Cognitive Science Foundations of Integer Understanding and Instruction

This chapter considers psychological and neuroscience research on how people understand the integers, and how educators can foster this understanding. The core proposal is that new, abstract mathematical concepts are built upon known, concrete mathematical concepts. For the integers, the relevant foundation is the natural numbers, which are understood by reference to a mental number line (MNL). The integers go beyond the natural numbers in obeying the additive inverse law: for any integer x, there is an integer −x such that x + (−x) = 0. We propose that practicing applying this law, such as when students learn that the same quantity can be added or subtracted from both sides of an equation, transforms the MNL. In particular, perceptual mechanisms for processing visual symmetry are recruited to represent the numerical symmetry between the integers x and −x. This chapter reviews psychological and neuroscience evidence for the proposed learning progression. It also reviews instructional studies showing that the hypothesized transformation can be accelerated by novel activities that engage symmetry processing compared to conventional activities around number lines and cancellation. Ultimately, these instructional insights can guide future psychological and neuroscience studies of how people understand the integers in arithmetic and algebraic contexts.