This book is about children's needs for intellectual rigor, as seen through the work and discourse of five undergraduate elementary education students who constructed a number system in base five and then reflected on their experience in the light of what they know about young children's mathematics. The deeper purpose of developing a number system is to explore precisely what it means to perform an act of learning. It is concluded that people reconstruct their culture's mathematics as they learn it. (Contains 49 references.) (ASK)