Students’ Units Coordinations

Units coordination describes how people construct and organize units to solve problems. People construct units by using their unitizing operation, a mental action of creating singularities out of their experiences. Composite units are also the result of the unitizing operation where people take a collection of discrete items as one unit. In the most basic units coordination, people distribute the units of one composite unit across the units of another, a basis for whole number multiplication (Steffe, Learn Individ Differ 4(3):259–309, 1992). However, units coordination has become a way of understanding students’ thinking that has explanatory power far beyond whole number multiplication: It has been found to influence students’ fractions knowledge, understanding of integers, algebraic reasoning, proportional reasoning, and combinatorial reasoning. In this chapter, we define and characterize units coordination, and associated constructs, as a framework for modeling students’ mathematical thinking. We give examples of its use in research and discuss why units coordination is useful for making second-order models of students’ mathematics. Second-order models are explanatory accounts of students’ mathematical reasoning and learning that can be used to orchestrate future mathematical interactions with other students. Then, we describe standards of evidence for making claims about students’ units coordination. Finally, we discuss what is known about the numbers of students at different units coordination stages in elementary and middle school and how students advance across stages.