Kindergarten and First-Grade Students’ Understandings and Representations of Arithmetic Properties

We present a study that explores Kindergarten and first-grade students’ understandings and representations of arithmetic properties. The 16 students participated in a classroom teaching experiment designed to explore children’s algebraic understandings, including their understandings and symbolic representations of three arithmetic properties: additive identity, additive inverse, and commutativity. We characterized students’ understandings in terms of Skemp’s framework of instrumental (rules without reason) and relational (knowing what to do and why) understandings. Following Vergnaud, we analyzed the types of additive relationships (transformation, comparison, or combination) and representations used by students. Our findings show that students’ understandings developed in sophistication over time. We observed the least sophisticated understandings for the commutative property, particularly among Kindergarten students who exhibited instrumental understandings even after instruction.