Young children's approaches to solving conceptually linked addition problems

The research of inter- and intra-individual differences in arithmetical development has revealed notable discrepancies and dissociations between the three main forms of arithmetical knowledge, namely procedural, factual and conceptual knowledge, and has supported the view that arithmetical ability is not unitary but involves the quite often variable development of different components (e.g. Dowker, 1998; Cowan, 2003). The present pilot study explored the conceptual, procedural and factual knowledge that 6-7 year old children of different abilities in arithmetic employ when solving conceptually linked numerical addition problems. Conceptually linked addition problems have been previously used in research on children’s understanding of addition principles (e.g. Dowker 1998; Canobi, 2005). The focus of this study was particularly on studying and comparing different children’s ability to use and combine all three aforementioned types of knowledge in their problem solving approaches either spontaneously or after explicit prompt. Thirty three children of different abilities in arithmetic, as assessed by their teachers, took part in individual problem solving sessions, each lasting 30 minutes. The children were presented with pairs of addition problems which involved either two or three terms and which were related by the principles of commutativity, associativity and additive composition. Children’s approaches to the first of each pair of problems were indicators of their ability to use calculation procedures or factual knowledge to work out the answers. Children’s approaches to the second problem were indicators of their attempt and/or ability to use their conceptual knowledge in order to derive the answer from the first problem solution (erroneous or not) either spontaneously or after an explicit prompt. The paper discusses the similarities and differences observed in the problem solving approaches of two particular cases: an able child and a child who is underachieving in arithmetic. The analysis of these two cases shows notable differences in the mastery and use of procedural knowledge but similar levels of factual and conceptual knowledge accompanied by noteworthy differences in the spontaneous use of these latter two types of knowledge in problem solving.