In this paper four programs are described in which children learn multidigit number concepts and operations with understanding: (1) the Supporting Ten-Structured Thinking projects, (2) the Conceptually Based Instruction project, (3) Cognitively Guided Instruction projects, and (4) the Problem Centered Mathematics Project. The diversity in these programs indicates that learning with understanding is possible under a variety of conditions; however, a critical feature shared by the four programs is that they engage students in building connections. In each of the programs, treating the development of arithmetic procedures as a problem-solving activity and asking students to share and explain their answers encourages students to reflect on procedures and on the properties of the whole number system and to learn these topics with understanding. Contains 16 references. (MKR)