support and justification for different procedures. Rather, the research provides evidence that, at any one time, most children use a small number of procedures and that teachers can learn to identify them and help children learn procedures that are conceptually more efficient (such as counting on from the larger addend rather than counting all).²²

Mathematical proficiency with respect to single-digit addition encompasses not only the fluent performance of the operation but also conceptual understanding and the ability to identify and accurately represent situations in which addition is required. Providing word problems as contexts for adding and discussing the advantages and disadvantages of different addition procedures are ways of facilitating students’ adaptive reasoning and improving their understanding of addition processes.

Subtraction follows a progression that generally parallels that for addition (see Box 6–3). Some U.S. children also invent counting-down methods that model the taking away of numbers by counting back from the total. But counting down and counting backward are difficult for many children.²³

Box 6–3 Learning Progression for Single-Digit Subtraction