A strong national effort is currently under way in the United States to reform school mathematics instruction, so that children become autonomous problem solvers (National Council of Teachers of Mathematics [NCTM], 1989, 1991, 2000). One recommended change is encouraging students to invent their own solution procedures for problems or calculation tasks. This is important for fostering their mathematical power or proficiency, which involves an understanding of mathematics, the thoughtful and fluent use of procedures, the capacity to engage in mathematical inquiry, and a positive disposition toward learning and applying mathematics. (For discussion of how, see Ambrose, Baek, & Carpenter, chap. 11, this volume; Baroody, chap. 1, this volume; Baroody, with Coslick, 1998.)

There is presently, however, an insufficient research base for understanding the range of solution procedures children can invent in a socially and cognitively supportive environment or of what supporting roles teachers should play in promoting such inventions. Most previous work on children’s invented multidigit addition and subtraction procedures was limited to two-digit numbers (for reviews, see Beishuizen, Gravemeijer, & van Lieshout, 1997; Fuson, 1990; Fuson & Smith, 1997; Fuson, Wearne, et al, 1997; Labinowicz, 1985). Unfortunately, many of these procedures do not generalize well to larger numbers or depend on special solutions for the decades.