Examined is the situation in which pupils invent mathematics on their own and teachers' reactions to this situation. The assimilation of students' original ideas into correct mathematical concepts is discussed. (CW)

Discussed are preservice elementary teachers' misconceptions and inconsistent beliefs about multiplication and division with decimals. Sources of inconsistencies and recommendations for overcoming inconsistencies are included. (KR)

This paper describes the BUD project which surveyed childrens' conceptions of division, and of fractions and decimals. The lack of connection between counting skills and conceptual understanding is discussed. The expectations for new algorithms and the basic idea in planning the BUD project are summarized. Some previous studies on counting, the…