Grade six students' multidigit arithmetic strategy adaptivity and flexibility: evaluating a novel tri-phase assessment tool

In this paper we present a novel adaptation of a tri-phase assessment tool, originally devised to investigate students' linear equations-related strategy flexibility, to evaluate Danish grade-six students' multidigit arithmetic-related strategy adaptivity and flexibility. Participants, 731 students, median age 12 years and drawn from 20 demographically different schools, completed a series of positive integer addition, subtraction and multiplication tasks designed to elicit shortcut strategies. First, students solved each task by means of their preferred strategy, with those using a shortcut strategy being defined as showing adaptivity for that task. Second, students solved the same tasks by means of whatever alternative strategies they could devise; those that offered at least two strategies were defined as showing flexibility for that task. Third, for each task, students were asked to indicate which of their strategies they believed was optimal. Statistical analyses, based on the total number of tasks on which students demonstrated adaptivity or flexibility, found students to be more flexible than adaptive, with 55% of students showing moderate or high levels of flexibility but only 21% for adaptivity. Both strategy adaptivity and flexibility, which differed according to operation, were implicated in students' solution accuracy, with adaptivity being a stronger predictor than flexibility.