larities with number words are difficulties posed by the complexity of the system for writing numbers. As we said in chapter 3, the base-10 place-value system is very efficient. It allows one to write very large numbers using only 10 symbols, the digits 0 through 9. The same digit has a different meaning depending on its place in the numeral. Although this system is familiar and seems obvious to adults, its intricacies are not so obvious to children. These intricacies are important because research has shown that it is difficult to develop procedural fluency with multidigit arithmetic without an understanding of the base-10 system.^{64} If such understanding is missing, students make many different errors in multidigit computations.^{65}

This conclusion does not imply that students must master place value before they can begin computing with multidigit numbers. In fact, the evidence shows that students can develop an understanding of both the base-10 system and computation procedures when they have opportunities to explore how and why the procedures work.^{66} That should not be surprising; it simply confirms the thesis of this report and the claim we made near the beginning of this chapter. Proficiency develops as the strands connect and interact.

The six observations can be illustrated and supported by examining briefly each of the arithmetic operations. As is the case for single-digit operations, research provides a more complete picture for addition and subtraction than for multiplication and division.

The progression followed by students who construct their own procedures is similar in some ways to the progression that can be used to help students learn a standard algorithm with understanding. To illustrate the nature of these progressions, it is useful to examine some specific procedures in detail.

The episode in Box 6–6 from a third-grade class illustrates both how physical materials can support the development of thinking strategies about multidigit algorithms and one type of procedure commonly invented by children.^{67} The episode comes from a discussion of students’ solutions to a word problem involving the sum 54+48.

The episode suggests that students’ invented procedures can be constructed through progressive abstraction of their modeling strategies with blocks. First, the objects in the problem were represented directly with the blocks. Then, the quantity representing the first set was abstracted, and only the blocks representing the second set were counted. Finally, the counting words were themselves counted by keeping track of the counts on fingers.