Students' conceptualisations of multiplication as repeated addition or equal groups in relation to multi-digit and decimal numbers

Multiplicative understanding is essential for mathematics learning and is supported by models for multiplication, such as equal groups and rectangular area, different calculations and arithmetical properties, such as distributivity. We investigated two students' multiplicative understanding through their connections between models for multiplication, calculations and arithmetical properties and how their connections changed during the school years when multiplication is extended to multi-digits and decimal numbers. The case studies were conducted by individual interviews over five semesters. The students did not connect calculations to models for multiplication, but showed a robust conceptualisation of multiplication as repeated addition or equal groups. This supported their utilisation of distributivity to multi-digits, but constrained their utilisation of commutativity and for one student to make sense of decimal multiplication