This thesis seeks to characterise how instruction progresses from task to task in highly responsive, one-to-one arithmetic instruction. Within a broader design research program developing arithmetic instruction for low-attaining students, the study addresses the content domain of basic multiplication and division, moving students from counting-based strategies to more advanced multiplicative strategies. The study seeks to characterise the instructional progressions involved in this domain.
Several mathematical dimensions of instructional progression are investigated, such as progressions from lower to higher multiples, and progressions from visible to screened materials. The aims of the study are: to identify key dimensions of instructional progression, to describe how each dimension progresses, and to characterise how responsive instruction progresses in terms of the dimensions. The study contributes to three areas of significance: instruction for multiplicative thinking; instruction for low-attaining students; and the design of instructional frameworks.
Using a teaching experiment methodology, data was drawn from an experimental arithmetic intervention program in which 10- and 11-year-old students participated over 20 weeks, with all assessments and lessons recorded on video. One student’s instruction in one topic—basic facts involving 5s—was selected as a case of progression from counting-based to multiplicative strategies. The case comprised segments from eight lessons. Task-by-task analysis tracked adjustments on dimensions of instruction, alongside progressions in the student’s thinking, revealing a rich tapestry in the instructional progression.
Five key dimensions are identified. The range dimension adjusts between lower and higher multiples, and also between even and odd multiples. The orientation dimension adjusts which parts of a multiplicative task are posed as unknowns. The setting dimension adjusts between having visible, screened, and absent materials. The notation dimension adjusts between informal and increasingly formal notating. Finally, the structuring and strategies dimension includes a range of comments drawing students’ attention to mathematical relationships.
The instructional progression overall is characterised as a strategic, interwoven calibration of the five dimensions. Features of instructional progressions include: progressions are typically multidimensional; progressions are highly recursive along each dimension; different dimensions share distinctive relationships; and different lesson segments can be characterised by distinctive calibrations of the dimensions. The study informs an instructional framework organised in terms of the five dimensions, and reflects on the potential of such a framework as a form of instructional design.