Bridging Informal and Formal Reasoning

Children’s intuitive understandings of mathematical ideas—both correct, generalizable strategies alongside misconceptions—showcase the complexity of their thinking. However, recognizing children as complex thinkers is one thing but it is another thing altogether to leverage their ideas to plan for and carry out mathematics instruction. The purpose of this article is to describe our efforts to make students’ thinking explicit in ways that lead to generalizable procedures for solving problems. Furthermore, we provide a set of principles and practices that we have found productive in our planning and implementation of mathematics tasks. As an example, we draw on our work in a classroom of fourth-grade students, where a classroom teacher, a university faculty member, and preservice teachers regularly focus and reflect on children’s mathematical thinking (Hodges and Mills 2014).