Using the above observations, as well as Fuson‘s model of developing numerical competence, it may be possible to distinguish several progressive stages of ability in solving the partitioning task:
1) Identify a single solution
To make sense of a question asking how to break a number in different ways, children need to know that a number can be broken. This understanding is reflected in Fuson‘s Breakable chain level. Without it, children may find a question asking them how many ways a number such as ‗7‘ can be partitioned quite difficult, although they may understand the question if it is presented using physical materials. Children know that a collection of objects can be separated into two groups and may therefore reason that a specific collection, e.g., 7 objects, can be partitioned and each part enumerated. Indeed Canobi, Reeve & Pattison (2003) demonstrated how children were more likely to notice how addends could be decomposed when problems were presented using objects than with symbols. However, as demonstrated by Hughes (1981), although children may have difficulty with symbolically presented problems, they may not actually need physical materials but simply a reference to concrete objects to solve problems. Presenting the
113 partitioning problem in a story context may therefore be sufficient to help children recognise how an amount can be partitioned into two parts.
If children recognise that a number can be broken, they need to identify a strategy for enumerating each part. Jones et al (1996) describe how a child was able to count back to identify one part. However, this was possible because the interviewer provided the first part. Therefore, without support, a key demand for children is to identify this first part. For this, children need to identify that a part must be any positive number less than (or equal to) the whole.
The fact that Jones et al refer to the last pair of children ―figuring out pairs... mentally‖
suggests that the other children used materials to support them. Indeed, the authors describe earlier in the paper how manipulatives were given. Consequently, children could use these materials to reduce the calculation demands – hence reflecting the use of perceptual items in Fuson‘s Breakable chain level. Perceptual items may support children‘s counting by helping them maintain correspondence between the count word and object, and to keep track of the last object counted. Once children have counted out the total amount to partition, they can then use this to identify a first part simply by counting a selection of these objects. They can then enumerate the other part by counting the remaining objects. If children physically partition objects into two spatial groups, this may help by a) providing perceptual clues as to which objects need to be included in which group when counting and b) creating smaller collections of objects that can be enumerated by subitising. Considering the small amounts used by Jones et al (5, 8 and 10), this may be highly relevant as most parts will be less than five.
114 2) Identify multiple solutions (but less than half)
Central to the partitioning problem is the fact that there is more than one solution and children need to recognise that a number can be decomposed in more than one way. This may be unfamiliar to children as many numerical problems only have one solution, but the idea is central to the concept of additive composition and seems to reflect Fuson‘s Bidirectional level. Fuson even refers to children ‗knowing each number as all the combinations‟.
Following Resnick‘s (1992a) arguments, if children are familiar with the way that collections of objects can be decomposed in different ways (protoquantitative understanding), this may help them recognise how numbers can also be decomposed in different ways. Indeed, it is possible to apply Martin & Schwartz‘s (2005) theory of Physically Distributed Learning to this problem. If children have incipient knowledge of additive composition (or protoquantitative understanding), physically manipulating objects may help them to develop numerical ideas (quantitative understanding). This may occur because children‘s understanding of the problem may be sufficient to constrain their actions to partitioning objects into two groups. Children are then able to count objects in each group to identify a correct solution. Then, through simply physical actions afforded by the materials, children can create different configurations that they can enumerate.
By acting on the representation physically, children are hence able to identify multiple partitioning solutions. This process may help them map their protoquantitative and quantitative understanding of decomposition or, alternatively, may help them develop numerical understanding of decomposition simply through the experience of identifying repeated numerical combinations (the ‗application before evaluation‘ process described by Bisanz, Sherman, Rasmussen & Ho (2005)). As argued in the literature review, both accounts might be possible: where children‘s understanding is developed
115 through an iterative process of both building on former knowledge and gaining experience by identifying numerical solutions.
3) Identify multiple solutions (more than half)
In order to solve the partitioning problem, children need to identify multiple solutions, but they also need some idea of the range of solutions possible – the problem space. Of course, children can just continue to identify different combinations independently of one another, but the greater the number of solutions that are identified, the greater the chance that children will repeat a solution if they have no means to track what solutions have been given. Without a strategy, this would certainly require substantial memorising.
Certain representations, such as paper, may help children by providing a record of previous solutions. If annotations reflect previous configurations, children can use this to determine what solutions have been given as well as an indication of what solutions remain. Physical materials do not provide such a record – they are confined to the ‗eternal present‟ (Kaput, 1993). However, physical materials may still help children by providing a visual (and tactile) representation: children then have an additional source of information to recall past solutions (in addition to remembering verbal solutions given), although this may still be cognitively demanding. More likely perhaps is that physical objects help children by fostering the use of efficient strategies for keeping track of solutions. The external representation may help children recognise a simple strategy of progressing through different configurations such as moving one object at a time from one group to another.
In order to identify more than half the solutions, children will need to identify
‗commutative‟ solutions - those that have the same parts in different orders. This may be
116 difficult for children working mentally because these solutions may sound highly similar – they have the same numerical parts, although reordered. Physical objects may help children recognise the difference, as it may be clearer to see that objects in a different order present a different configuration.
4) Identifying all solutions
Again it is possible for children to identify all solutions simply through repeatedly identifying solutions independently of each other. Without a strategy this may however be quite laborious, nor is it clear how easily children will recognise solutions with zero in one part and the whole in the other. Contextual clues may help, for example, by presenting the validity of choosing to put all biscuits in one bag and none in the other;
although this may seem quite unpragmatic (why have the other bag?).
It is possible that certain strategies help children recognise that ‗all and none‘ is a possible solution. For example, by moving one object at a time from one group to another, children will eventually reach this configuration. It seems however that without previous experience of identifying such a solution children would probably need support to recognise its validity.
2.1.4 Summary
This section has described Fuson‘s model of numerical development and how it relates to the development of children‘s strategies in problems such as addition. This model was then adapted to consider how children‘s numerical development might influence their
117 ability to solve the partitioning problem, and described four possible levels of increasing ability.
It was highlighted in this section how children‘s success may be significantly supported by the use of external materials. More specifically, it was discussed how certain properties of physical objects may support strategies: for example, by helping children create spatial configurations through simple actions or the use of tactile information to offload counting demands. Alternatively, there may be certain limitations: they might not be as easy to manipulate as fingers and, unlike material such as paper, provide no trace of previous solutions.