Conceptual and procedural understanding of counting by pupils with severe learning difficulties
This thesis concerns the ability of pupils with severe learning difficulties (SLD) to develop both the skills and an understanding of counting. To date, research in this area has been limited, one strong reason being the difficulties the researcher encounters in this field. For example, there is no single defining criteria of severe learning difficulties, the population is small and heterogeneous, and the nature of the children's learning difficulties provide a series of challenges to the collection of data first hand. The implications of these factors for the design of research are introduced in chapter one, together with a brief over-view of different approaches to understanding the nature of SLD. In chapter two, a review of the literature on how typically developing preschool children learn to count is presented. For this, the work of Gelman and her colleagues is taken as a starting place. Her five principles of counting and the distinction she draws between conceptual and procedural understanding form the basis of the discussion. Within this framework, different views of the order in which conceptual and procedural understanding occur, are considered. These include Gelman's view that children are guided in acquiring the procedures by an implicit understanding of the principles of counting as well as the opposing view that the child has first to acquire the procedures for counting, before they are able to deduce the principles. From this analysis it is suggested that neither extreme position is viable. Instead, it is proposed that the process of acquisition should be viewed as interactive with the acquisition of procedures leading to understanding which in turn leads to refinements in performance. Little research has taken place in the field of counting with pupils with SLD. In chapter three, a review of the existing literature considers studies which investigate overall performance levels, those which make comparisons between pupils with SLD and other children with a similar mental age, and studies which describe attempts to improve performance through intervention. Although the performance studies reveal a variation of ability with overlap between pupils with severe and moderate learning difficulties they do suggest a probable ceiling on the attainments children and young people can reach. What predicts these levels of performance, however, is unclear. The comparison studies provide contradictory evidence concerning the equivalence of preschool children and those with SLD and the review of intervention studies reveal that improvements in performance are possible but are uninformative with respect to the processes which underlie learning. Overall the review revealed that the research with children with SLD has been limited, both in the questions which it addresses and in its theoretical orientation. The primary aim of this thesis is to investigate the extent of pupils attainments in counting, both in relation to performance and understanding. An essential part of this objective is to first establish whether children with SLD are able to understand the principles of counting and second to investigate the temporal order to this. A third aspect is to explore selected factors which might contribute to differences between pupils' attainment. A series of three studies are undertaken which examine the acquisition of procedures and conceptual understanding in counting in both children with severe learning difficulties and preschoolers. These are preceded by an exploratory study which sets out to establish a satisfactory way of studying counting in this particular group. Once this has been achieved the first proper study addresses the question of which comes first for children with SLD, the acquisition of counting skills or understanding. It reveals that some children are clearly able to demonstrate understanding and that this is followed (but was not solely dependent on) successful performance on counting tasks. The second study then compares the attainments of a group of pupils with SLD and a group of typically developing preschoolers. Performance on both counting and error detection tasks is found to be poor in these two groups but no overall differences are found. The study does, however, reveal some qualitative differences in the responses of the two groups, with children with SLD using fewer count words and using a smaller repertoire of responses to a cardinal question than the preschoolers. The final study explores some of the reasons for these qualitative differences between the groups by comparing various of aspects of the number environment in which children with SLD and nursery children learn to count. The study has three parts and is carried out with children from two SLD schools matched on two different measures with children from two nurseries. Firstly, questionnaires are used to examine staff attitudes towards counting activities. Whilst staff in both settings are found to have a variety of different general aims, they share similar views of the importance of counting and acquiring the number word sequence. More precisely, they don't see either activity as very important. The second part of this study is based on observations of children's experiences of number words in both nursery and SLD settings across curriculum areas/activities. Observations reveal considerable variation in the practices of the four settings with none providing all the characteristics identified as important by those investigating mother-child interactions. Conversely, no setting provides all the characteristics identified by those favouring mechanistic approaches. Additionally, no setting provides evidence of staff individualising childrens' experiences. The final part of the study into children's counting performance reveals (unsurprisingly) that overall the best predictor is mental age. In chapter nine the data from all three studies is pooled for further analysis, revealing that children can be seen as falling within four groups of attainment, the largest being those who are in the process of acquiring the procedures and with an equal number falling in the two extreme groups of non-counters and error detectors. The remaining small group of children are described as being transitional, able to count proficiently but not able to demonstrate understanding. Throughout the studies all children who demonstrate understanding are proficient counters. Whilst no difference is found between typically developing and SLD children in the distribution of children across the four groups, the pattern of acquiring the procedures appears to be different with the SLD group acquiring competence in one:one correspondance first, followed by stable order (and then cardinality) and the preschoolers adhering to stable order and then one:one correspondance. Chapter ten explores the theoretical and practical implications. The process of acquisition of skills and understanding is characterised as a gradual one with children's thinking developing from being based on a series of rules which operate largely independently in restricted contexts, to being based on an understanding of the principles enabling the generation of strategies for problem solving. It is suggested that a likely mechanism for change is the child's awareness of a mismatch between the expectation and the outcome of a particular action or sequences of action but that further investigation is needed focusing on particular transition times.