Before we examine the concepts involved in multiplication and the array representation in particular, let us first outline the theoretical model of understanding of mathematical concepts that we shall work with. In doing so, we wish to clarify not only our views of ‘understanding’ and ‘reasoning’ that make up the subject of this paper, but also the model will point to implications for developing and demonstrating understanding, which in turn will guide us in our research work, suggesting how to examine children’s understanding of and reasoning within multiplication. The model for understanding that we have adopted emphasises the importance of the connections between internal or mental representations of a concept. In the literature, we find a host of examples of understanding being defined with respect to these connections. Skemp (1976) described the process of learning relational mathematics as “building up a conceptual structure” (p. 14). Nickerson (1985) also referred to the connections between concepts: “The richer the conceptual context in which one can embed a new fact, the more one can be said to understand the fact” (p. 235–236). Hiebert and Carpenter (1992) specifically defined mathematical understanding as involving the building up of the conceptual ‘context’ or ‘structure’ mentioned above: