Capítulo 3 Puesta en práctica del procedimiento
3.1 Catálogo de requisitos
The findings of this study suggest that the advantages of physical materials over no material in Study 2 were not simply attributable to an external representation of units. In this study, children were able to identify more correct solutions when they were able to physically manipulate the representation. There are several possible reasons why physical materials provided this advantage.
Firstly, it seemed that the materials were more accessible for children with little domain knowledge. Whilst six children in Reception and three in Year 1 failed to identify any solutions at all in the Pictorial condition, no child failed to do so in the Physical condition. This may be because physical objects limit how the representation can be acted upon. Therefore, with little understanding, children in this condition may have been more likely to use the materials appropriately by creating two spatial groups, than to annotate the pictorial materials appropriately by creating lines to separate two groups.
193 The physical materials may also have supported the procedure of identifying two amounts accurately. By creating two spatially separate groups, not only are the boundaries of each group clearer but children‘s enumeration may be supported through subitising. Tactile feedback and moving objects may also have supported counting.
Indeed, there were significantly more incorrect solutions in the Pictorial condition.
However, although there were more incorrect solutions in the Pictorial condition, the reason for there being less correct solutions was because far fewer verbal solutions were provided.
It is predicted in PDL that children will identify more solutions using physical materials because they allow children to act on the interpretation with ease, creating spatial configurations that can be interpreted to support new ideas. This study did not record the number of changes to the representations children made, but informal observations indicated that children created more configurations than they identified – suggesting that the representation did help children to explore the range of solutions possible, and allowed children to identify new solutions by first acting on the representation and then enumerating (interpreting) the resulting solution.
4.4.3 Motivation
Another possible reason that children in the Pictorial condition identified fewer solutions may be motivational. The advantages of physical materials may not be so much that children become aware of the greater number of solutions possible, but rather that the lower demands of manipulating the representations (and/or familiarity) motivate them to continue. Physical materials are more easily and quickly manipulated than pictorial materials as well as providing both tactile and visual stimuli. Significantly, unlike pictorial materials, manipulation does not leave a trace, so that children may be less concerned
194 about ‗going wrong‘. Even in the Physical Trace condition, a trace was only made after children had identified a solution.
It is not clear how PDL accounts for possible motivational effects. Motivation might encourage children to adapt the materials more, thereby leading them to develop more ideas, but it is difficult to isolate motivation as a factor. Nevertheless, there was reason to believe that the advantage of the materials in this study was not purely motivational. Firstly, there were no clear signs of loss of motivation in either condition (e.g., loss of visual concentration). Secondly, sessions were relatively short (around 12 minutes on average), especially for the older children where the advantage of physical materials was still clear. Finally, if children were losing motivation, a fall in performance over the three problems might have been expected, yet there were no such differences in either condition. Therefore, although it is not possible to rule out motivation as a key factor in differences between conditions, it is unlikely to be the only factor.
4.4.4 Strategies
Differences in the ways children identified successive solutions in each condition provide a window onto how the representations may have influenced problem solving. Using the coding scheme developed in Study 2, it was possible to compare and contrast strategies that children may have used. Similarly to Study 2, a large number of solutions were related, seemingly derived from the previous solution, with older children identifying more related solutions. As this suggests developmental progress, it is possible to argue that because children identified more related solutions using physical objects, this representation fosters the use of more developed strategies for partitioning. However, children in the Physical condition also identified a greater amount of other solutions.
195 Therefore, it seems that the greater number of related solutions reflects a general effect of identifying more solutions overall using physical objects.
The possibility that the larger number of related solutions in the Physical conditions may simply reflect the larger number of solutions found overall does not itself negate the benefits of this representation: a greater number of solutions identified this way means that children will have more experience of such strategies and hence a possibly greater chance of developing related numerical ideas. However, it does suggest that the manipulative or perceptual properties that may have fostered certain strategies in Study 2 are not unique to physical objects. In other words, it may simply be the external, linear representation of objects that helps children identify related solutionsIt should be noted, however, that difference in the number of commutative solutions identified between conditions appeared more substantial. This raises the possibility that the manipulative or perceptual properties of physical objects do foster this strategy. Indeed, with cubes, it is easy to change a configuration (e.g., 2 and 5) into a unique but symmetrical configuration of (i.e.. 5 and 2) through simple actions: grabbing a group with each hand and then swapping over hands. Unfortunately, without video data it is difficult to conclude that such actions were indeed responsible for the greater use of this strategy.
4.4.5 Summary
This study has shown that children are able to identify more ways to partition a number when using physical than when using pictorial representations. This supports PDL and suggests that the implementation costs of having to annotate paper does not result in more planful behaviour, as has been found with adult studies (O'Hara & Payne, 1999).
This may not be surprising considering the evidence suggesting that children find planning cognitively challenging and unappealing (Ellis & Siegler, 1997).
196 The problem used in this study required children to keep track of which solutions they had given and which solutions still remained to be identified. The prediction that children would utilise a trace of their solutions to meet these demands was not supported.
This suggests that the cognitive demands of interpreting these previous solutions to inform strategies were too high, although it is not clear how children would respond to more explicit instruction before starting the problem solving.
Although the trace may have been of limited value to children, the value of representational trace for the teacher should not be ignored. Having a trace of children‘s actions allows the teacher to see the child‘s progress without constant supervision. Whilst children‘s use of pictorial materials provides this trace, physical materials do not.
Developing our understanding of the value of a trace of solutions is important, not only in evaluating this representational characteristic but also in evaluating novel digital technologies that can provide a trace of solutions even when objects are manipulated spatially. Indeed, the ability for computer based manipulatives to provide a trace of actions is referred to as a key advantage that these materials have over physical objects (Clements, 1999; Kaput, 1993).
This study also examined the effect of using the different representations on children‘ strategies, and showed that children in the Physical conditions identified more solutions that related to the previous one. Although this could simply be a reflection of a greater number of solutions having been found overall, it appearedthat one particular strategy (commutativity) was more likely to be used in the Physical condition. More focused video analysis of children solving partitioning problems may help to explain which representational characteristics of the materials encourage the use of this strategy.
Similarly, there are other questions raised in the studies conducted so far that might be addressed using more qualitative analysis. These include questions about the relationship between children‘s actions and the solutions identified, and how certain unique
197 properties of physical materials such as sensorimotoric information may play a role. It might also be possible to identify which behaviours are related to children‘s use of the most efficient strategy for solving this problem (compensation) and possibly thereby begin to suggest ways in which the materials themselves can be adapted to foster such a strategy.
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