent and 76 per cent; χ2

(1, N = 17) = 30.943; p<.05, respectively). This indicates that children in both age groups, representational ability increased over the 30 week period.

Session one revealed that children represented more base problems in the problems without an auxiliary task compared to the problems needing an auxiliary task. The same was found in session two, χ2

(3, N = 17) = 27.997; p<.05 (92 per cent and 43 per cent, respectively). This supports the results found in session one and confirms the suggestion that the relations in the problems needing an auxiliary task may be too difficult for children of this age to represent. Children’s representation in both problems without an auxiliary task and with an auxiliary task increased over sessions. Children’s ability to represent problems without an auxiliary task increased by 33 per cent and in the problems needing an auxiliary task, 19 per cent. Chi square analysis revealed significant difference across sessions (without auxiliary - χ2(1, N = 17) = 29.436; p<.05 and with auxiliary - χ2

Table 12: The Percentage of Children’s Descriptions of the Base Analogue in the Video Condition as a Function of Problem Type and Age

___________________________________________________________________________________________________________ Without auxiliary task Auxiliary task

3-yr-olds 4-yr-olds 3-yr-olds 4-yr-olds

No response/Irrelevant 13 0 21 11

Problem only 0 0 13 15

Solution only 4 0 33 22

Problem and solution 83 100 33 52

Table 12 shows that both 3- and 4-year-olds found representation easier in the problems without an auxiliary task compared to the problems needing an auxiliary task. Chi square analysis revealed significant differences in 3-year-olds’

representational ability between the problems with and without an auxiliary task χ2 (1,

N = 8) = 7.261; p<.01 (83 per cent and 33 per cent, respectively). Significant

differences were also observed in 4-year-olds between both problem types χ2

(1, N = 9) = 6.644; p<.02 (100 per cent and 52 per cent, respectively). This reiterates the difficulty faced by both age groups in representing the problems needing an auxiliary task, and confirms that the relations within the base problem were conceptually difficult for children to understand.

As stated, children’s ability to represent the problems without an auxiliary task was high across both sessions. The figures were also significantly different for both 3- and 4-year-olds across session (67 per cent and 83 per cent: χ2

(1, N = 17) = 6.827; p<.05 and 52 per cent and 100 per cent χ2

(1, N = 17) = 63.158; p<.05, respectively). No significant differences between sessions for 3-year-olds (session one- 25 per cent, session two- 22 per cent) was found. However, differences emerged in the problems needing an auxiliary task for 4-year-olds between sessions χ2

(1, N = 9) = 7.386;

p<.05 (session one - 33 per cent, session two - 52 per cent). Thus, just over half of 4-

year-olds were representing the base problem compared to only a third in session one.

Of interest is the similarity of 3-year-olds representational ability at session two with 4-year-olds ability in session one (28 per cent and 20 per cent, respectively). Since the 3-year-olds in session two were at similar ages as the 4-year-olds in session one, the similar performance suggests that this is the optimal representational ability in 4-year- olds. However, this cross-sectionalfinding does not match 3- and 4-year-olds target problem solution. In fact, 3-year-olds were choosing the analogy based object with adaptation more in session two than 4-year-olds in session one. Therefore, this could not be due to representational ability because their performances were similar. It is more likely to be due to practice with tasks in session one.

In relation to children’s lack of representation, 3-year-olds ability to represent the solution only was similar across session one and two (37.5 per cent and 33 per cent).

Since these were from the same children indicates that their representational ability remained the same 30 weeks on, in that they could only represent the solution and not both problem and solution. In contrast, the 4-year-olds’ representational ability increased significantly and the dominant representation of solution only, in session one, significantly decreased in session two χ2

(2, N = 9) = 40.996; p<.05 (67 per cent and 22 per cent, respectively). The dominant description in both 3- and 4-year-olds in the problems without an auxiliary task was both problem and solution. Similarly, in 4- year-olds problems needing an auxiliary task, the problem and the solution was the dominant strategy. However, 3-year-olds gave problem and solution and solution only to a similar degree. Thus, 4-year-olds representational ability increased considerably in session two to include both the problem and solution.

The results of children’s representational ability, indicates that two thirds of children were unable to represent and understand what had occurred in the base problem. Whether or not children were able to utilise this information to form the analogy based object with adaptation in the target problem was explored next. Since children’s representational ability improved significantly in session two, then it was expected that children’s use of representation on the target problems would also increase.

Session one revealed that 40 per cent of children were able to form a representation and use this information to solve the target problem. Because more children were able to represent the base problem in session two, than in session one, it was expected that the percentage of children able to use the representation the base problem and solve the target problem would be greater in session two. Results revealed children who were able to form a representation of the target problem were more likely to solve the target problem. Of the 69 correct representations 51 per cent were used to solve the target problem. Thus, just over half went on to use this information to solve the target problem. This is an 11 per cent increase from session one.

A chi square analysis was carried out to examine children’s target problem

performance as a function of their base representations. Children who represented the base problem were more likely to solve the target problem (52 per cent) than those who did not represent the base problem (12 per cent) χ2

As in session one, this suggests that children benefited more from being able to represent the base problem than those who could not. However, not all children who represented the base analogue went on to solve the target problem. Eight correct representations (12 per cent) were followed by the level 2 errors strategy in the target problem and 15 (22 per cent) were followed by analogy based object but without adaptation. Thus, just because children were able to form representations of the base problem, does not mean that they go on to solve the target problem using the analogy based object with adaptation suggesting a problem of access or transfer. However, considerably more children in session two were able to use the representation in their target solution (52 per cent and 40 per cent) χ2

(1, N = 17) = 2.899; p<.05. Thus, children were benefiting more from their representations in session two than in session one.

Since results reported earlier revealed differences in representational ability between 3- and 4-year-olds, but no age differences in children’s ability to solve the target problem, no age prediction of whether children would use the representation in the target problem was made. The figures reveal that 50 per cent of the 3-year-olds who gave correct representations went on to solve the target problem with the analogy based object with adaptation and 54 per cent of 4-year-olds. These differences between 3- and 4-year-olds did not reach significance χ2

(1, N = 17) = .321; p>.05. However, significant differences were found between age group and session. Three- year-olds represented more base analogues and used them in the target problem in session two than in session one (50 per cent and 24 per cent, respectively) χ2

(1, N = 8) = 14.500; p<.05. The same was found in 4-year-olds (54 per cent and 17 per cent, respectively) χ2

(1, N = 9) = 29.894; p<.05. This suggests that both 3- and 4-year-olds improved in their ability to utilise their representation in the target problem over session.

As in session one, children found the formation of representation difficult in the problems needing an auxiliary task. Children were able to form a representation and extract relations from the base problem and use them in the target problem more in the problems without an auxiliary task than in the problems needing an auxiliary task (66 per cent and 9 per cent, respectively) χ2