Multiplicative thinking is a critical stage in mathematical learning and underpins much of the mathematics learned beyond middle primary years. Its components are complex and an inability to understand them conceptually is likely to undermine students' capacity to develop beyond additive thinking. Of particular importance are the ten times…

Introduction: In the present-day didactic process of transition from natural numbers to integers, a certain kind of measure is involved which is neither easy nor appropriate to integrate into the familiar numerical systems. We refer to measures and comparisons for which we need a third numerical system--what we will call "relative…