Making sense of fractions in different contexts

This report is based on a study of 9-10 year old pupils in two classroom episodes. In Episode 1 the pupils were working on a practical task, preparing batter for waffles. The mathematical content focused on was how to obtain 15 dl of milk, using boxes marked 1/4 litre. One litre measuring beakers, with marks for each decilitre, were available. In Episode 2 the pupils were presented with drawings of imaginary milk boxes, red and blue boxes defined to contain 1/4 litre and 1/3 litre, respectively, and asked to consider the amounts of milk in total in the following cases: A: three blue boxes; B: four blue boxes; C: four red boxes; D: three red boxes. Standard notation for fractions was used here. Some of the problems presented were: ''Which box, red or blue, contains the largest amount of milk?'' and ''Determine how many red boxes would be needed to get 15 dl of milk''. The comparison between Episode 1 and Episode 2 shows that two situations that are mathematically equivalent are treated differently depending on the available mediating artefacts. Depending on the choice of signs, the focus of the task may shift so that the "concept" in the Epistemological Triangle becomes a different one than intended. See (Rønning 2010, 2013) for further analyses and examples.