Developing concreteness through connections with mathematical representations

Abstract

This paper presents a new framework called the Concretion-Based Learning Framework (CBLF). It is a tool created to support teachers in planning learning environments that promote conceptual understanding through the interactions that students have with multiple representations. Drawing from my teaching experiences, this project identifies gaps in existing models, such as the CRA model and the ACT model, specifically with how conceptual understandings are explicitly planned for. Drawing on research from Wilensky (1991), Hattie et al. (2017), and Wiggins and McTighe (2005), the project challenges previous understandings of concrete and abstract knowledge in mathematics and suggests that the process of concretion is the missing component when developing conceptual understandings. The CBLF challenges a more traditional linear instructional approach and suggests a cyclical planning process where goals, assessments, and instruction are intentionally aligned so that we can monitor the connections that are being made by students. In response to this need, a Teacher Clarity Tool was developed to guide the instructional design process to promote alignment to achieve the goal of developing concreteness and mathematical proficiency.