This exploratory study compares key features of the Singapore and U.S. mathematics systems in the primary grades, when students need to build a strong mathematics foundation. It identifies major differences between the mathematics frameworks, textbooks, assessments, and teachers in Singapore and the United States. It also presents initial results from four pilot sites that introduced the Singapore mathematics text book in place of their regular text books. Analysis of these evidentiary streams finds Singaporean students more successful in mathematics than their U.S. counterparts because "Singapore has a world-class mathematics system with quality components aligned to produce students who learn mathematics to mastery." These components include Singapore's highly logical national mathematics framework, mathematically rich problem-based textbooks, challenging mathematics assessments, and highly qualified mathematics teachers whose pedagogy centers on teaching to mastery. Singapore also provides its mathematically slower students with an alternative framework and special assistance from an expert teacher. "The U.S. mathematics system does not have similar features." It lacks a centrally identified core of mathematical content that provides a focus for the rest of the system. Its traditional textbooks emphasize definitions and formulas, not mathematical understanding; its assessments are not especially challenging; and too many U.S. teachers lack sound mathematics preparation. At-risk students often receive special assistance from a teacher's aide who lacks a college degree. As a result, the United States produces students who have learned only to mechanically apply mathematical procedures to solve routine problems and who are, therefore, not mathematically competitive with students in most other industrialized countries. Appended are: (1) References; (2) Charts Referenced in Chapters 3 and 6; and (3) Professional Development Case Study. (Contains 46 exhibits.) [This study was supported by the U.S. Department of Education Policy and Program Studies Service (PPSS).]