Adaptive number knowledge: Exploring the foundations of adaptivity with whole-number arithmetic

Abstract Adaptivity with arithmetic problem solving is a key aspect in the long-term development of mathematical skills, and the knowledge of numerical characteristics and relations is a core component of adaptivity with whole-number arithmetic. The present studies represent the first attempt to investigate the nature of the adaptive number knowledge, referring to the well-connected knowledge of numerical characteristics and arithmetic relations between numbers, which underlies adaptivity with arithmetic. A new measurement tool, the Adaptive Number Knowledge Task, was developed to capture individual differences in primary school students' and adults' number knowledge. Three groups of participants – 3rd to 5th graders, university students, and 6th graders – completed the measure, which required participants to produce arithmetic sentences equaling a target number by using any combination of four to five given numbers and arithmetic operations. These studies revealed substantial individual differences in the quantity and quality of participants' adaptive number knowledge. Adaptive number knowledge was found to be related to arithmetic fluency and knowledge of arithmetic concepts in 6th graders. Results suggest that the Adaptive Number Knowledge Task was able to capture individual differences in recognizing and using numerical relations in whole-number arithmetic problem solving, and that these differences may help explain differences in adaptivity with arithmetic.