This paper synthesizes research from three separate studies, analysing how different representations of a mathematical concept may affect young children's engagement with mathematical activities. Children between five and seven years old engaged in counting objects, identifying triangles and completing repeating patterns. The implementation of three counting principles were investigated: the one-to-one principle, the stable-order principle and the cardinal principal. Children's reasoning when identifying triangles was analysed in terms of visual, critical and non-critical attribute reasoning. With regard to repeating patterns, we analyse children's references to the minimal unit of repeat of the pattern. Results are discussed in terms of three functions of multiple external representations: to complement, to constrain and to construct.