Exploring the Connection Between Sixth Grade Students’ Adaptive Number Knowledge and Proportional Rules in Balance Scale Problems

Abstract Adaptive number knowledge represents the understanding and use of numerical characteristics and relations. However, there is a lack of knowledge about the role of adaptive number knowledge in solving problems that require the recognition of patterns. Therefore, we studied 61 sixth grade students’ adaptive number knowledge and its connections to solving a problem where recognizing proportional relations is relevant. The students’ adaptive number knowledge was measured using a paper-and-pencil test. In addition, the students participated in a balance lesson where they collaboratively generated balance rules and solved balance scale problems in a digital learning environment. We analyzed the students’ balance rules based on whether they were proportional or not. Students' adaptive number knowledge and the generality of their proportional balance rules were related: students with higher adaptive number knowledge were more likely to express proportional rules for multiple situations. The examination of the development of proportional balance rules revealed that students mainly expressed either proportional rules with certain ratios or the general multiplication rule where torques are calculated. In addition, students with lower adaptive number knowledge seemed to struggle more when faced with more complex ratios, while the students with higher adaptive number knowledge expressed proportional rules regardless of the complexity of the task. These results give new insight into the role that adaptive number knowledge plays in expressing proportional rules in balance scale tasks.